Proceedings of the International Conference
SPT 2004 Symmetry and Perturbation Theory
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Proceedings of the International Conference
SPT 2004 Symmetry and Perturbation Theory Cala Gonone, Sardinia, Italy
30 May - 6 June 2004
Edited by
Giuseppe Gaeta Universita di Milano, Italy
Barbara Prinari Universita di Lecce, ltaly
Stefan Rauch-Wojciechowski Linkoeping University, Sweden
Susanna Terracini Universita di Milano-Bicocca, Italy
NEW JERSEY
-
K World Scientific LONDON
SINGAPORE
-
BElJlNG
*
S H A N G H A I * HONG KONG
TAIPEI * C H E N N A II
Published by
World Scientific Publishing Co. Re. Ltd.
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SYMMETRY AND PERTURBATION THEORY Proceedings of the International Conference on SPT2004 Copyright 0 2005 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereoJ may not be reproduced in any form or by any means. electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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FOREWORD
Previous conferences on “Symmetry and Perturbation Theory” were held in Torino’ (1996), Roma2 (1998), and in Cala Gonone, on the eastern coast of Sardinia (2OOl3l5 and 20024). Comforted by the success of previous conferences in the series it was decided to organize a new SPT conference, again in the magnificent scenery of Cala Gonone, in late spring of 2004; this was organized by three of us (GG, BP, ST) together with Prof. A.Degasperis from the Department of Physics of the “Universith La Sapienza” in Rome and with the help of the SPT2004 Scientific Committee. On this occasion, we also added a new feature, i.e. a satellite workshop on separable systems; this was organized by one of us (SRW) together with S. Benenti (dipartimento di Matematica, UniversitL di Torino), F. Calogero (dipartimento di Fisica, Universith di Roma), and F. Magri (Dipartimento di Matematica, Universith di Milano-Bicocca); this is turn followed an earlier workshop on the same subject’ SRW organized in Linkoeping in January 2004. In SPT2004 we focused on several topics and, for the first time in this series of conferences, had to resort to parallel sessions due to the number of high-quality proposed contributions. In particular, we would like to point out we had two intensive sessions on the quite new - and hot topic of “choreography” solutions of the N-body problem, organized by ST; these were, for the quality and quantity of talks, nearly a conference in the conference. Other special sessions - beside the general sessions held in the mornings - have been devoted to algebraic and geometric integrability, to geometry and symmetry of dynamical systems, to integrable evolution equations, to different kind of perturbation theories and bifurcation analysis, and again to the geometrical theory of separation of variables. In organizing these we received the helpful assistance of Profs. Degasperis, Fels, Kuznetsov, Verhulst and Walcher. The papers collected here should give some flavor of the many topics
V
vi
discussed, and results presented, a t the conference (unfortunately, for several reasons, contributions in the field of algebraic integrability were not provided for this volume). We preferred, in line of the interdisciplinary nature of the whole SPT conference series and as in previous SPT conference volumes, to present all of them together, i.e. without a separation in different topics. We would also like to point out that, as already happened in 199g2 and 20015, there were some “tutorial papers” prepared in connection to the conference; these will appear separately6. We hope the reader will enjoy the paper collected here and will find them useful to gather a picture of the recent progress in the fields our conference touched upon. If this is the case, the merit is of course not ours but of the authors.
References 1. D. Bambusi and G. Gaeta eds., Symmetry and perturbation theory (SPT96), Quaderni GNFM-CNR, Firenze 1997 2. A. Degasperis and G. Gaeta eds., Symmetry and perturbation theory - SPT98, World Scientific, Singapore 1999 3. D. Bambusi, M. Cadoni and G. Gaeta eds., S y m m e t r y and perturbation theory - SPT 2001, World Scientific, Singapore 2001 4. S. Abenda, G. Gaeta and S. Walcher eds., S y m m e t r y and perturbation theory - SPT 2002, World Scientific, Singapore 2002 5. G. Gaeta ed., special volume of Acta Applicandae Mathematicae 70 (2002) 6 . G. Gaeta ed., special volume of Acta Applicandae Mathematicae, forthcoming (2005) 7. State-of-the-art of classical separability theory f o r diflerential equations. See the web site http://www.itn.liu.se/ krzma/SEPARABILITY/konf.html
ACKNOWLEDGEMENTS
A number of people and Institutions also helped us in the organization and running of the conference, and we would like to thank all of them here. First of all, the Scientific Committee, consisting o f Simonetta Abenda (Bologna), Dario Bambusi (Milano), Giampaolo Cicogna (Pisa), Antonio Degasperis (Roma), Giuseppe Gaeta (Milano), Vadim Kuznetsov (Leeds), Giuseppe Marmo (Napoli), Peter Olver (Minneapolis), Juan Pablo Ortega (BesanGon) , Stefan Rauch-Wojciechowski (Linkoeping), Esmeralda SousaDias (Lisboa), Susanna Terracini (Milano), Ferdinand Verhulst (Utrecht), Sebastian Walcher (Aachen), Boris Zhilinskii (Dunquerque). Special thanks should also go to a number of persons involved in nonscientific aspects of the conference: all the personnel of the Hotel Palmasera, where the conference took place, and of the Dorgali Tourist office; as well as the staff of TIVIGEST (the society running Hotel Palmasera) with a special thank to Dr. Enrico Belli; and - first and last contact with this beautiful region for many participants, the pool of taxi drivers who safely ran us from and to Olbia airport. Last but by no means least, the conference received a substantial financial support, which made it possible and which we most gratefully acknowledge here, by several Institutions: GNFM-INdAM (Gruppo Nazionale di Fisica Matematica, Istituto Nazionale di Alta Matematica), by the Dipartimento di Matematica and by the Rettore of Universita’ di Milano, and by the Dipartimento di Fisica of Universita’ di Lecce.
Giuseppe Gaeta Barbara Prinari Stefan Rauch-Wojciechowski Susanna Terracini vii
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CONTENTS
Foreword .............................................................
v
Acknowledgements ...................................................
vii
Contents .............................................................
ix
Papers 1. A cc-chain map for the G invariant De Rham complex.. . . . . . . . . . . . .. l
I.M. Anderson and M.E. Fels 2. New examples of trihamiltonian structures linking different Lenard chains .....................................................
13
C. Andrci and L. Degiovanni 3. Wave propagation in an elastic medium: GDS equations.. . . . . . . . . . . 2 2 C. Babaoglu and S. Erbay 4. Parametric excitation in nonlinear dynamics .......................
.27
T . Bakri 5. Collisionless action-minimizing trajectories for the equivariant 3-body problem in R2 .............................................
35
V. Bamtello 6 . The Lagrangian and Hamiltonian formulations for a special class of non-conservative systems ..................................
43
S. Benenti 7. Shadowing chains of collision orbits for the elliptic 3-body problem .......................................................... S. Bolotin
ix
.51
X
8. Similarity reductions of an optical model ........................... M.S. Bruzdn and M.L. Gandarias 9. Fold, transcritical and pitchfork singularities for time-reversible systems ...........................................
59
.67
C.A. Buzzi, P.R. Silva and M.A. Teixeira 10. Homographic three-body motions with positive and negative masses ...........................................................
75
M. Celli 11. Remarks on conformal Killing tensors and separation of variables ...................................................... C. Chanu and G. Rastelli
.83
12. A regularity theory for optimal partition problems.. . . . . . . . . . . . . . ..91 M. Conti, G. Verzini and 5’. Terracini 13. Lambda and mu-symmetries. .....................................
.99
G. Gaeta 14. Potential symmetries and linearization of some evolution equations .......................................................
,106
M. L. Gandarias 15. Periodic solutions for zero mass nonlinear wave equations.. . . . . . . .115
G. Gentile 16. Fundamental covariants in the invariant theory of Killing tensors .................................................. .124 J . T. Homood, R. G. McLenaghan, R. G. Smirnov and D. The 17. Global geometry of 3-body trajectories with vanishing angular momentum .............................................
.132
W.Y. Hsiang 18. The relation between the topological structure of the set of controllable affine systems and topological structures of the set of controllable homogenuous systems in low dimension. . . . . . . .141
A . Kadem 19. On preservation of action variables for satellite librations in elliptic orbits with account of solar light pressure . . . . . . . . . . . . . . ..151 1.1. Kossenko
xi 20. An explicit solution of the (quantum) elliptic Calogero-Sutherland model ......................................
159
E. Langm ann 21. An application of the Melnikov integral to a restricted three body problem.. ................................................. J . Llibre and E. Perez-Chavela 22. Reductions of integrable equations and automorphic Lie algebras ....................................................
.175
.183
S. Lombard0 and A . V. Milchailov 23. Geometric reduction of Poisson operators.. ......................
.193
K. Marciniak and M . Btaszak 24. Closed manifolds admitting metrics with the same geodesics.. . . . .198 V.S. Matveev 25. A transcritical-flip bifurcation in a model for a robot-arm
. . . . . . . . 209
H. G.E. Meijer 26. Alignment and the classification of Lorentz-signature tensors . . . . . 215
R. Milson 27. Renormalization group symmetry and gas dynamics.. . . . . . . . . . . . .223 S. Murata 28. Refined computation of hypernormal forms.. ....................
.229
J. Murdoclc 29. New order reductions for Euler-Lagrange equations. . . . . . . . . . . . . . .236
C. Muriel and J.L. Romero 30. Regularity of pseudogroup orbits.. .............................. P. J. Olver and J. Pohjanpelto
.244
31. Relaxation times to equilibrium in Fermi-Pasta-Ulam system., . . ,255
S. Paleari and T. Penati 32. Energy cascade in Fermi-Pasta-Ulam models .....................
263
A . Ponno and D. Bambusi 33. On Birkhoff method for integrable lagrangian systems.. . . . . . . . . . .271 G. P u cacco
xii
34. Symmetry of singularities and orbit spaces of compact linear groups ...................................................
.279
G. Sartori and G. Valente 35. Symmetric solutions in molecular potentials
.....................
.291
L. Sbano 36. Variational approach to soliton generation and stability analysis of multidimensional nonlinear Schrodinger equation. .... .300
V. Skarka, N .B. Aleksic and V. Berezhiani 37. Differential invariants for infinite-dimensional algebras. .......... .308
I. Yehorchenko
Conference information Conference program ................................................
313
List of participants .................................................
318
List of communications .............................................
321
Papers appearing in previous SPT proceedings .....................
.326
A CO-CHAIN MAP FOR THE G INVARIANT DE RHAM COMPLEX
I.M. ANDERSON, M.E. FELS Department of Mathematics and Statistics, Utah State University, Logan Utah, USA, 84322
[email protected],
[email protected] 1. Introduction
In this note we characterize the Lie group actions for which there exists, at least locally, an evaluation map that defines a cochain map from the differential complex of invariant forms on a manifold to the de Rham complex for the quotient. This problem is motivated by the principle of symmetric criticality [4]. Before giving any specific definitions we would like to illustrate the notion of such an evaluation map with a simple example. Consider the two dimensional Abelian Lie group G = IR2 with coordinates ( a , b ) acting on IR3 by
( a ,b) *
(2,
Y,z ) = (z, Y + a,
+ b).
If (Y E R2(IR3)Gand v E R3(IR3)G, where we use the convention that a group superscript denotes the invariants of the group, then cy and u are necessarily of the form cy
= a(z)dz A dy
+ b(z)dzA d z + c(z)dy A d z
and
v = A(s)dzA dy A dz.
The Lie algebra of infinitesimal generators of this action of G is generated by {ay,a,} and it is easy check that evaluation on the generators cy(ay,
a,)
=
4 . )
a,, -)
~ ( a y ,
= A(s)dz
defines a cochain map from R*(IR3)G to R*-2(IR), that is, (dCY)(ay,a,, -) = d(cy(d,,
a,)) = c(z)’dz.
As we shall see, not all group actions admit cochain evaluation maps.
1
2 2. Lie group actions and invariant vector fields
Let G be a pdimensional Lie group which acts effectively on an ndimensional manifold M with multiplication map p : G x M + M . We write gx instead of p(g,x). For x E M and g E G , we define p, : G -+ M and pg : M + M t o be the maps P X ( 9 ) = Pg(X) = gx.
For any g E G , pg is a diffeomorphism of M . We let G, denote the isotropy subgroup of G at x,
G,
=
(9 E G J g x= x}.
For each x E M I the map b, : G / G , h/lgiven by jl,([g]) = gx is a oneto-one immersion which is also G equivariant with respect t o the canonical action of G on the coset space GIG,. The Lie algebra g of the Lie group G is the Lie algebra of right invariant vector fields on G. The action of G on M induces a Lie algebra homomorphism r : g X ( M ) of g t o the vector fields on M whose image is the Lie algebra of the infinitesimal generators of the action of G on M [7]. We write I? = r(g). Because the action of G on M is assumed effective, the map r is injective. Let I' c T M denote the (integrable) distribution generated by I?. The action of the Lie group G on M is said t o be regular if the space of orbits is a manifold &f = M / G such that the quotient map --f
--f
q:M+&f is a submersion. We will assume from here on that all actions are regular. For regular actions the orbits all have the same dimension which we assume t o be q and so the isotropy subgroup G,, for any x E M , will have dimension p - q . Let Vert M -+ M be the sub-bundle of q vertical vectors in T M , so VertM = kerq, = I'. We also have the important property
( b x ) * ( T [ e ] G / G= x ) VertxM. The action of G on M defines an action of G on T M using the differential (pg),: T M + T M .
For each g E G,, equation (1) gives ( p g ) +: T,M
+
T,M
which defines the linear isotropy representation of G, on the tangent space
T,M.
3 Suppose now that X is a G invariant vector field, that is,
(Pg)*Xx = x g x .
(2)
If g E G,, then equation (2) implies that
x, E (T,M)G?
(3)
This observation leads us to define the following subset of T M ,
n(TM) =
u
K ( T , M ) , where n(T,M)
=
(TxM)GJ.
xEM
Equation (3) implies that every G invariant vector field X takes values in the subset K ( T M )c T M . Since q o p g = g , the action of G on T M restricts to an action on Vert M and the linear isotropy representation of G, also restricts t o a representation on vertical vectors (,ug), : Vert,M
-+
Vert,M,
g E G,
Thus a G invariant vertical vector field takes values in the set K(
VertM) =
u
K(
Vert,M) , where
n( Vert,M) = ( Vert,M)Gr
xEM
In the next theorem we give conditions which guarantee the existence of invariant vector fields. This is a special case of the general construction given in [2] or on p. 657 in [3].
Theorem 2.1. If n ( T M ) c T M is a vector sub-bundle, then for each x E M and Y E n(T,M) there exists a G invariant vector field X on M such that X , = Y . The analogous statement holds for G invariant vertical vector fields i f K ( Vert M ) c Vert M is a vector sub-bundle. Remark 2.1 For the rest of this article we assume that all group actions are regular and that K ( T M ) ,and K ( VertM) are bundles. 3.
Lie algebra cohomology
Given a Lie group G and a Lie subgroup K c G, with corresponding Lie algebras t c g, define the vector space of K relative forms on g by
A'(g, K ) ={ cy E A T ( g )I ~ , a= 0, b' v E t and Ad*g. cy
= cy
, b' g E K },
where A'(g) are the alternating r-forms on g and Ad* denotes the co-adjoint representation of G on A T ( g ) .
4
The usual exterior derivative d on A*(g)restricts t o make A*(g, K ) a differential complex whose cohomology is denoted by H * ( g ,K ) , the Lie algebra cohomology of g relative t o the subgroup K . If K c G is a closed Lie subgroup, let H * ( R * ( G / K ) G be ) the dcohomology of the G invariant forms on G / K .
Lemma 3.1. If K C G is closed, then R T ( G / K ) GN A T ( g , K )and H T ( R * ( G / K ) GN) H T ( g ,K ) . See Theorem 13.1 in [6] for a proof of this Lemma. It is well-known [8], that if G is connected and compact and K closed, then H * ( g ,K ) computes the de Rham cohomology of the homogeneous space G / K . It is useful t o note that if K2 = gKlg-' are conjugate subgroups of G then Ad(g) induces an isomorphism A*(& K1) = A*(&K2).
(4)
Example: Consider the 2 sphere S2 and the projective plane RP2 as the homogeneous spaces S 0 ( 3 ) / S 0 ( 2 ) and S 0 ( 3 ) / 0 ( 2 ) . Letting X I , X 2 , X 3 be a basis for so(3) with X 3 the basis for so(2) (which particular so(2) is actually irrelevant because of (4)) and letting a', a 2 ,a3 be the dual basis, we find A'(so(3), SO(2)) = ( 0 ) and
A2(so(3),SO(2)) = {a'
A
a'}.
Therefore H2(so(3),S O ( 2 ) )is generated by a1 A a2. On the other hand, there is a reflection in O(2) which maps X1 to - X I and X2 to X 2 so that A1(so(3), O(2)) = ( 0 ) and
A2(so(3),O(2)) = ( 0 )
and therefore H2(so(3),O(2)) = 0. Of course, these computations reflect the fact that S2 is orientable whereas IRP2 is not. 4.
A map on the G invariant de Rham complex
In this section we generalize the evaluation map from the introduction by studying the problem of defining a map
pxIC . s2k(M)G-+ R"q(M) '
which shifts form degree by the orbit dimension q of G on M . To begin, we define A,( VertM) + M t o be the vector bundle of vertical q-chains on VertM (alternatively, the bundle of vertical multi-vectors of degree q ) . Given that the orbits of G have dimension q it follows that about each point
5
M there exists an open set U and vector fields X I , X 2 , . . . , X , in I? which define a local frame for Flu = VertU. Consequently if X is a section of A,( VertM) then X(u can written as
x
E
XI"
=JX,AX2A...AXq,
where J E C"(U). The action of G on VertM described in section 2, induces an action of G on A, ( Vert M ). Given a G invariant q-chain X : M -+ A, ( Vert M ), we now define a map L X : O k ( M )-+ S2:C4(M) where
Cl2,*,(M)= {w E R*(M) I L
~ = W 0
are the q semi-basic forms on M . The map
LX
for all X E r) is defined by setting
(LXW)x(Yl,Y2,-. , Y k - q ) = ~ x ( ~ x , ~ l l ~ 2 1 . . . l ~ k - q )
for w E 0 2 " ( M and ) Y, E T x M . If w E S2k(M)Gthen L X W is q semi-basic, and since X is G invariant, L X W is G invariant and so G basic. By this last statement L X W E S2:,q(M)G, and therefore by Lemma A.3 in [l],we find there exists a unique ( k - q)-form on satisfying q*(W) = LXW. The sought after evaluation map p x is then defined by
LXw
Note that for each invariant X we have a map p x .
Theorem 4.1. If there exists a non-vanishing G invariant vertical q-chain X on M , then
As(g,G,) # 0
for all x E M .
(6)
Conversely, i f for each x E M , Aq(g,G I ) # 0 then about each xo E M there exists a G invariant open set U and non-vanishing G invariant vertical qchain X on U . Proof. Let X be a non-vanishing G invariant vertical q-chain. Let x E M and let 2 be the restriction of X to G/G,, so that (jix)*2= X . By the equivariance property of jix the q-chain 2 is G invariant. Now let a E W ( M ) satisfy a(X) = 1. The form Q: is not unique, and it is not necessarily invariant. We claim the form ji;a defines a non-zero element of Oq(G/Gx)G.We compute (g*ji:cy)(q = a ( ( L ) * g * X ) = cr((jix)*%)= a(X)= 1.
Thus ji:cy is a non-vanishing G invariant form of top degree on GIG, and so, by Lemma 3.1, Aq(g,G,) # 0.
6
We now prove the converse part of the theorem. Let
u
K(A,(VertM)) =
K(h,(Vert,M)),
,EM
where K(A,(Vert,M)) = (A,(Vert,M))G=. We shall show that Aq(g,G,) # 0 implies n ( A , ( V e r t M ) ) is a line bundle. Then the existence of a G invariant q-chain is guaranteed (in a similar manner to Theorem 2.1) by Theorem 1.2 in [2]. If Aq(g,G,) # 0 then by Lemma 3.1 there exists a non-vanishing 6 E s2q(G/Gz)G. Let % be the invariant q-chain defined by ti(%) = 1. Then x, = (p,),5&1 E A,(Vert,M))G= by the equivariance of f i x , and is non-zero. Thus A,(Vert,M))G7 = A,(Vert,M) and so K(A,(VertM)) = A,(VertM) is a line bundle. 0 5. The cochain condition In this section we find necessary and sufficient conditions on the action of G on M that determine whether we can choose a G invariant q-chain X so that the map px : s2*(M)G 4 R*-,(M) defined in (5) is a cochain map, that is, P X ( d 4 = dPX(W).
(7)
Granted that the action of G on M satisfies the conditions in Remark 2.1, the solution to this problem is given by the following theorem.
Theorem 5.1. If there exists a non-vanishing invariant q-chain X such that the map px in (5) defines a cochain map, then H4(g,Gx) # 0 for all x E M . Conversely, i f HQ(g,G,) # 0 for all x E M then about each xo E M there exists a G invariant open set U and a non-vanishing G invariant vertical q-chain X on U such that px s2*(U)G R*-q(U/G) is a cochain map. ---f
In order t o prove this theorem, we need a number of preliminary results. The first of these is the important observation that the cochain condition ( 7 ) , which is a condition that involves the quotient manifold can be expressed as a condition entirely on M .
a,
Lemma 5.1. if and only i f
A G invariant, vertical q-chain X defines a cochain map p x ~xdw= (-1)4d(LXu)
for all w E s2*(M)G.
(8)
7
Proof. If q is any G basic form, then dq is also G basic. Let f j be the unique form on H such that q*(fj) = q. Then, since q*(dfj) = d q * ( f j ) = dq
the two forms dfj and & pullback by q t o the same form and must therefore be equal. Since X and w are both G invariant, we can apply this observation t o the G basic form L X W t o deduce that d ( w ) = d(Lxw).
The cochain condition (7) can therefore be expressed as ~~
(9)
(-1)'LXd(w) = d(LXW).
But two G basic forms on M are equal if and only if the corresponding forms on are equal and so (9) proves the equivalence of (7) with (8).
Lemma 5.2. If (8) holds for all G invariant ( n- 1)-forms, then (8) holds for all G invariant r-forms, r 2 q .
Proof. Suppose (8) holds true for all G invariant ( n - 1)-forms. Let w be a G invariant r-form, where q 5 r < n - 1. Then, if Q is any G basic ( n - r - 1)-form, w A a is a G invariant ( n - 1)-form and therefore we can use (8) t o write LXd(w A a ) = (-1)'d
(LX(W
A
a))
Because CY (and hence d a ) is G basic, the expansion of both sides of this equation bwives ' (Lxdw) A
= (-l)'d(LXW) A CY
Since Q is an arbitrary G basic form and LXdw and d(Lxw) are both G basic this implies
Lemma 5.3. Let p be a G basic ( n - 4)-form on a G invariant open set U . Let X be a non-vanishing, G invariant, vertical q-chain on U and let Q be any q-form such that a ( X ) = 1. Then
v =QAp is a G invariant n-form on U .
8
Proof. For any g E G, we compute [ P ; ( a ) l ( x ) = 4(PUs)*(X))=
=1
and therefore L x [ P ; ( 4 1 = LX[P;(a) A PI = CL = L X V .
This suffices to prove that p i ( u ) = u.
0
Lemma 5.4. If X is non-vanishing vertical q-chain and R is a G invariant vector field then CRX = X R X ,
where XR is a G invariant function. Proof. Let X I , . . . ,X q be vector fields in r which form a local basis for VertM in some neighborhood about the point x. Then X = J X 1 A X2 A . . . A X, and, since [ R, X , 1 = 0 ,
Theorem 5.2. If X is a non-vanishing, G invariant, vertical q-chain, then the map px : C2*(M)G-+ O * - q ( m ) is a cochain map i f and only i f
LRx = 0 for all G invariant vector fields R on M
(10)
Proof. We start by assuming ( 1 0 ) . Then by Lemma 5.2 it suffices to prove (8) for G invariant ( n - 1)-forms. Given the non-vanishing G invariant vertical q-chain X , let x E M and use Lemma 5.3 t o construct a non-vanishing G invariant n-form u = a A p on an invariant open set U about x. Let w E C2n-1(M)G,then restricted to U there exists a unique G invariant vector field S on U such that w u = LSU.
(11)
Let R be a G invariant vector field on M which agrees with S on an invariant open set V c U of x so that W v
(12)
= LRU.
With wv given by ( 1 2 ) , we compute on V [d(Lxw)IV = ~ ( L X L R U = ) (-l)'d(LRLXU) = (-1)'d(LRP)
[ L X C ~ W= ] ~ L X ~ ( L R V=) L X L R ( V=) L R ( P )- L
and
~ ~ ( X ) U .
9
But it is easy t o check that if p is a G basic ( n- q)-form, then d p = 0 and therefore (13)
[d(LXW)- ( - l ) q L x 4 , = ( - l ) q L L R ( x ) v .
Evaluating ( 1 3 ) at x E V shows that if (10) holds a t x then (8) holds a t x for all G invariant ( n - 1)-forms w . Since our original point x E M was arbitrary, equation (8) holds on M . To prove that (8) implies (10) we reverse the argument above. Let R be a G invariant vector field on M and let x E M . Choose a G basic ( n - q)-form p which doesn't vanish at x. Then the form w = LRv, where v = Q A p with Q ( X ) = 1, is a G invariant n - 1 form on M . Equation (13) (evaluated at x) coupled with Lemma 5.4 shows that (8) implies (10) at x. 0 But x was arbitrary so (10) holds on M . We are now in a position t o prove Theorem 5.1. Proof. We begin the proof by first noting that the condtion H q(g, G,) # 0 is equivalent to the following: i] For each x E M there exists a non-vanishing 6 E S2q(G/Gz)G; and ii] for all f j E S2q-1(G/G2)G, dfj = 0.
Suppose there exists a non-vanishing G invariant q-chain X such that p x is a cochain map. Let a E P ( M ) with a(X)= 1. Then as was shown in Theorem 4 . 1 , given any x E M , &a E S2Q(G/G,)G and is non-vanishing. Thus condition i] is true. is Let f j E S2q-1(G/G2)G. Then f j can be written f j = L ~ P where ~ Q a G invariant vector field on G/G,. Now (p2).qe1 E ts(Vert,M) and by the hypothesis on invariant vector fields (Theorem 2.1), there exists a G invariant vector field Y on M such that Y, = ( p , ) . q e l . Thus f j = & ( ~ y c r ) . In order to calculate dfj we let p be a G basic ( n- q)-form which doesn't vanish at x so that by Lemma 5.3 cr A p is G invariant n-form which doesn't vanish at x. It is simple t o check that L X [ ~ ( L Y Q= ) ]0 ~ if and only if LX[d(LYO)A 4 2 = 0. By using the fact p is d-closed and by applying equation (8) to the invariant one-form L Y ( QA p) = ( L Y Q )A p , we find LX[d(LYQ)A PI2 = LX[d(LYQA P)12 = Thus
L X [ ~ ( L Y C Y )= ] , 0.
d f j ( X ) [ e ]=
[d(LXLYQ)A P ) ] , = 0.
Now computing
[ Q W L Y[4Q=)[(P~~ ~) (]L Y Q ) ( Q=] [[ ~~( L]Y & ) ( X ) I ,
=0
(14)
10
and using the invariance of i j we get dij = 0. This proves ii] and therefore Hs(8,GX) # 0 . To prove the converse, choose xo E M . Then by Theorem 4.1 the hypothesis Aq(g,G,) # 0, implies there exists a non-vanishing G invariant q-chain X O on an invariant open neighbourhood U of XO. Suppose that the rank of K ( VertM) is s and that the rank of K ( T M )is T . Let Y,, a = 1,.. . s be a local frame about 20 for K ( V e r t M ) consisting of invariant vector fields. Choose invariant vector fields Z,, t = s 1 , .. . , r which together with Y, form a local frame about xo for n(TM). Refine U so all these objects exist on an invariant open set which we again call U . We now show that if H q ( g ,G,) # 0 then Cx,Xo = 0. First we compute
+
( l Y a a ) ( X o )= Y a ( a ( X 0 ) ) - C Y ( . C Y a X O ) = - a ( l Y a X 0 ) . Expanding out the left side of this equation we get [d(Lyaa)+ ~ y - d a(]X O )
=
-a(.CxzXo).
(15)
Immediately LXoLyada= 0, because Y, is vertical, while condition ii] implies by the argument used above that[d(~y,a)](XO)= 0, and so (15) along with Lemma 5.3 implies Ly,Xo = 0. To finish the proof of the theorem we now show there exists an invariant K which doesn't vanish at xo so that X = KXo satisfies equation (10) for Y,,Z,. Using the fact that CxLX0= 0, it is easy to check Cx,(KXo) = 0. The conditions CZ,(KXo) = 0 leads to the differential equations for K
Lz,X
=
+
( Z , ( K ) KXz,)Xo = 0
where XZ, are determined as in Lemma 5.4. The functions K , XZ, and the vector fields 2, are all invariant so letting 2,= q*Z,, this equation for K can be written on as
Z,(K) + RXz, = 0.
(16)
The integrability conditions for K or K can be easily verified by a computation using Lemma 5.4. Therefore there exists an open neighbourhood of 10 and a non-vanishing which is a solution to (16). Consequently K X =~ ( q * K ) X o satisfies (10)on q-l(V).
v
6. Examples
Example 1. As our first example consider the two dimensional solvable group G = IR' x IR with coordinates ( a ,b) acting on IR x IR*x IR by (a,b)*(X,Y,z) = (az+b,ay,z).
11
This is a free action and so H 2 ( g ,G,) = H 2 ( g ) and one easily computes H 2 ( g )= 0. We proceed to check Theorem 5.2 for this example. The most general G invariant vertical 2-chain X is of the form
x = K ( z ) y 2 a x A a,. The invariant vector fields are
Computing C,a,X
we get
CYa,X =
A
and so, consistent with Theorem 5.2 and Theorem 5.1 there is no choice of non-zero K ( z ) so that (10) is satisfied, and no cochain map exists.
Example 2. Consider the action of the two dimensional Abelian group G = IR2 with coordinates ( a lb ) on IR2 given by (a,b)*
(ZlY) =
(s+aY +b,Y).
The fact the group is Abelian implies H1(g,G,) # 0, for all z E IR2, so a cochain map exists by Theorem 5.1. The G invariant vertical 1-cochains are given by
and the invariant vector fields are
R = a(y)& . So every cochain X in (17) satisfies (10). This examples demonstrates the fact that the q-chain may not be unique, and by a further simple computation, that the cochain map p x may not be surjective. As a final remark we state a theorem on the surjectivity of p x .
Theorem 6.1. Let X be a G invariant vertical q-chain such that px defines a cochain map. Then px is surjective if and only i f there exists Q E a q ( M ) such that a(X)= 1 and Q is G invariant. See [4] and [5] for other examples
12
References 1. I.M. Anderson and M.E. Fels, Exterior Differential Systems with Symmetry, submitted, Acta. Appl. Math., SPT. 2004. 2. I.M. Anderson and M.E. Fels, Topology and its Applications, 123 , 2002, pp. 443-459 3. I.M. Anderson and M.E. Fels, Commun. Math. Phys., 212 , 2000, pp. 653-686 4. I.M. Anderson and M.E. Fels, Amer. Jour. Math., 119 , 1997, pp. 609-670 5. M.E. Fels,C.G. Torre, Class. Quantum Grav., 19,2002, pp. 641-675. 6. C. Chevalley,S. Eilenberg, Trans. Amer. Math. SOC.,63,1948, pp. 85-124. 7. P.J. Olver, Applications of Lie groups to differential equations,Springer-Verlag, 1993 8. M. Spivak, A comprehensive introduction to differential geometry Vol. 5, Publish or Perish.1979
NEW EXAMPLES OF TRIHAMILTONIAN STRUCTURES LINKING DIFFERENT LENARD CHAINS
c. ANDRA, L. DEGIOVANNI Dipartimento d i Matematica, Universitci di Torino via Carlo Alberto 10, 10123 Torino, Italy degioodm. unito. it The extension of bihamiltonian systems allows to realize a recursion relation between different Lenard chain. In particular extensions of either three-particle periodic Toda lattice, or systems on the Euclidean plane separable in parabolic coordinates or in elliptic-hyperbolic ones are presented.
1. Introduction
The classical concept of Hamiltonian vector field X h , generated by the Hamiltonian function h, can be generalized using a Poisson tensor P and formulas
Xh(f) = P ( d h , d f ) = {h, f}. One of the advantages of this generalization is the possibility t o construct many Poisson tensors on the same manifold and then to associate more vector fields to a given Hamiltonian, or to associate more Hamiltonians to a vector field. Moreover, in this way it is possible to deal with degenerate Poisson brackets too, i.e. brackets that admits non-constant functions (called Casimir functions of the bracket) in involution with all the others. Two Poisson tensors P and Q are said t o be compatible (in Magri’s sense) if Q - XP is a Poisson tensor for any A; this Poisson tensor is called the Poisson pencil of the two tensors. Two compatible Poisson tensors define a bihamiltonian structure. A vector field is called bihamiltonzan if it is Hamiltonian with respect to two compatible Poisson tensors. A particular importance in the study of integrable systems is given to bihamiltonian vector fields X , belonging to a Lenard chain, i.e. such that their Hamiltonians satisfy the Lenard-Magri recursion relations Pdh, = Qdh,+l
13
(1)
14
for a bihamiltonian structure. The recursion relations (1) can be represented by the diagram
An important theorem of the bihamiltonian theory states that all the Hamiltonians associated to the vector fields of a Lenard chain are in involution with respect t o both the Poisson structures. A quite effective method to construct a Lenard chain is to look for a Casimir function of the Poisson pencil generated by the two Poisson tensors: all the coefficients of the powers expansion with respect to X of such function satisfy the recursion relations (1). In many cases a Poisson tensor has more than one Casimir function, therefore multiple Lenard chains exist. Moreover, there are well known examples of bihamiltonian systems admitting three or more Hamiltonian formulation. However, in almost all these classical examples the further Poisson structures give a recursion between vector fields already belonging to a certain Lenard chain; if there are different Lenard chains, they will remain unconnected. Then it becomes very interesting the case of bihamiltonian systems with first integrals organized in multiple Lenard chains for which a third Poisson structure exists and establishes a recursion relations between distinct Lenard chains. In this way a “two-dimensional’’ recursion scheme is constructed, instead of the one-dimensional one (2). Some general properties of these kind of systems were investigated and a class of examples was constructed7, but until now very few systems that are trihamiltonian in this sense are known. In this work some new examples of this kind of systems are presented, and moreover they are related to wide used techniques, like separation of variables and R-matrix theory.
2. Trihamiltonian framework
A trihamiltonian structure on a manifold is given by three mutually compatible Poisson structure P , Q and R. In this paper only trihamiltonian structures admitting a common Casimir function are considered. A common Casimir function for a trihamiltonian structure is a function f (depending on two parameters X and p ) such that (Q-XP)df=O, (R-pP)df=O.
15
Not every trihamiltonian structure admits such a function7, and it plays a role analogous t o Casimir functions of the Paisson pencil in the standard bihamiltonian framework: the coefficients of its powers expansion f = C fijX',uJ obey the recursion rules
representable by the following recursion diagram
...
x 1 2
x22
Y Y Y Y Y Y f12
...
f32
f22
1.
1.
1.
x 1 1
x 21
x3 1
Y Y Y Y Y Y Y fll
1.
...
f21
1.
...
f31
1.
...
Similarly to the bihamiltonian case, the recursion relations ( 3 ) imply that the coefficients f i j are mutually in involution with respect t o each of the three Poisson structures7. 3. Trihamiltonian extension of Toda lattice
A Lax representation for the periodic Toda lattice can be constructed" on the Kupershmidt associative algebra K , of formal power series in the shift operator A with coefficients in the ring, equipped with the component-wise operations, of n-periodic sequence a = ( a l l , . . . , g i n ) . On this algebra the relation Aku = aIklAk,where = uli+k, holds; moreover the linear ) A P k uand the traceform involution * ( a A k =
are defined. An operator L E K , is skewsymmetric if * L = -L, and it is symmetric if *L = L. A symmetric element is of degree k if it contains
16
powers of A only until k, then it can be represented as
L = A-'ok On
+ . . . + oo + . . . + okAk
K,, using the associative product, the two natural operations [A,B ] = AB - B A AB+ BA A.B= 2
and the ad-invariant scalar product ( A ,B ) = tt(AB) are introduced. The Lie algebra K, can be split into two subalgebras: g+ of skewsymmetric operators and g- of operators with powers of A not greater than 0. Using the ad-invariant scalar product a second decomposition K, = g+@gis obtained, where g+ = (g+)l is the vector space of symmetric operators and g- = (8-)' is the subalgebra of operators with powers of A strictly negative. The space of symmetric operators of degree k is indicated with g z . Therefore an R-matrix is given, as usual, by
r=rI+-nand one can prove1~2~12 that the linear and quadratic Poisson brackets
if7 g } p = ( L[r(Vf),091 + [Of,r(Vg)I) {f,dQ= ( L[ r ( L oO f ) , 091 + [Of,r ( L Vg)I) are reducible by restriction on g t for any k. Finally, an isomorphism of associative algebras between K, and the algebra of formal Laurent series with coefficients in g [ ( n ) exists''. The ordinary periodic Toda lattice with n particles is set in the subspace &. A Lax matrix for the three-particles case (the only one considered from now on), in Flaschka coordinates (bi, ai), is given by"
A Casimir function of the pencil Q - XP is given by any coefficient in C of det(L - A€), so in this case one has two distinct Casimir functions, that are polynomial in the parameter A. The coefficients of these two polynomials are independent and mutually in involution first integrals, enough in number to prove the complete integrability of the system.
17
A natural extension is obtained considering the subspace g t , in this case the Lax matrix of the system, in the coordinates (bi, a i , c i ) , is
and also in this case any coefficient in
< of
is a Casimir function of the pencil Q - XP'22. This extension admits a trihamiltonian structure: introducing the Poisson tensor
R=
and reorganizing the previous functions in the polynomial
a common Casimir function for the two pencils Q - XP and R obtained','. The corresponding recursion scheme is
-
p P is
18
0
0
0
4. Trihamiltonian extension of natural separable systems
A further class of examples is given by separable systems in the Euclidean plane. Here only systems separable in parabolic coordinates or elliptichyperbolic ones are considered, the general case can be worked out in a similar way6. A natural Hamiltonian is a function of the form
H = -1g z 3. p. . apj. + V ( q ) 9
defined on the cotangent bundle of a Riemannian manifold ( Q , g ) . The separability, trough point transformation, of this kind of Hamiltonians on the Euclidean plane is characterized by the existence of a second function K (related to a special3 Killing tensor for the metric g ) that satisfies the so called Stuckel relations. In each coordinate system there is a specific form for the two functions H and K and for the Stackel relations5. The two Hamiltonian functions in parabolic coordinates are:
They satisfy the Stackel relation
19
The two Hamiltonian functions in elliptic-hyperbolic coordinates are
H= K=
(;P:
+ dl(S)) ( s 2- k 2 )
(;P;
-
- (;P; s2 - d2
ip:
+ d 2 M ) ( d 2 - k2)
+ 4 2 ( 4 - di(s)) (s2 s2
-
-
k2)(d2 - k2)
d2
and satisfy the Stackel relation
Using the symplectic transformations for the parabolic case A1
= u, A2 = v,p1 = pu, p2 = Pv
and respectively for the elliptic-hyperbolic case P S
= s2 - k2,X2 = d2 - k 2 , p l = - , p 2
2s
Pd
=-
2d
in both cases the Stackel relations become:
where, respectively for parabolic and elliptic-hyperbolic coordinates, the functions r i ( X , p ) are
The two relations (4) are very similar to the separation equations obtained, in the general trihamiltonian framework, starting from a common Casimir function of the trihamiltonian structure7, except the fact that the canonical Poisson structure on the cotangent bundle doesn't have any Casimir function. The trihamiltonian scheme
20 is the simplest one that the vector fields associated to H and K fit, respecting polynomial relations (4). In fact, it is associated to the common Casimir function f = i? ~ i ; i x2c1 pc2 ~ p c 3 . The diagram ( 5 ) cannot be constructed using a non degenerate Poisson tensor. One needs to enlarge the phase space of the system to T*Q x R 3 (with coordinates Xi, pi and c,). This is a Poisson manifold with the trivial extension of the canonical Poisson structure on T*Q:
+
+
+
+
On the other hand, fi and are functions that reduce to the Hamiltonians H and K on the symplectic leaf c, = 0. The explicit expression for H and is found by imposing the separation relations (in Sklyanin's sense):
K
K + x,H + X ? C ~ +pzc2 + Xzpzc3 = rz(x,,p z ) ,
(6)
that generalize the relations (4), and by solving this linear system in the two unknowns H and i?. The two further bivectors Q and R are finally defined on T*Q x R3 by formulas a Q= a apz A X, A =,
c,x
&+
a a R = z a pzap, -A -+ X E ax,
A
a acz + X H-
A -2acs.
As final result of this construction, the three bivectors P , Q and R are Poisson tensors, the function f is a common Casimir function for the two pencils Q - XP and R - p P and therefore generates the recursion scheme (5), for any choice of the functions r, . The presented extension procedure is analogous to the techniques used in the bihamiltonian separability theoryg>'', and the relations (6) correspond to the separation curve recently introduced4. Acknowledgements The authors wish to thank Prof. Guido Magnano for his suggestions and supervision, both in analyzing the problem and in writing the paper. References 1. ANDRAC., Una struttura trihamiltoniana per il reticolo di Toda, BSc thesis, University of Torino 2004 2. ANDRA C., DEGIOVANNI L., MAGNANO G., A trihamiltonian extension of Toda lattice, in preparation. 3. S. BENENTI, Separability in R i e m a n n i a n manifolds, to appear in Philos. Trans. R. SOC. Lond. Ser. A.
21 4. M. BLASZAK, Degenerate Poisson pencils on curves: New separability theory, J. Nonl. Math. Phys., 7 (2000), 213-243. 5. C. CHANU,Separation of variable and Killing tensors in the Euclidean threespace, PhD thesis, University of Torino 2001. L., Trihamiltonian extensions of separable system in the plane, 6. DEGIOVANNI submitted for publication, nlin.S1/0407030. 7. DEGIOVANNI L., MAGNANO G., %-hamiltonian vector fields,spectral curues and separation coordinates, Rev. Math. Phys ., 14 (2002), 1115-1163. 8. FALQUI G., MAGRIF . , PEDRONI M., Bihamiltonian geometry and separation of variables for Toda lattices, J. Nonlin. Math. Phys., 8 (2001), 118-127. M. PEDRONI, Separation of variables for bi-hamiltonian systems, 9. G. FALQUI, Math. Phys. Anal. Geom., 6 (2003), 139-179. 10. A. IBORT, F . MAGRI, G . MARMO,Bihamiltonian structures and Stackel separability, J. Geom. Phys., 33 (2000), 210-228. and the periodic Toda 1 11. MOROSIC., PIZZOCCHERO L., R-matrix Theory, Formal Casimirs and the Periodic Toda Lattice, J. Math. Phys., 37 (1996), 4484-4513. 12. OEVELW., RAGNISCOO., R-matrices and Higher Poisson Brackets for Integrable Systems, Phys. A, 161 (1989), 181-220.
WAVE PROPAGATION IN AN ELASTIC MEDIUM: GDS EQUATIONS
C. BABAOGLU Department of Mathematics, Faculty of Science and Letters, Istanbul Technical University, Istanbul, Turkey
[email protected]. tr
S. ERBAY Department of Mathematics, Faculty of Arts and Sciences, Isik University, Istanbul, Turkey
[email protected]. t r
Wave propagation in a bulk medium composed of an elastic material with couple stresses is considered. Using a multi-scale expansion of quasi-monochromatic wave solutions, it is shown that (2+1) (two spatial and one temporal) dimensional wave modulation is governed by a system of three nonlinear evolution equations which will be called the ”generalized Davey-Stewartson (GDS) equations”. Some special solutions of the GDS equations are also presented.
1. Introduction
It is well-known that the envelope of a ( l f l ) dimensional quasimonochromatic wave train is governed by the single nonlinear Schrodinger
(NLS)
ZAt +PA,,
+ q1AI2A = 0,
(1)
where t is time, x is the spatial coordinate and A denotes the complex amplitude. The NLS equation appears t o be a generic equation describing unidirectional wave modulation. If modulations transverse to the wave propagation direction are also allowed, second spatial coordinate effect should be taken into account and new (2+1) evolution equations should be derived [1,2]. A natural way t o obtain two-dimensional modulations of nonlinear waves is simply to replace the one dimensional dispersive term with a two dimensional dispersive term [3,4]
22
23 However, in many two dimensional systems both short waves and long waves may co-exist and the modulation of such a system can be characterized by the Davey-Stewartson (DS) equations [5,6,7]
iAt + P A , ,
+ TA,, + qlAI2A = bAd,,
dm + m d y y = (1A12)z,
(3)
where A is the complex amplitude of the short wave and 4 is the long wave amplitude. The DS system is a model for the evolution of weakly nonlinear packets of water waves that travel in one direction but in which the amplitude of waves is modulated in two spatial directions. The main purpose of the present study is to extend the analysis of Davey and Stewartson to describe (2+1) dimensional wave motion in a bulk elastic medium.
2. Two-Dimensional Wave Packets The micromorphic elastic solids, roughly speaking, are the classical elastic solids that admit micro deformations of the micro volume elements about the center of mass of the volume element [8]. The deformation tensors are defined as 2ek1 = u k , l f Ekl = @ k l
U l , k f um>kum,l
+ U l , k f um,k am1
r k l m = @/cl,mf un,k @nl,m
(klI , m,
= 1,2 , 3 )
(4)
where subscript k after comma denotes partial differentiation with respect to space variable xk and summation convention is valid over repeated indices. Here u k , l is the displacement gradient, e k l is the macro deformation tensor characterizing the relative displacements of the mass center of macro L r k l m are new micro deformation tensors of the micromorvolume, E ~ and phic theory. In the present study, a simplified form of the micromorphic elasticity l 0 and the product am1is omitted, i.e., the theory is used where ~ k = macro- and microdeformation tensors take the form
In addition, the kinetic energy is assumed to be
24
where po is the mass density in the reference configuration and subscript t denotes partial differentiation with respect to time variable t. The strain energy density function in terms of the invariants of the deformation tensors is given by [9]
c = x- e k k e i i + pekiekl+
A
+ Bekieikemm + 2pm2(rkimrkim + V r k i m r i k m ) , -eklemiekm
3
C (7) 3 where A, /I are linear elastic constants, A, B and C are second-order elastic constants, and v and m are new constants characterizing the microstructure. The governing equations of ( 2 + l ) dimensional problems are obtained from the variational problem +-ekkeiiemm
s J Ldt = s JJJ Ldxdydt = 0, where L represents the Lagrangian, and L = T - C. The Euler-Lagrange equations of the variational problem are obtained as
If the strain energy density function C and the kinetic energy density function T are substituted into equation (9), the field equations are obtained (see [lo], for details). From the linearized form of the field equations, we have the dispersion relations as D ~ ( K , w=)w2 - C ~ -K 4(1+ ~ v)c;m21c4, D ~ ( K , L=J D) 3 ( K , W ) = W 2 - c;K2 - 4Czm 2 2 K4 ,
(10)
where K' = kf + k z . In equation (lo), D1 represents the dispersion relation corresponding t o the longitudinal displacement mode associated with u1. Also, D2 and 0 3 represent the dispersion relation corresponding to the transverse displacement modes associated with u2 and u3, respectively. As is seen from equations (lo), both the longitudinal and the transverse modes are dispersive. At this step, by using the reductive perturbation method, the evolution equations characterizing the nonlinear wave interaction of a short transverse wave, a long transverse wave and a long longitudinal wave will be obtained.
25 To this aim, the slowly varying amplitude of the short wave and amplitudes of the long waves will be assumed to be functions of the slow variables
5
r] = €9,
T =E
2
t,
(11) where E is a small parameter measuring the weakness of dispersion and nonlinearity, cg is the group speed of the short transverse wave, and x, y and t are fast space and time variables; E , r] and r are slow space and time coordinates in a frame of reference moving with the group speed of the short transverse wave. Here the direction of the wave propagation is taken along the x axis and it is assumed that the field variables depend on the transverse coordinate y as well. Since we deal with the nonlinear interaction of the quasi-harmonic transverse wave and the zero harmonic transverse and longitudinal waves, it is convenient to expand the components of the displacement vector in asymptotic power series of E as follows: =z
E(X
- Cgt),
+ 2[iZ(~, r ] , T>ezis+ c.c.] + . . . = ~ c j z ( 6r,] , r ) + t2[iG(t, r ] , r)eZie+ c.c.] + . . . 213 = €[A([, r ] , r)eie + c.c.] + E~[G([, r], T)e2ze+ c.c.] + . . . u1 = €41( E , r], r ) 212
(12)
Here 6 = kx - w t is the phase, k is the wave number of the carrier wave, w is the frequency, C.C.stands for the complex conjugate of the preceding term. In expansion (12), A is the complex amplitude of the free short transverse wave mode whereas 41 and 4 2 are the free long longitudinal and free long transverse wave modes, respectively. Now, the scale transformation (11) together with power series solutions (12) are substituted into the field equations. If the coefficients of like powers of E are equated, a hierarchy of perturbation equations is obtained. For order c 3 , using the results of lower order perturbation equations, leads one to the evolution equations. The non-dimensional form of these equations may be given as
+ 6AZZ + A,, = XIAI2A + b(41,Z + 42,y)A, 41,ZZ+ mz41,,, + n42,%,= (IA12)Z1 A42,zz + m 1 4 2 , y y + n41,zy = (IAI2)y, 2-4
(13) where (A- 1)(m2- m l ) = n2 and the non-dimensional coefficients are given in the form
26 3. Special Solutions of the GDS System In order to find the travelling wave solutions of the GDS equations (13), the following solutions are considered
A
41 = g(C),
= f(C)eie,
42 =
h(0,
(15)
+
where B = l l x 12y - Ot, and the amplitudes of the short wave f, and the long travelling waves g and h are real functions of 5 = k l x + kzg - 2 ( S k l l l + k2l2)t. Here we assume that all the derivatives of f, g and h tend t o zero as C + 3 ~ 0 0 . Substituting the solution (15) into the system (13) leads t o a set of coupled ordinary differential equations (see [lo], for details). By solving these equations, the solution functions f , g and h are found in terms of Jacobian elliptic functions whose special cases involve the following two sets of solitary wave solutions:
ffl
1
ff2
1
g(C) = - ; P 2
h(C)= - - P z
ff
where a
< 0 , P > 0 and
where a
> 0 , P < 0.
tanh[PlC
+ 011,
tanh[piC
+ 011,
References 1. V.E. Zakharov, Sov. Phys. J . Appl. Mech. Tech. Phys. 4,86 (1968). 2. D.J. Benney and G.J. Roskes, Stud. Appl. Math. 48,377 (1969). 3. J. Pouget, M. Remoissenet and J. M. Tamga, Phys. Rev. B47, 14866 (1993). 4. B. Collet and J. Pouget, Wave Motion. 27, 341 (1998). 5 . A. Davey and K. Stewartson, Proc. Roy. SOC.London Ser. A338,101 (1974). 6. V. D. Djordjevic and L. G. Redekopp, J . Fluid Mech. 79, 703 (1977). 7. M. J. Ablowitz and H. Segur, J. Fluid Mech. 92, 691 (1979). 8. E. S. Suhubi and A. C. Eringen, Int. J . Engng. Sci. 2, 389 (1964). 9. V. I. Erofeyev and A. I. Potapov, Int. J . Non-Linear Mech. 28,483 (1993). 10. C. Babaoglu and S. Erbay, Int. J . Non-Linear Mech. 39,941 (2004).
P A R A M E T R I C EXCITATION IN N O N L I N E A R D Y N A M I C S
T. BAKRI Mathematics Institute, Utrecht University, PO Box 80.010, Utrecht, TA 3508, The Netherlands bakriomath. uu.n1 Consider a onemass system with two degrees of freedom, nonlinearly coupled, with parametric excitation in one direction. Assuming the internal resonance 1:2 and parametric resonance 1:2 we derive conditions for stability of the periodic solutions by using the normal form method of averaging. We found an attracting torus with large amplitudes by a Neimark-Sacker bifurcation. The results on the NeimarkSacker bifurcation obtained by the numerical software package CONTENT and by averaging are compared. In all cases we have good agreement. Continuation of the torus was done by averaging and by the numerical software package TORCONT. In all cases we have good agreement. The torus grows towards a homoclinic structure around the origin. The normal form method of averaging proves in general to be very effective in studying this type of continuation problems.
1. Introduction
Nonlinear vibrating systems often consist of two - or even more - subsystems, where one of them is excited, the Primary System, and the other ones are coupled through nonlinear terms; they are forming the Secondary System or Excited System. The Primary System is an oscillator which can be excited externally, parametrically or by self-excitation, while the Secondary System is excited indirectly through the nonlinear coupling. In the present paper we shall consider a single-mass system, but with parametric excitation in the Primary System and a nonlinear coupling expressed by second degree terms in the differential equations. The parametric excitation acts, for example, due to kinematic excitation of the supports through nonlinear springs; see Figure 1. The equations of motion are
For the damping coefficients we have b,bo,rc
27
2
0, furthermore c
> 0,
28
\
/ / /
M
\
1_J
/ c
&C
cos2q t
H
-&C
cos2qt
I
'L X
Figure 1. Two degrees of freedom single mass system with simply parametric excitation in the presence of a force field parallel to the y-direction.
y 2 0,and
E is a small positive parameter. Apart from linear damping we have assumed the presence of progressive damping to ensure a limited vibration amplitude, even at parametric resonance. In section 1 we perform a scaling and first order averaging to the equations of motion. This leads to explicit results on the stability of the trivial solution, see In section 2 we find families of periodic solutions. Their stability is well determined in l . Because of the complexity of the expressions this is quite surprising. In section 3 the stable periodic solution is continued and becomes unstable. This instability is triggered off by a Neimark-Sacker bifurcation of the periodic solution. This results in an attracting torus with fairly large amplitudes. The Neimark-Sacker bifurcation was first pinpointed by using the numerical bifurcation program CONTENT. An unusual feature is that this bifurcation can also be identified and analysed using the normal form method (averaging). In section 4 the resulting torus is then continued using averaging as well as the software package TORCONT. A homoclinic structure is pinpointed with good accuracy by the normal form.
'.
29
1. Scaling and first order averaging We shall now use a normal form method (averaging) which enables us to obtain more detailed information about the solutions and for which precise error estimates are known. As before we consider system (1)in the vicinity of the origin. To make this more explicit we introduce the following scaling. For more details about this typical scaling see I.
x
y = E 112y, a=
=E'/'%, -
60
b = &'I2&,K = E E , 6 = 8, y = 7, and
=&60.
Introducing this scaling into system (1) and omitting the bars yields:
i+
x + (1 + ECCOS 277t)x + ~ 6 0 2+ Eaxy + &(6x2i+ yx3) = 0, q2y EKY + Ebx2 = 0.
ji
(2)
+
The next steps are the usual ones in averaging approximations; see for instance 2 , chapter 11. We introduce the following transformation:
i
+ 4(t)) +$(t))
+ +
x ( t ) = R l ( t )cos(t $(t)) k ( t ) = -Rl(t) sin(t y(t) = R2(t) cos(2t $ ( t ) ) $(t)= -2R2(t) sin@
Averaging the resulting system in the near resonance case (i. e. q = 4 E u , 77 = 1 ~ p yields: )
+
+
Rla = E R ~ ~ { ;sin(24 R ~ ~- $) f sin24 - iRSa - $} cos(24 - $) f cos 24 :-jaya -p} = &{
+
4,
+
R2a
= $ R ~ ~ { - ~ I R Y ~ sin(24 / ( ~ R~ ?I) ~ )- K }
"i'a
= ;{bRYa/(4R2a) COS(2d -
+
(3)
$1 + i(ff - 8P))
2. The periodic solutions Equating the right-hand side of system (3) to zero yields the nontrivial critical points of these equations which correspond with 2~-periodicsolutions of the original system (2). Without loss of generality we assume R2, > 0. For notational simplicity, we introduce the following quantities:
a=
+ 02) + abup
-3 y ( 4 ~ ~
l b l c , / q
and ufi = u - 8p.
'
'=
+
+
6 ( 4 d u;) 2 a b ~ 60 ,z=zp l b l c d q C 1
= ;
I-1
30 The results for the non-trivial critical points corresponding with periodic solutions are summarised as follows:
R;a =
I/
2 4k2+uE R2a
Ibl
sin(24 - 4) =
Rta =
- 2sgn(b)
K
J4.2+.:.,
- (apz
+
sin24 = PR2a 22
+ 4azJ + da2 + P 2 a2 + P2
-
4(za - 2z,P)2
>O
The stability of these periodic solutions in the case of exact resonance as well as in the near resonance case is discussed in l. 3. The Neimark-Sacker bifurcation
We have made use of the software package CONTENT t o pinpoint this bifurcation; see 4 , chapter 10, appendix 3 for more information on the available software packages and where to download them. We continued the stable periodic solution when ab < 0 with b as a control parameter and monitored its multipliers. CONTENT’Sresults are presented below.
3.1. Numerical data generated b y CONTENT The following parameters, with respect to system (2), have been used in all our numerical analysis. E = 0.1, c = 1, a = 0.5/JE, 7 = 1, 60 = 0, K = 1, 6 = 0.4, y = 0.2, and u = 0.8 ( q = 2.02). A Neimark-Sacker bifurcation has been found at the critical value b, = -0.179576/Jz. Note that, because of the scaling introduced in section 1, to obtain the original values of b corresponding t o system (l), we have t o multiply with &. The corresponding multipliers computed by CONTENT, presented in the modulus-argument form, are as follows: p1 = 1, 41 = 0.207607, p2 = 1, 4 2 = -0.207607, p3
= 1, 4 3 = 0, p4 = 0.623983, 4 4 = -0.163293,
p5
= 0.623983, 4 5 = 0.163293, pf3 = 3.4873
46
= 0.
31 We conclude that when the control parameter b goes below the threshold value b, = -O.l79576/fi a Neimark-Sacker bifurcation takes place, see Figure 2. Numerical results shown in Figures 15-17 of Ref. 1 confirm this very clearly. We can see from these figures that as the parameter b decreases, it takes longer for the periodic solution to stabilise. When b drops below b, the periodic solution looses its stability in the bifurcation. Note that system (2) has to be made autonomous in order to able to spot the bifurcation with CONTENT. Its dimension becomes therefore equal to 6.
,
,
,
,
,
,
3
Figure 2. Figure generated by CONTENT. The cycles (prior t o the bifurcation) are projected on the (x/&, y/&) plane. The outer cycle with dots is the one that bifurcates. NS stands for Neimark-Sacker bifurcation.
3.2. Averaging method results
Remarkably enough, one can also track down the Neimark-Sacker bifurcation by looking for a Hopf bifurcation of the nontrivial critical points of the averaged system (3). We have done this in the case So = 0 and compared the averaged results with the more accurate data obtained by CONTENT. The averaged system has a nontrivial critical point which undergoes a Hopf bifurcation at the critical value b, 2i -0.171/+. This is a rather good first order approximation of the more accurate value b, = -0.179576/& computed by CONTENT. The relative error in b is about 5%. The precision will improve if we take E smaller.
32 4. Continuation of the torus Continuation of tori is in general not an easy task to perform from the numerical point of view. However, as we know, the torus in the original system corresponds with a periodic solution in the averaged system. One can now easily continue the cycle in the normal form and try t o predict the behaviour of the torus in the original system. However, from the theorems we know that this is possible if the torus is normally on averaging, see hyperbolic. If this is not the case, there is no guarantee the integral manifold in the averaged system will correspond with one in the original system. As in our case the periodic solution is hyperbolic and we have parallel flow, we can proceed with the continuation without complications. 516,
4.1. The averaged system To avoid hitting the singularity at the origin, we introduce the following transformation and average the resulting system keeping the parameters as in section 3.
i
z ( t )= q ( t ) c o s t + z 2 ( t ) s i n t y(t) = yl(t)cos2t
k ( t ) = -z1(t)sint+z2(t)cost
+ +~2(t)sin2t $(t)= -2yl(t)sin2t
+y2(t)cos2t
A homoclinic orbit connecting the origin to itself has been detected by CONTENT at b N -0.1864/&. See Figure 3 below.
:I -0 2
-0 3
-0 4 -0 5
I
Y
-08
Figure 3.
-06
-04
02
0
, 02
04
06
08
Towards a homoclinic orbit at b = -0.1864/fi
XI
33 4.2. The original system
For the continuation of the torus in the original system we use the experimental software package TORCONT by Frank Schilder7. The torus grows towards a homoclinic structure a t b e -0.1965/&. See Figure 4 below.
Figure 4. 3-D plot of the torus in the (z,k,y)space at b = -0.1965/& has completely closed around the origin yielding a homoclinic structure.
. The torus
The relative error is here again about 5%, which is satisfactory. The torus looses gradually its smoothness and then disappears yielding this homoclinic structure. This phenomenon of loss of smoothness of tori is quite interesting as it leads to the destruction of the torus which gives rise to interesting dynamics. See for references '. 5 . Conclusions
(1) In most cases Neimark-Sacker bifurcations are studied by numerical means. Interestingly, we can also analyse this bifurcation in our problem by using the normal form method of averaging. The results are in good agreement. (2) The normal form method of averaging is a powerful tool which can easily be used for the continuation of tori without use of experimental software. Numerical results show here good agreement as well.
34 (3) Period-doubling as well as loss of differentiability of the torus (which we did not discuss in this paper) are well detected by this normal form.
Acknowledgment The author would like to thank F. Verhulst for his remarks and contribution.
References 1. T . Bakri, R. Nabergoj, A. Tondl, F. Verhulst, Parametric Excitation in Nonlinear Dynamics, International Journal of Non-Linear Mechanics 39 (2004) 311-329. 2. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer, Berlin, Heidelberg, 1996. 3. The program CONTENT is available via ftp from ftp.cwi.nl in the directory /pub/CONTENT. 4. Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd Edition, Springer, New York, 1995. 5. J.K. Hale (1969), Ordinary Dzfierential Equations, Chapter 7. 6. F. Verhulst, Invariant Manifolds in Dissipative Dynamical Systems, SPT2004Symmetry and Perturbation Theory, Acta Applicandae Mathematicae. 7. F. Schilder’s homepage: http://eis.bris.ac.uk/.-enxfs/
COLLISIONLESS ACTION-MINIMIZING TRAJECTORIES FOR THE EQUIVARIANT 3-BODY PROBLEM IN lR2
V. BARUTELLO*
Dipartimento d i Matematica e Applicazioni, Uniuersitci d i Milano Bicocca, via R. Cozzi, 53 20125 Milano, ITALY uiuinaQmatapp.unimib. it
We consider periodic and quasi-periodic solutions of the G-equivariant 3-body problem in R2 with a-homogeneous potential, a > 0, from the point of view of equivariant calculus of variations. We show that local symmetric minimizers for the Lagrangian action are always collisionless, without any assumption on the group G other than the fact that collisions are not forced by the group itself.
1. Introduction In the past few years some variational methods have been exploited in the search of new periodic solutions as symmetric minimizers for the n-body problem. The major problems in the variational approach are the following: first the minimimum has to be achieved and this requires a condition of coercivity of the action functional on the space of symmetric loops. Second, one has t o prove that the minimizer is collisionless. As we will see in Section 3, coercivity depends on the group G acting on the loops space and possibly on the angular velocity of the rotating frame. Concerning collisions, we obviously exclude the case when collisions are forced by symmetries. A group will be termed bound to collisions if every equivariant loop has a collision. To avoid collisions, in [3] the authors proposed a class of symmetry groups with the property that all local minimizers are collisionless (groups with the rotating circle p r o p e r t y ) , for the general a-homogeneous n-body problem in dimension d 2 2. In spite of its *This work is partially supported by M.I.U.R. project "Metodi Variazionali ed Equazioni Differenziali Nonlinear?'.
35
36 generality, the main theorem of [3] cannot be applied to some relevant symmetry groups for the planar 3-body problem, such as the symmetry group of the Chenciner-Montgomery eight-shape orbit (see [2]). In Sections 4 and 5 , we show that local minimizer for the planar 3-body problem are collisionless for all classes of symmetry groups and we prove the following
Theorem 1.1. Let G a symmetrg group of the Lagrangian in the 3-body problem (in a rotating frame or not). If G is not bound t o collisions, then any local minimizer is collisionless. We would like to enphsize that, in the proof of Theorem 1.1,we do not need action estimates on colliding trajectories, indeed our approach is purely local: we shall exhibit local variations around parabolic ejection-collision solutions (which, as shown in [3], are the blow-ups of possible colliding minimizers) that make the action functional decreases. Moreover, Theorem 1.1 holds for homogeneous potentials of degree -a for every a > 0. For the complete theory of the planar equivariant 3-body problem and a detailed proof of Theorem 1.1,we refer to [l].
2. Setting and notations Let E = R2 E C the 2-dimensional Euclidean space and 0 E R2 its origin. Let m l , m2, m3 be positive real numbers and X the configuration space of three point particles with masses m i respectively with center of mass in 0. Let a be a given positive real number. We work in a uniform rotating plane with angular velocity w and we consider the potential function (opposite to the potential energy) and the kinetic energy defined respectively on X and on the tangent bundle of X as
where J is the complex unit. We suppose that the origin (which coincide with the center of mass) is the point of the plane fixed by the rotation. The Lagrangian is
L,(z,?)
+U(2).
= K,(Z,?)
Let T c R2 denote a circle of length T = I T1 and A = H’(T, X ) the Sobolev space of T-periodic L2 loops T + X with L2 derivative.
37 The action functional is the positive-defined function A, : A defined by
-+
IR U co
JT
We say that x E A has a collision at the collision time f E T if there exist i # j such that z,(q = z3(E).The action functional & is of class C1 on the subspace of collision-free loops of A; hence collisionless critical points of A, in A are T-periodic C2-solutions of the associated Euler-Lagrange equations mixi = 2JwXi
au + w2xi + axi
i = 1,2,3.
3. Symmetries and coercivity
In this section we breafly explain how the action of a finite group G on the loops space A is defined, recalling that a group G is said to act on a space X if there exists a map G x X -+ X, (9, z) H gx, such that (g1,922) H ( 9 2 . g I ) z , where . is the internal operation in G. As described widely in [1,3],we consider the orthogonal group representations T , p : G -+ O(2) and the group homomorphism c : G -+ &. By T and p, G acts on the time circle T c IR2 and on the Euclidean space E , respectively; by c it acts on the index set n = { 1,2,3} satisfying the property Vg E G : ( a ( g ) ( i ) = j
+ mi = mj)
Given p and o with property (l),G acts orthogonally on the configuration space X by (gx)i = p(g)zo(g-l)i = gxg-li, for every g E G, where in the last espression we understand the action of G as a space linear transformation or a permutation on the index set. Using the representation T , we are now able to define the action of G on the set of loops A as follows Vg E G, V t E T,Vx E X ,
(9.x ) ( t )= gx(g-'t).
Given a group G acting on the loops space A, we define the closed linear subspace A 3 AG := {z E A : ( 9 . z ) ( t )= z(t),W E T , g E G } , as the set of G-equivariant loops. Thanks to the Palais Principle of symmetric criticality, the analysis of critic1 points for the action functional can be restricted to sets of equivariant loops and the existence of collisionless critical points will be guaranted by some properties on the group G . Obviously, since we are looking for collisionless minimizers of the action functional &,, we are
38 not interested in those groups G such that every equivariant path has at least a collision, we will call bound t o collisions a group of this kind. A: is We will term A: the restriction of the action functional &I*G; termed coercive on AG if A:(z) diverges to infinity as the H1-norm of x goes to infinity in AG. Coercivity is the fundamental property to guarantee the existence of minimizers on the set AG for the functional d,".Remark that when w $ Z,for every finite group G acting on the loop space the action functional A: is coercive on AG; moreover, when w = 0, AGis coercive on AG if and only if X G = {z E X : ga: = x,V g E G } = { ( 0 ,0, 0)) (see Proposition (4.1) in [3]).
4. Symmetry constraints for collision trajectories As we have already explained in Section 3, the existence of minimizers for the action functional A: is guaranted by a condition on the action of the finite group G on the configuration space. In this section we deal with the much more complicate problem of avoiding collisions for a local minimizer of the action functional A:. Fortunatly, we can deeply simplify this problem using some well known results. At first, we recall that in [4] W. Gordon remarked that when Q 2 2, local minimizers for the action functional are collisionless, since the action level of colliding trajectories is infinite. Now on, we will then suppose Q E (0,2). In 4, D. Ferrario and S. Terracini, inspired by the Marchal's Principle exposed in 2 , 6 , propose a property of the action of the group G (the rotating circle property) to avoid collisions in local minimizers for the restricted action functional A,". Concerning the three body problem in the plane, a finite group G acts on n = { 1,2,3} (resp. k,Ikl = 2) and E with the rotating circle property (r.c.p.) if for every g E G the determinant det(p(g)) = 1 and if there exist at least two different indices i l , i 2 E n such that V g E G, ( g i l = il V gig = 2 2 ) 3 p ( g ) = 1 (resp. there exists at least one index i l E k such that V g E G : gil = il 3 p(g) = 1). We now start the proof of Theorem 1.1;let z E AG be a local minimizers for A:. Suppose that z has a n isolated collision at t = 0 (Section 5 in [3] ensures that we are allowed to suppose a collision being isolated). Theorem (10.7) in [3] suggest the study of ker 7: for the planar 3-body problem, we proved in Proposition (5.5) of [l]that when kerT is not trivial, local minimizers are collisionless. We are then left with the situation of trivial ker 7; the T-isotropy subgroup, Go := { g E G : T(g)(O) = 0}, correspondent to the collision instant is then maximal. Theorem (10.10) in [3] yields to
39 the study of Go which is a group of order two (since G T(G) c O(2)) generated by go and ensures that if a ( g 0 ) acts trivially on the index set or if the group Go has the r.c.p., then no collision can occur at t = 0. Without loss of generality we can suppose that a ( g 0 ) = (1,2). As to p ( g o ) , three cases are possible: either p(g0) is the identity, the antipodal map or a reflection with respect to a line I . In the first two cases Go has the r.c.p., so we are left with the latter case. In the sequel we will refer to G as a group containing Go = ( g o ) , where pl := p(g0) is a reflection with respect to a line 1, a ( g 0 ) = (1,2) and .r(go) is a reflection with respect to t = 0. The G-equivariant loops satisfy then a t least the symmetry constraint
that we name the Isosceles S y m m e t r y . In particular a t t = 0, the third body lies on 1 and the positions of the first and the second are symmetric with respect to 1. 5. Collisions with the Isosceles Symmetry
In this section we conclude the proof of Theorem 1.1, showing that a local minimizer x E AG for dz,satisfying the Isosceles Symmetry in (2), can not have a collision at t = 0. By the sake of contradiction, suppose that z, satisfing (2), has a collision at t = 0 and let k c n = { 1,2,3} be the colliding cluster; the symmetry constraint (2), of course, after relabeling the indices, implies that either k={1,2}ork=n. A parabolic collision trajectory for the cluster k C_ n is the path qi(t) = lt12/(2f")(i, i E k, t E R
where ( = ((i)iEk is a central configuration with k bodies (i.e. a critical r n i 1 ~ i 1=~ 1). An escaping point for the potential function restricted to path for the cluster k C n is a path of type y = q f ' p , where q is a parabolic collision trajectory for the cluster k and 'p E I$;@); we say that an escaping path is go-equivariant if y(g0t) = g o y ( t ) , for every t E R. We term Lw,k the partial Lagrangian function; when x = ( z , ) i E k , L w , k ( x ) is the Lagrangian restricted on the bodies of the cluster k and we say that a parabolic collision trajectory, q = ( q i ) i E k , is a go-equivariant minimizing parabolic collision trajectory if for every go-equivariant escaping
xi
40
path y = q
+ 'p
L
+W
A& :=
[ L w , k ( q + 'p) - L w , k ( q ) ] d t 2 0.
Let q ( t ) = z ( t ) - z o ( t ) , where zo(t) is the center of mass of the bodies in k, and let qx be defined by q y t ) = P/(Z+a)q(Xt).
As shown in Section 7 of [3], there are sequences A, 0 such that qXn(t) converges to a go-equivariant parabolic minimizing collision trajectory. Let 6 E (R2)k, k = 2,3, and T > 0 a real number. The standard variation associated to 6 and T is defined as --f
ws(t)
=
{:
(T -
if 0 5 Jtl5 T - 161 if T - 161 5 It1 5 T if It[2 T.
t)h
Remark that when the vector 6 is fixed by go, 6 = 906, then d ( t ) is a go-equivariant escaping path. To conclude the proof of Theorem 1.1, we show that, if q is a minimizing parabolic collision trajectory, then there always exists a go-equivariant standard variation such that AAw < 0 and q w6 does not have a collision a t t = 0. This would imply the existence of a go-equivariant collisionless trajectory whose action level lower the one of the parabolic collision trajectory. Consider the function
+
where
E , 6 E R2. Let t9 E
[ 0 , 2 ~ such ] that cost9 =
I, 6 then )
(It1
hence the function S ( [ , b ) depends only on the mutual positions of the vectors E and 6. It is easy to verify (see (6.12) in [l])that the function +W 1 1 - T d t , defined continously Qa(4 = 4 2 t"+z (t* - 2c0s19t* + 1) on ( O , ~ T ) , has a symmetric plot with respect to the line 19 = T , decreses on [O,T) and, when a! < 1, Q m ( O ) = Qa(27r) is finite, while, when a! 2 1, Qp,(19) = +cm. limo+o+ Qa(t9) =
41
the basic tool to estimate the variation of &;in fact i = 1,.. . , k is a parabolic collision trajectory if q = { q } i = {t2/(2+a)1i}, and v6 a go-equivariant standard variation, then for S -+ 0 The function
@a is
We start studying binary collisions; since there exists just one cenTo make the actral configuration with two bodies, thus t 1 = -&. tion functional decreses, we consider the standard variation associated to the vector 6 = ( 6 1 , - - 6 1 ) , such that 61 is orthogonal to the line 1 and 19 = arccos -61) lies in the interval [;,TI (see Figure 1 (a)).
(H,
Figure 1. Collisions from collinear central configurations.
On the other hand a triple collision can take place in two different ways: from collinear configurations or from the Lagrange configuration. Since m l = m2, we are left with three possible situations: two collinear configurations (the second or the third body in the middle) and a Lagrange coniguration. When a collision occur from a central configuration, we construct a standard variation similarly as we made for the binary collision and we refer to Figure 1 (b)-(c). Consider now the 3 particles at the vertices of a regular triangle moving on a parabolic collision trajectory that makes them collide at t = 0 in their center of mass. Since m l = m2, the center of mass lies on the line h represented in Figure 2 . Our aim is to show that there always exists a vector 6 E S', such that, when replacing 13 with & 6, the interaction potential decreases (the sum of the two variations is negative even thought they are not necessarily both negative) and therefore A& is negative. In the following we refer to Figure 2. and we call y the angle in [0,7r/2]with edges the lines h and 1. To conclude we prove the inequality
+
42
Figure 2.
Triple collisions from Lagrange configuration.
The monotonicity of the function implies that it is sufficient to prove ( 4 ) Using the /3 function, we can write (see (6.17) in [l]) when y =
5.
%(f) +aa(;) 1 +-
1
a+2 a t 2
=a - / -3 (2T , T ) +
a + 4 k /3(,+k,= a+2 ~ ~ + 4 k - 2
4
+k
)
.
(5)
k=l
To conclude we use the property of the function
/3
and the inequality (see (6.27) in [l])
< 0, ifx E (1/2,1). k=l
Inequality ( 4 ) ,with y = $, follows replacing (6) and (7) in ( 5 )
References 1. V. Barutello, D. L. Ferrario and S. Terracini, preprint. 2. A. Chenciner, ICM August 2002, Peking. 3. A. Chenciner and R. Montgomery, Ann. of Math. 152 881 (2000). 4. D. L. Ferrario and S. Terracini, Invent. Math. 155 305 (2004). 5. W. Gordon, Trans. Amer. Math. SOC.204 113 (1975). 6. C. Marchal, Celes. Mech. Dyn. Astr. 83 325 (2002).
(7)
THE LAGRANGIAN AND HAMILTONIAN FORMULATIONS FOR A SPECIAL CLASS OF NON-CONSERVATIVE SYSTEMS*
SERGIO BENENTI. Department of Mathematics, University of Turin, via Carlo Albert0 10, 10123 Torino - Italy sergio.
[email protected] This is an outline of the major results contained in a n extensive tutorial paper presented at SPT-2004 and dedicated to a special kind of symmetric two-tensors which appear, in the recent and in the old literature, in connection with special kinds of mechanical systems and with the theory of the separation of variables in the Hamilton-Jacobi equation.
1. Posing a question
A holonomic system is a mechanical system whose configurations form a set Q endowed with a differentiable manifold structure with finite dimension n (the number of degrees of freedom).a This manifold is in turn endowed with a positive-definite metric tensor g = ( g i j ) determined by the expression of the kinetic energy K = gij qiqj w.r.to any natural coordinate system (92, qi) on the tangent bundle TQ. The active forces are then represented by a vector field F = (Fi) on Q or by a one-form Fi = gij Fj.b For such a system, represented by the triple (Qnlg,F),the dynamics is completely determined by the second-order Lagrange equations
which assume the Newtonian form d2qi
--g+I?'
.
hj
dqh d q j dt dt
a
--=Fi
a=F,
* This work is supported by the Dept. of Mathematics, University of Turin and by INDAM-GNFM. a Here we consider only the case of time-independent constraints. The forces are here assumed to be time-independent and velocity-independent.
43
44
being . dLqi . dqh dqj a"-++' -dt2 hi d t d t the absolute acceleration of the dynamical system and I'ij the Christoffel symbols i.e., the coefficients of the Levi-Civita connection associated with the metric tensor. This last second-order system is equivalent t o the first-order dynamical system
.
.
on the tangent bundle T Q , with coordinates (q,g) = (q', w 2 ) = (qz,#). If the force F is potential (or "conservative") i.e., it is the gradient of a potential energy V (a real-valued smooth function on Q ) ,
F i = - g i j a .3 v ,
F = -VV
then we have a Lagrangian system. Eqs. (1) assume the well-known form
where L = $gij(i.iqj - V is the Lagrangian function on T Q . In this case we can apply alternative methods of integration by passing to the Hamiltonian function on the cotangent bundle T * Q , with coordiV . This function gives rise to a nates ( q , p ) = ( q i , p j ) , H = i g i j p i p j HamilFoiian system (i.e., to first-order Hamilton equations) and to a Hamilton-Jacobi equation. Let us pose the question: m a y the Lagrangian and t h e Hamiltonian methods be extended t o systems with non-potential forces?
+
2. Equivalent systems
For posing this question in a more precise way we make use of the notion of equivalent systems introduced by Painlev6 (1894) and Levi-Civita (1896). Definition 2.1. Let ( Q ,g, F) and ( Q ,g, F) be two holonomic systems with the same configuration manifold Q. Let
45 be the corresponding dynamical systems of T Q with coordinates (qi,w i = $). They are said to be equivalent or correspondent, if there exists a function of f : T Q + R, such that, for any solution
of the first system (3), by a change of the time-parameter of the kind
-d t_dt
1
f
(P"t),
+w)
(4)
we get a solution of the second system,
This means that the trajectories on the configuration manifold Q of the two systems coincide, up to a change of the time-parameters given by (4) (in other words, the trajectories are the same, but covered with different velocities) .' A special but fundamental case is that with F = F = 0. It concerns with geodesics:
Definition 2.2. Two metric tensors g and g on the same manifold Q are said to be equivalent if they have the same unparametrized geodesics. After this definition our question can be reformulated as follows: which non-potential systems are equivalent t o Lagrangian systems? However, as we shall see in the next section, at the state-of-the-art we are able to answer this question only by adding a further condition on the definition of equivalence. 3. The equivalence theorem of Levi-Civita
Levi-Civita was able to prove
(p. 272)
Theorem 3.1. Two systems (g,F) and (g,F) are equivalent zf and only i f there exist functions p and c i j , depending o n the coordinates only, such In the definition of Levi-Civita the function f is considered depending also on t. However, he proves that in fact this function is independent of time, under the assumption that the forces depend only on the coordinates (p. 269).
46
that the following equations are satisfied p2 Fi = F“, wh (dailogp
+ cij
id) + (I?$
Remark 3.1. In the special case
p2Fi = p ,
cij
Fh. 23 =
-
r$ + ~h
c i j ) vivj = 0 ,
(5)
= 0 conditions (5) reduce to
rh. 23 - 1 2.
+bjhPi).
This last equation shows that the two metrics are equivalent and suggests the notion of geodesically equivalent dynamical systems: they are equivalent holonomic systems whose underlying Riemannian metrics are also equivalent.
4. Main theorems Theorem 4.1. A dynamical system ( Q ,g, F) is geodesically equivalent t o a Lagrangian s y s t e m i.e., t o a system ( Q , g , F ) where g i s a n equivalent metric and F = - V V , i f and only i f the fundamental metric g admits a non-singular special conformal Killinq tensor J such that
F = -A- ‘VV, A = c o f J .
(6)
Definition 4.1. A special conformal Killing tensor (SCKT) is a symmetric two-tensor Jijsatisfying the equation VhJij =
i
(“a
gjh
+ “j gih),
(7)
where ai are the components of a suitable one-form.d We denote by a boldface letter the corresponding (1,l)-tensor, J = ( J j ) , Jj = gihJhj. Such a tensor is strictly connected with other special symmetric twotensors, here denoted by A, B and L and called for simplicity A-tensor, B-tensor and L-tensor (then a SCKT will be also called J-tensor). The first two-tensors are defined by the differential equations
4
VhAij = Ph Aij - ( P j Ahi f Pi Ahj) > VhBij = - ( P j Bhi Pi B h j ) ,
4
+
(8)
This kind of tensor has been introduced and studied by Crampin and Sarlet (20002003).
47 where pi are the components of a suitable one-form. The definition of L-tensor will be given in 55 below. All these tensors are related by several equations, used for proving Theorem 4.1 and other theorems. For instance,
A = cof J = p J-', J = PA-', B = J-' = L A , c o f B =z A-', P p = det J.
(9)
Hence, to look for a J-tensor with u , # 0 is the same as to look for a non-singular A-tensor or B-tensor. It is rather surprising that systems satisfying the condition (6) of Theorem 4.1 have been recently introduce in the literature: they have been called cofactor systems.e Hence, we can restate Theorem 4.1 as follows: A dynamical system ( Q ,g,F) is geodesically equivalent t o a Lagrangian system i e . , i f and only i f at is a cofactor system. As a consequence of Theorem 4.1 it is clear that when we have a cofactor system then we can apply to the equivalent Lagrangian system the Hamiltonian methods, including the integration (possibly by separation of variables) of the Hamilton-Jacobi equation. Then we describe the motions of the original system simply by changing the time-parameter according to the formula
which follows from (4) and ( 5 ) . The components of the new metric tensor g, with which we can write the Lagrangian and Hamiltonian function of the equivalent system, =
1 -. 2
Qa,
,
-a-j 2,
V
-
v,
&,
=
jj =
12 -g2 3
pap,
+v,
dqa u z = -.
dt
(11)
are given by
Ba3,
9'3
=p
5'3
(12)
Note that the operation of raising and lowering indices is always performed by the basic metric g. Note that the equivalent metric (12) may not be positive-definite. So, we are led to consider pseudo-Riemannian manifolds also in connection with problems of classical mechanics. A test for finding a cofactor-system-structure is given by the following Rauch-Wojciechowski, Marciniak, Lundmark, 1999, et a1
48
Theorem 4.2. A dynamical system (Q,g, F) is a cofactor system i f and only if g admits a non-singular A-tensor A = (Aij) such that
d(Aij F j dqi) = 0.
(13)
This theorem has a local character. For a global meaning, the vector field AF = (AjFj) must be a gradient. Hence, Eq. (13) must be replaced bY
A F = -VV.
(14)
This formula has the advantage of giving the potential V of the equivalent system. A remarkable fact concerning the cofactor systems (or the bi-cofactor systems, see below) is that, under certain condition on the eigenvalues of the special tensors involved, the Hamilton system defined by (11) is an orthogonal separable system of special kind, called L-system.f To see this we need the notion of L-tensor. 5 . L-tensors, L-sequences, L-systems
Definition 5.1. A L-tensor L on a Riemannian (or pseudo-Riemannian) manifold is a torsionless conformal Killing two-tensor with pointwise simple and real eigenvalues. The following theorem shows the interest of this definition.
Theorem 5.1. Let L = ( L i ) be a symmetric two-tensor. Then the tensors (K,) = (KO,K1,. . . ,Kn-l) defined by the L-sequence KO= I,
K,
=
t r (K,-IL) I - K,-1 L,
are n independent Killing tensors with common normal and only i f L is a L-tensor.
a g
>1
(15)
eigenvectors i f
Indeed, n independent Killing tensors with common normal eigenvectors define a Killing-Stackel space K, containing the metric tensor and whose elements commute in the Schouten brackets of symmetric tensors; in other words, they give rise to a n-space of geodesic quadratic first integrals in involution. The geodesic flow is then completely integrable. Furthermore, Orthogonal separable systems are also called Stiickel systems. This means that each eigenvector field is orthogonal to a family of surfaces (a foliation) of submanifolds of codimension 1. All these foliations form an orthogonal web. g
49
the orthogonal web determined by the common eigenvectors is a separable web, in the sense that any orthogonal coordinate system q = (qi) adapted to this web is separable (it separates, additively, thegeodesic Hamilton-Jacobi equation): these coordinates are such that the web is locally represented by equations qz = constant or equivalently, the vector fields 8, = d/dqi are common eigenvectors (the one-forms dqi are common eigenforms). Let us call L-system any Stackel system of this kind. It is remarkable the fact that for a L-system all the Killing tensors underlying the separation, and forming the Killing-Stackel space K ,are all constructed algebraic way by means of the L-sequence (15). The general theory of the orthogonal separable systems shows that a potential V is separable w.r.to a Killing-Stackel space K if and only if for any arbitrary element K E K with simple eigenvalues the one-form K dV is closed (may be exact),
d ( K d V )= 0,
K d V = dU.
If this condition is satisfied, then by taking a basis (K,) of functions Va such that K, dV = dV,,’ then the functions
H
L
a
LKij
-2
a
PiPj
K
and the
+Va
form a system of independent first integrals in involution. All the results above applied to a L-system, show that, for instance, K1 in the L-sequence is a characteristic Killing tensor, so that a potential V is separable in a L-system if and only if j
d ((tr L) dV - L dV) = 0.
(16)
6. Cofactor and bi-cofactor systems are L-systems
The sentence which is taken as a title of this last section is true under certain conditions.
Theorem 6.1. Let (Q,g,F) be a cofactor system, whose J-tensor J has pointwise simple eigenvalues. T h e n the geodesically equivalent Hamiltonian system is a L-system generated b y the L-tensor B, the B-tensor associated It can be proved that such a tensor, called characteristic tensor, always exists. Indeed, it can be proved that if K d V is closed or exact, the same happens for all elements of K . j For Q = R” and in Cartesian coordinates, this is known as the Bertrund-Durboux equation.
50 with J, i f and only i f F = - VW (i.e., the cofactor system is itself a Lagrangian system).
Definition 6.1. A bi-cofactor system (or cofactor-pair system) is a holonomic system (Q, g , F) which is a cofactor-system in two distinct ways:
F = -A-'VV
= -A-'Vv,
-
(17) -.
where A = cof J and = cof 5, being J = ( J j ) and J = ( J j ) two nonsingular (and non-trivial) J-tensors w.r.to the metric g .
Theorem 6.2. If a bi-cofactor system is such that the tensor J = 3 J-' has pointwise real and sample eigenvalues, then the equivalent Hamiltonian system ( Q ,g, H ) , where g is the equivalent metric determined by J , is a L-system generated b y the L-tensor J . References 1. S. Benenti, Special Symmetric two tensors, Equivalent Dynamical Systems, Cofactor and Bi-cofactor systems, Tutorial Papers SPT-2004, Acta Applicandae Mathematicae, to appear. 2. Levi-Civita, Sulle trasformazioni delle equazioni dinamiche, Ann. di Matem. 24 (1896).
SHADOWING CHAINS OF COLLISION ORBITS FOR THE ELLIPTIC 3-BODY PROBLEM*
SERGEY BOLOTIN Department of Mathematics University of Wisconsin Van Vleck Hall, Madison 53706, USA
We consider a Hamiltonian system modelling the plane restricted elliptic 3-body problem with one of the masses small and prove the existence of periodic and chaotic almost collision orbits. Periodic orbits of this type were first studied by Poincar6 who named them second species solutions. The proofs are based on variational methods.
1. Introduction
Let D be an open set in R2 containing 0 and let T = R/2nZ. Consider a Hamiltonian system ( H E )with phase space P = ( D \ (0)) x R2 x T and time periodic Hamiltonian
depending on a small parameter E . Let u,,V, E C 3 ( D x T). For E > 0 the potential has a Newtonian singularity at q = 0 which disappears for E = 0. System ( H E )is a singular perturbation of system ( H o ) . An example is:
Elliptic restricted 3-body problem. Suppose Sun of mass 1, Jupiter of mass E 0 such that the pair C i - l ( t i ) , & ( t i ) belongs to the set
53 Theorem 1.1. For any nondegenerate periodic collision chain c of system (Ho), there exists EO > 0 such that for any E E ( O ’ E O ) , there exists a unique 2nm-periodic orbit yE : E% -, D \ (0) of system ( H E )which is O ( E ) shadowing c. More precisely, IyE(t)- ci(t)l 5 for ti-1 5 t I ti. The proof is based on variational methods and will be published elsewhere. An analog of Theorem 1.1 holds for collision chains which correspond to topologically nondegenerate critical points, but then the shadowing orbit will be non-unique in general. In a subsequent paper we will check the conditions of Theorem 1.1for the elliptic non-circular 3-body problem. A result analogous to Theorem 1.1 holds for autonomous systems.’ Then nondegeneracy of a collision chain is defined for fixed Hamiltonian h in terms of the Maupertuis-Jacobi action functional. The corresponding shadowing orbits all lie in the same level H E = h , and the period T depends on h. Our Theorem 1.1 gives shadowing orbits with given period 2 n m and doesn’t apply to autonomous systems: the nondegeneracy condition fails. The circular restricted 3-body problem with eccentricity E = 0 and small E > 0 has autonomous Hamiltonian HE of the form (1) in the rotating coordinate frame. In fact H E = J - the Jacobi integral. One can check the nondegeneracy conditions,’ and obtain for each h E (-312, &) an infinite number of orbits periodic with respect to the rotating coordinate frame and having the Jacobi integral J = h. See also earlier publication^.^^^^^ Orbits of the circular problem obtained in these papers are periodic with respect to the fixed coordinate frame when Tl27r is rational. Then the periodic orbits fill a torus 72c P . By Poincark’s Theorem,’ for 0 < E 0 provided that the changing direction condition holds. The conditions of Theorems 4.1-4.2 hold for the elliptic almost circular 3-body problem with small eccentricity E . For this problem, hi corresponds to the value of the Jacobi integral a t the collision at t = ti. In a subsequent paper we will show that every h E (-3/2,&) is contained in an infinite number of appropriate intervals J k l such that the conditions of Theorem 4.1 hold in Jkl. In particular, we get lots of periodic and chaotic shadowing orbits for 0 < E 0 there exists an orbit ro of X , that approaches r as E J. 0 , with respect to the Hausdorff distance. Proof. Consider the auxiliary vector field X:(x, y , ~ )= (f (z, y,e), T = (0,yo) on r n F i x ( R ) . Such a point exists because r connects R-symmetric invariant manifolds on S . Let 1 = { ( x , Y , E ) : x = 0 , y = yo, 0 I E < E O } , where €0 is some sufficiently small positive number. Thus 1 is a segment of a curve transverse to Fax ( R )at r . Denote by Cl the saturate of 1, that is the closure of the union of segments of orbits of X : through the points of 1 and taken between the first intersection of this curve with Fix (R) in negative time (Cy) and in positive time (CT). The orbits on Ci are the orbits of X,. Ci crosses F i x (R) transversally because the singular orbit r does. Finally, the reversibility of E g ( z , y , E ) , 0 ) on R3. Take
70
Y
X
Figure 1. Singular Orbit J? and Center Manifold Cl
X, implies that C r nFix ( R )= C: n Fix (R) . To complete the proof it is enough to choose r Eto be any orbit contained on Ci. 0 We observe that if p and q, as in the QG-condition, are hyperbolic singular points of the slow system then the family obtained in Theorem A is composed of heteroclinic cycles. Moreover if (0,0,O) (0,0,0 ) < 0, r is a singular orbit of Xo passing through (0,O) and connecting R-symmetric points on the slow manifold then there exists a sequence of periodic orbits Fa, of X,, that approaches I? for E j, 0. In fact, if X, is R-reversible then we have
.%
f (Z,Y,E)
= 'p ( x 2 , Y , E )
9(Z,Y,E)
= x1C,(x2,Y,E)
for some smooth functions 'p and 1C, on R2. The singular points for are given by the equations:
4-J (&Y,E)
E
>0
=0
'p ( Z 2 , Y , E ) = 0
By the implicit function theorem there is a smooth function y = Y ( E ) pro(O,O, 0) # 0. Moreover vided that
2
71
Finally, the reversibility condition ensures that the singularity ( 0 ,y ( E ) ,E ) is a center for X,. 3. Fold, Transcritical and Pitchfork Singularities
Consider the singularly perturbed ordinary differential equations given by (1). Assume that (0,O) E S is a non-normally hyperbolic singular point. a generic classification of slow manifolds in 4,5 a generic classification is given by the exhibition of three normal forms:
3- Pitchfork Singularity X, :
x’ = xy - x 3
g(0,O) # 0. Yl = EL7 (x,Y ) These singularities have been analyzed by several authors in different contexts. We mention the works If SO c S is a normally hyperbolic submanifold, then it persists for sufficiently small E > 0 as a nearby locally invariant slow manifold S,, for E 1 0. In 4,5, the case where SOcontains non normally hyperbolic points was studied. The main question is to understand the behavior of S, at these points. This analysis is made by the inspection of the transition maps defined on transversal sections to slow manifolds at points where the normal hyperbolicity occurs. 9
495.
Theorem 3.1. (see
415)
a) A s s u m e that (0,O) is a fold singularity and g(0,O) < 0 . Let C,, i = 1 , 2 , be transversal sections t o S at pi = (x,,y,) , with x1 < 0 and 2 2 > 0. Consider n : C1 + C2 the transition m a p and K c S a normal hyperbolic part of the slow manifold with pl E K. T h e n K,, invariant manifold of X, converging t o K ,crosses transversally C2 f o r E 1 0. Moreover the map n is a contraction with constant 0 (e-’) f o r some constant c > 0. (see illustration o n figure %(a)). b) A s s u m e that (0,O) i s a transcritical singularity and g (0,O) > 0. T h e n X, i s topologically equivalent t o X(x,y ) = (-y2 x2 X E , E ) . Moreover let C1 be transversal t o y = x at p l = (51,X I ), with x1 < 0 and C2 transversal t o y = -x at p2 = (xz,-x2) i f X > 1, 2 2 < 0 and t o y = 0 at pi = (x;,0 ) if X < 1 and x; > 0. Consider n : C1 -+ C2 the transition m a p and K c S a normally hyperbolic part of the slow manifold with pl E K . T h e n as before
+ +
72
K, crosses C2 transversally f o r E
1 0. (see figure
Case I
Figure 2.
2-(b)).
Case h < 1
1
Transition Maps.
We recall that the cases treated are not reversible with respect to
W z ,Y) = Y). Extending Slow Manifolds with Reversibility: (-7
In our approach we treat the case when g (0,O) = 0.
Definition 3.1. (0,O) is a simple reversible singularity if it is either a fold or a transcritical singularity and (0,O) # 0.
2
2
If (0,O) < 0 the dynamics is not interesting (see figure 3-(A)) and its analysis will be omitted. The next result is a immediate consequence of the Theorem A. Theorem B Let X, be given by X,(z, y) = ( f (z, y), E g (z, y ) ) . A s s u m e further that (0,O) is a simple reversible singularity with (0,O) > 0. Let C1 be a transversal section t o S at p = (a,b) with a < 0 , b > 0 and C2 = R(C1). T h e n either the a- limit set of p i s {(O,O)} or the orbit of X , through p , O ( p ) , satisfies O ( p )n C2 = {R ( p ) } . We observe that the pitchfork singularity is a degenerate singularity. Its +E). It is not R-reversible. generic canonical form is X(z, y ) = ( z y - z3, We consider the generic reversible pitchfork singularity. It is represented z4, E X ) , and all possible by the following normal form X ( z , y ) = ( - s 2 y fast and slow dynamics are illustrated in figures 3-(E) and 3-(F). Figure 3 describes the dynamics of some situations that occur for reversible singular perturbation problems. They are:
+
(A) f(z,y ) = -y + x2 and g ( 0 , O ) < 0. (B) f (z, y ) = -y + z2 and A(0,O) > 0. (c)f ( z ,y) = -y2 + z2 an2&(0,0) > 0. F (D) f ( z , y ) = -y2 + z2 and &(O,O) < 0.
73
Figure 3.
Fast and slow dynamics.
In what follows we give an example of a reversible singularly perturbed and vector field XE,xpossessing a 1-parameter family of singular orbits a 2-parameter family of regular orbits rs,xhaving the following properties: (1) If X < 0, then r e , x are periodic solutions and re,x + rx as E -+ 0. (2) If X > 0, then rE,xare homoclinic solutions and rE,x -+ rx as E -+ 0. Consider the following family of time-reversible singularly perturbed systems X ( z ,y) = (-y3 z2 XY,EZ). This family is time-reversible with respect to the involution q5(z,y) = (-z, y). The slow system and the fast system are given, respectively, by
+ +
0 = -y3 y=2
+G
+ xY
X=-y3+52+Xy y = 0.
We show how the dynamics of XE,xvaries in figure 4.
(3)
74
E=O
kO
Figure 4. Bifurcation
References 1. C.A.Buzzi, P.R.Silva and M.A.Teixeira. Singular Perturbation Problems for
2. 3.
4.
5. 6.
7.
Time-Reversible Systems, Technical Report IMECC-UNICAMP 33/04, 2004. To appear in Proc. Amer. Math. SOC. F.Dumortier, R. Roussarie. Canard Cycles and Center Manifolds, Memoirs of the A.M.S, V. 121, 1996. N.Feniche1. Geometric Singular Perturbation Theory for Ordinary Differential Equations, J. D. E 31, 53-98, 1979. M.Krupa and P.Szmolyan. Extending geometric singular perturbation theory to nonhyperbolic points: fold and Canard points in two dimensions, SIAM J. Math. Anal. 33, 2, 286-314, 2001. M.Krupa and P.Szmolyan. Extending slow manifolds near transcritical and pitchfork singularities, Nonlinearity 14, 1473-1491, 2001. J.S.W. Lamb and J.A.G. Roberts. Time-reversal symmetry in dynamical systems: a survey. Phys. D, 112(1-2):1-39, 1998. Time-reversal symmetry in dynamical systems (Coventry, 1996). P.Szmolyan. Transversal Heteroclinic and Homoclinic Orbits in Singular Perturbation Problems , J.D.E 92, 252-281, 1991.
HOMOGRAPHIC THREE-BODY MOTIONS WITH POSITIVE AND NEGATIVE MASSES
M. CELL1 Institut de Me'canique Ce'leste et de Calcul des Ephe'me'rides UMR 8028 d u CNRS, Observatoire de Paris 77, avenue Denfert-Rochereau 75014 PARIS. F R A N C E
[email protected] Laboratoire Analyse, Ge'ome'trie et Applications UMR 7539 du CNRS, Institut Galile'e, Universite' Paris 13 99, avenue Jean-Baptiste Cle'ment 93430 VILLET AN E USE. F R A N C E
[email protected] This note describes some homographic solutions of the N-body problem with positive or negative masses. Some of them are natural generalizations of the solutions of the two-body problem with vanishing total mass. The configurations which generate these motions are described in the case: N = 3. A case of integrability of the collinear three-body problem with vanishing total mass is shown. A similar case of integrability was known for four vortices satisfying Helmholtz's equations [j].
1. The vector of inertia Let us consider N bodies with masses m l , ...' m N , whose sum is M . The mi do not vanish and may be positive or negative. Let us denote by 41, ..., q N the positions of the bodies. The q i ( t ) belong to a euclidean space, whose scalar product is denoted by (.I.). The Newtonian potential is defined by:
Newton's equations are defined by:
75
76
For M
# 0, the center of inertia is defined by:
For M = 0, the vector of inertia is defined by: N
X(ql,...,qN) = x m i ( q i - Q ) ’ i=l
The definitions do not depend on Q. Whereas the motion of the center of inertia is uniform rectilinear for M # 0, taking the sum of Newton’s equations when M = 0 provides the following result. Proposition 1.1. For any solution of Newton’s equations with M = 0 , the vector of inertia X i s an a f i n e function of time.
It is known that for N vortices with vanishing total vorticity which satisfy Helmholtz’s equations ([’]), we can define, in the same way, a vector of vorticity. This vector is a first integral. Let us write: X ( q l ( t ) ,...,qN( t ) )= tP+Q, where P and Q are constants. The previous result and Leibniz’s equality give the following relation:
c
mimj/Iqj(t )qi(t)/I2 =
- - / l X ( q ~ ( t...,qN(t))1t2 )~ = -(ltP+
&/I2
1
I
< 42 < q3. As q3 - q2 < q3 - ql,we have:
1 + 4d2 (q3 -
1 (q2 -
-
1
(43
-
41)2
>
1 (q2 - q1)2
> 0.
So [ # 0. According to Prop. 3.1: X(q1, q2,q3) = 0. Conversely, for a configuration such that X(q1, q2,q3) = 0,we have:
-
m242
-41
m 3 q3 - 41
72(q1,q2,!73)-71(41,q2,q3) - Y2(Ql,q2,Q3) -71(41,q2,43) q2
-
41
q2
-
41
Let us now consider a non collinear complex central configuration. Let us suppose 5 # 0. Then, according to Prop. 3.1, X(ql,q2,q3) = 0. So the
81
configuration is collinear, which is impossible. So E = 0, which entails: O=
1 = (I142
-
-
41113
42 - 41 1142 - 41Il3 1
)
+
43 - 42 1143 - 42\13
+
41 - 43 1141 - 43113 1
1
1143 - 42113 (42-q1)+( 1143 - 42113
-
3 )
1143 - 4111
(43-41)'
Thus the configuration is equilateral. Conversely, we can easily check that the equilateral triangle is a central configuration with E = 0 and that it is not an absolute equilibrium. The next result follows from Props. 5.1 and 4.1.
Proposition 5.2. A three-body motion with M = 0 and X(q1, 42, 43) = 0 at any time is homographic, and there exists E E W\ ( 0 ) and a time-dependant complex number s, which is a solution of: s = Es/1sI3, such that: 4 j ( t ) - 4iW = s(t)(qj(O)- d o ) ) ,
For M # 0, we can have complex central configurations with (' $ R. By taking the cross products between Eqs. (1) and the qj - qi, we obtain the following proposition.
Proposition 5.3. The equilateral triangle is a central configuration f o r any system of masses. For any n o n collinear n o n isoceles three-body configuration, there is exactly one system of masses ( m l ,m2,m3) (up t o homotheties) such that the configuration is complex central with ( $ R. A non collinear and non equilateral isoceles configuration is complex central f o r n o system of masses. 6. A particular case of the three-body problem with M = 0
Proposition 6.1. For a three-body motion with M = 0 and constant vector of inertia X (a. e. P = 0), the vector 42 - 41 is a solution of a three fixed center problem. W e can take m3 E {-1,l) and r = mz/ml > 0. Then the centers 1)X, m3(r + 1)X. Their are o n the same line, with positions 0 , -m3(r-' masses are -m3, m3(r-l + I ) ~ m3(r , +I ) ~ .
+
.)I'[(
This phenomenon has been observed in the three-vortex problem
82
When X = 0, the three centers are equal, so 42 - q1 is a solution of a one fixed center problem. This is also a consequence of Prop. 5.2. In particular, we have just shown that for P = 0, the collinear threebody problem is integrable. As a matter of fact, for M = 0, the Poisson bracket { P , Q } vanishes, and for P = 0, Q = X does not depend on time and is a first integral. The same phenomenon occurs in the four-vortex problem, with vanishing total vorticity and vector of vorticity [ 2 ] ) . As a consequence of our remark in Sec. 1, the collinear three-body problem with Jacobi’s potential and M # 0 is also integrable ([‘]I).
(I4],
Acknowledgments I would like to warmly thank Alain Chenciner and Alain Albouy for proposing me to study this problem and for their advice, Susanna Terracini for inviting me, and all the SPT 2004 staff for giving me the opportunity to listen to such beautiful talks.
References 1. A. Albouy (2002), Homographic motions of N-body systems and N-vortex systems. Lecture for the Seminar “Astronomie et Systitmes Dynamiques” , Institut de Mkcanique Ckleste et de Calcul des Ephkmkrides. Observatoire de Paris. May 15th 2002. 2. H. Aref and M.A. Stremler (1999), Four-vortex motion with zero total circulation and impulse. Physics of Fluids, volume 11, 11’12, December 1999. pp. 3704-3715. 3. 0. Dziobek (1892), Mathematical theories of planetary motions. Dover publications, inc. New York, 1962. p. 70. 4. B. Eckhardt (1988), Integrable four vortex motions. Physics of Fluids, 31. pp. 2796-280 1. 5. H. Helmholtz (1858), O n integrals of the hydrodynamical equations which express vortex motion. Philos. Mag., 33, pp. 485-512. 6. C. G. J. Jacobi (1843), Vorlesungen uber dynamik. Gesammelte Werke. Chelsea. 1969. 7. N. Rott (1989), Three-vortex motion with zero total circulation. Journal of Applied Mathematics and Physics (ZAMP). Vol. 40, July 1989. Complementary note by H. Aref. pp. 473-500.
REMARKS ON CONFORMAL KILLING TENSORS AND SEPARATION OF VARIABLES
c. CHANU*, G. RASTELLI~ Dipartimento d i Matematica Universita d i Torino via Carlo Alberto 10, 10123 Torino, Italy
[email protected]. it
Properties of conformal Killing tensors employed in separation of variables theory are reviewed and new results concerning L-systems (Benenti-systems) are obtained.
1. Conformal Killing tensors and separation of variables
On a given Riemannian manifold (Q, G) a symmetric (contravariant) twotensor K is a conformal Killing tensor (CKT) if
(K,G]= X O G , where [., .] denotes the Schouten bracket, 0 the symmetrized tensor product and X is a vector field on Q. CKT’s are widely used in mathematical physics; for instance, in spacetimes they are associated with first integrals of null geodesics quadratic in the momenta. Moreover, R-separation of variables of Laplace equation is geometrically characterized in terms of “. .. involutive families of conformal Killing tensors ... which can be extended t o a family of commuting conformal s y m m e t r y operators...”’ We recall that two tensors K1 and Kz are in involution if [K1, K2] = 0. Special classes of CKT’s are often considered in the literature, for example trace-free CKT’S.’~~ In particular, we recall the following r e ~ u l t : ~
Theorem 1.1. For n > 2, the m a x i m u m number of linearly independent trace-free C K T ’ s is (n - l)(n 2)(n 3 ) ( n 4)/12 and it is attained in conformally flat manifolds. In these manifolds, all trace-free C K T ’ s are
+
+
+
*Work partially supported by GNFM and the Dipartimento di Matematica Universitk di Torino. +Work partially supported by Dipartimento di Matematica Universitk di Torino.
83
84
reducible i.e., linear combination with constant coeficients of symmetrized tensor products of conformal Killing vectors (CKV). Moreover, CKT’s are used to characterize orthogonal separation of variables for fixed values of the energy for Hamilton-Jacobi (HJ) equations associated with natural H a m i l t ~ n i a n s : ~
Theorem 1.2. T h e H J equation f o r the natural Hamiltonian H = $gajpipj + V i s separable in orthogonal coordinates o n H = E , f o r a fixed E E R,if and only if there exist n - 1 C K T ’ s (Ki) with real common eigenvectors, pairwise in involution, such that (G, K1,. . . Kn-l) are pointwise independent and satisfy
CKT’s satisfying (1) are called of self-gradient type. This means that they are KT’s for the conformal metric G/(E - V ) . In all the previous statements about separation of variables (as in the ordinary separation of variables theory for HJ and Schrodinger equations, see [5, 61 and references therein) orthogonal separable coordinates are determined by common eigenvectors of CKT’s in involution (or KT’s in involution for ordinary separation). Indeed, up to a closed singular set of Q, the eigenvector Ei is orthogonal to the coordinate hypersurfaces q’ = constant. Since we are interested in CKT’s eigenvectors, an equivalence relation naturally arises between CKT’s:
Definition 1.1. Let K1, K2 be CKT’s of order two; K1 and K2 are equivalent (K1 K2) if K1 - K2 = f G for a (almost everywhere C”) function f on Q.
-
Two equivalent CKT’s K1, K2 share the same eigenvectors and K1 has distinct eigenvalues if and only if K2 has distinct eigenvalues. This fact suggests that, instead of single CKT’s, equivalence classes of CKT should be used to characterize separation of variables in the above described cases. Unfortunately, involution is not compatible with -: the fact that [K1,K] = 0 does not imply in general [K2,K] = 0 for all K2 K1. It follows that tensors equivalent to self-gradient CKT’s are not necessarily of self-gradient type. Following [4],we introduce the notion of conformal involution.
-
85 Definition 1.2. Two symmetric tensors K1 and Kz are said to be in conformal involution if there exists a vector X12 such that [Ki,Kz] = Xiz 8 G.
-
Conformal involution is compatible with the equivalence relation . Indeed, CKT’s equivalent to CKT’s in conformal involution are in conformal involution. Moreover, properties of CKT’s in conformal involution are similar to those of KT’s in involution:
Proposition 1.1. (a) All CKT’s simultaneously diagonalized in orthogonal coordinates are in conformal involution. (ii) If n independent CKT’s with common eigenvectors Ei are in conformal involution, then Ei are normal (i.e., surface forming). Hence, for instance, we can restate Theorem 1.2 as f01lows:~
Theorem 1.3. The HJ equation for the natural Hamiltonian H = igajpipj V is separable in orthogonal coordinates on H = E , for a fixed E E R,if and only if there exist n- 1 CKT’s (Ki) with real common eigenvectors, pairwise in conformal involution, such that (G, K1, . . . , Kn-l) are pointwise independent and equation (1) is satisfied b y some K: Ki.
+
-
Any other (n- 1)-tuple of CKT’s equivalent to (K1,.. . ,Kn-l) defines the same web; some of them are made of KT’s for the conformal metric G / ( E - V ) and pairwise in involution. It is remarkable that3 a CKT is always equivalent to a trace-free CKT. Thus, by Theorem 1.1we conclude that
Theorem 1.4. I n n-dimensional conformally flat manifolds ( n > 2) the orthogonal coordinates characterized by CKT’s in conformal involution can be always characterized by using reducible CKT’s.
We remark that we can use the eigenvalues of CKT’s for finding the associated coordinate^,^ as shown for Killing tensors in [8]. 2. Conformal Killing tensors and L-systems
-
A further application of the equivalence relation in separation of variables theory concerns a special case of separable system, the L-systems (or Benenti-systems). We recall that5 a L-tensor L is a (1,l) CKT with vanishing Nijenhius torsion and pointwise simple eigenvalues. This means that there exist local
86
orthogonal coordinates ( q i ) such that = a/aqi are eigenvectors of L associated with eigenvalues ui = u i ( q i ) depending on a single coordinate. Moreover, in [9, 51 it is shown that a L-tensor generates a sequence of n (contravariant) KT’s pointwise independent and pairwise in involution:
KO = G,
1
K, = -tr (Ka-lL) G - K,-IL, U
( u = 1,.. . , n - 1).
(2)
which form a basis of the linear n-dimensional space of K T with common eigenvectors and in involution (KS-space); then the coordinates (42) are separable. An orthogonal separable system is a L - s y s t e m if it admits n independent KT’s recursively generated by a L-tensor by means of ( 2 ) . The presence of a term proportional to the metric tensor in each element of the sequence (2) suggests the possibility to construct a new sequence of tensors equivalent to K, in the sense of Definition 1.1.
2.1. K - s e q u e n c e s and C-sequences From a more general point of view, we consider a symmetric two-tensor L of type (1,l)on a n-dimensional Riemannian manifold ( Q ,G ) with (i) real eigenvalues (uz) and (ii) n independent eigenvectors (Ei). We remark that if the metric tensor is definite positive both requirements are satisfied by all symmetric tensors. By using L, we construct the sequences (K,) and (C,) made of (1’1) tensors in the following way
KO= I,
K, = i t r (K,-lL) I - K,-1L,
C1 = -L,
1 (C,) L - C,L, Ca+l = ;tr
(3)
where u = 1,.. . ,n - 1 and I is the identity matrix. We call (K,) the K-sequence and (C,) the C-sequence generated by L. We remark that if L is a L-tensor then the K-sequence (3)1 (rewritten in contravariant form) coincides with (2). Therefore, in the following we shall use the same notation for the two sequences. The relation between the sequences (3) is given by the following
Proposition 2.1. POTa n y u = 1,.. . ,n - 1, t h e elements of t h e sequences (3) satisfy
K,
2
1 --h(C,)I+C,. a
a7 0
Proof. By induction on the value a.
An alternative expression for K, and C, is based upon the algebraic invariants aiof L, which are the coefficients of the characteristic polynomial det(L-XI) = (-l)"X"+C~==l(-l)iaiX"-i (in particular we have 01 = t r L , an = det L). The functions ui are the elementary symmetric functions of the eigenvalues of L: ai = aa(uj)is the sum of all products of i distinct variables u j . We set 00 = 1. Further properties of the ai useful for our aims can be found in [9, 51.
Proposition 2.2. For any a = 0 , . . . ,n- 1, the elements of the K-sequence (3)1 satisfy a
K,
= C(-l)ha,-hLh.
(4)
h=O
For any a = 1 , . . . ,n, the elements of the C-sequence
(3)2
satisfy
a
c, = E(-l)hO,-hLh.
(5)
h=l
Proof. Equation (4) is proved in [9]. Due to Proposition 2.1, equation ( 5 ) follows. 0
Remark 2.1. Due to the Hamilton-Cayley theorem, it is immediate to see that both sequences (3) end after n steps. Indeed, we get K, = 0, C, = -(det L) I and K,+j, C,+j vanish for all j > 0. Proposition 2.3. (a) The tensors L, K, and Ca+l (a = 1 , .. . , n - 1) have the same eigenvectors. (ii) The tensors ( K O ,... , K,-1) are pointwise linearly independent i f and only i f L has distinct eigenvalues. (iii) The tensors ( ( 2 1 , . . . , C , ) are pointwise linearly independent i f and only i f L has distinct eigenvalues and det L # 0. Proof. Item (i) follows from (3). Let pk be the eigenvalues of the tensors K, and let XL be the eigenvalues of the tensors C, ( a = 1,.. . , n - 1). We have pb = 1 and A: = - d e t L for any i . According to [9], we have for that is the symmetric any a = 0 , . . . , n - 1: p i = a: where a: = a, functions of order a over the n - 1 variables ( u J )with j # i and det(ak) = J J J , 2 ( ~ z - - u J ) which is different from zero if and only if the u2are all distinct.
aa Thus, item (ii) follows. Moreover, by Proposition 2.2 we have X i = ui - 0,. Due to the properties of the determinants and being Xk = - det L we have det(Xd) = (-l)n+ldet(L)det(ai), so that we get (iii).
( a = 1, ...,n) 0
Remark 2.2. In order to avoid the restriction detL # 0, we can set C, = I without loss of generality. Indeed, generalizing the above proposition, we have that for any function f # 0 both the sets of n tensors (f1,K1,. . . K,-1) and (fI, ( 3 1 , . . . C,-l) are pointwise linearly independent if and only if L has simple eigenvalues. From Propositions 2.2 and 2.3, we get the analogous of a well-known property of the sequence defined by (2)5. For sake of simplicity we adopt the same notation for the tensors (K,) and ( C , ) written in (1,l)or in (0,2) form.
Theorem 2.1. If the tensor L has simple eigenvalues and normal eigenvectors then the tensors C1,. . . Cn-l, G are independent symmetric tensors simultaneously daagonalized in orthogonal coordinates. 2.2. C-sequences and L-systems We assume now that L is an L-tensor. We recall that in this case the Ksequence coincides with (2) and generates the KS-space of the L-system. For the C-sequence we get the following result:
Theorem 2.2. If L is an L-tensor for G then the tensors ((21,.. . Cn-l, G ) defined by (3)2 are independent conformal Killing tensors in conformal involution with respect t o any conformal metric G = f G and with common normal eigenvectors. The proof of the theorem requires the two following straightforward properties.
Proposition 2.4. If C is a C K T for G , then it is a C K T for any conformal metric G = f G where f is a function on Q. Proposition 2.5. If C1 and C2 are in conformal involution with respect to G , then they are in conformal involution with respect to any conformal metric G = f G where f is a function on Q.
89
Proof. Since L is a L-tensor, the K, are KT’s which generates a KS-space. Then, by Proposition 2.1 the C, are CKT’s equivalent to K,. Hence, by Proposition 1.1,the statement holds for the metric G . By Propositions 2.4 0 and 2.5, the theorem follows.
Example 2.1. The prolate spheroidal coordinates are obtained by rotating a planar elliptic-hyperbolic coordinate system around the line containing the foci. The three coordinate foliations are made of two-folded hyperboloids, ellipsoids, and meridian half-planes, respectively. It is easy to check that the prolate coordinates are an L-system generated by the Ltensor L, = r @ r - k2n @ n, where r is the position vector, n is a unit vector parallel to the rotational axis and k is half of the focal distance. The corresponding sequences K, and C, are
K1 = r2G - r @ r+ k2n@n+k 2 G c 1=
K2 = -k2(n x r) @ (n x r ) ,
C2 = k2[(n x
-Lp,
r) @ (n x r) + p 2 G ]
where p is the distance from the rotational axis.
Example 2.2. The Six-spheres coordinates. Given a Cartesian frame (0,5, y, z ) , each coordinate foliation of the six-spheres coordinates is a family of spheres all tangent in 0 with centers on the same coordinate axis (see [lo]). The spheres are obtained by inversion of the Cartesian coordinate planes and are orthogonal to the three independent CKV (inversions)
I,
= (-22
+ y2 + 2 2 , -22y,
I,
-22z),
I, = (-222, -2yz,
22
= (-2sy, 2 2
+ y2 - 22).
-
y2
+ 2 2 , -2yz),
We recall that in the Euclidean three-space the Cartesian coordinates are an L-system, with L-tensor given by L, = a, @ 8, 28, @ 8, 38, @ a,. Let us consider the tensor L, = I, @I, 2 I, 8 I, 3 I, @I,,whose eigenvectors are associated with the six-spheres coordinates. We get that L, is an Ltensor for the conformal metric G = f G , where G is the Euclidean metric and f = (z2 y2 z 2 ) - ’ . The corresponding C-sequence is
+
+
+
+
+ +
-Ls,
c1=
CZ
= L:
-
and the contravariant Cartesian components of c 2
= f-1
[
-411’ (z’ +5y2 +t2)--8(y’ f r ’ ) ’
,11
C3 = G
(trL,)L,, C2
are
22y(2’+7z2 -y2)
-2x2(3z2 +5yz -32’)
-9(x2+22)2-yz(14z2+9y2+2~2) -4y+(x2+2y2-2z2) I1 5(x2+y’)’+z2(22z2+26y2+5z’)
1
We remark that in this particular case the C, are KT’s in involution for the conformal metric G .
90 3. Conclusion The introduction of C-sequences in L-system theory allows to build the tensors characterizing L-systems with n - 2 instead of n - 1 iterations (C, = -L). Moreover, being a C-sequence is a property compatible with the natural equivalence relation between CKT’s. As seen in Theorem 1.4, the use of the equivalence classes of CKT’s allows, in conformally flat manifolds of dimension greater than two, to characterize associated separable coordinates by means of the finite dimensional linear space of reducible CKT’s. A C-sequence defined for G is a C-sequence for the whole family of metrics conformal to G. Example 2.2 suggests that, by using C-sequences, we can extend the notion of L-system, related to ordinary separable coordinates, to conformal L-system, related to conformal separable coordinate^,^ defined by C-sequences of CKT’s and generated by some suitably defined conformal L-tensor. On this subject the work is in progress N
Acknowledgments The authors wish to thank the organizers of the conference “Symmetry and perturbation theory 2004”.
References 1. 2. 3. 4.
E.G. Kalnins and W. Miller J . Phys. A: Math. Gen. 15, 2699-2709 (1982). K. Rosquist and G. Pucacco, J . Phys. A: Math. Gen. 28, 3235-3252 (1995). R. Rani, S.B. Edgar and A. Barnes, Class. Quant. Gravity 20 1929-1942 (2003) S. Benenti, C. Chanu and G. Rastelli, The separation of variables in the
Hamilton-Jacobi equation for fixed values of the energy Quaderni del Dipartimento d i Matematica, Universith d i Torino n. 18 (2003). 5. S. Benenti, Separability in Riemannian manifolds Phil. Trans. Roy. SOC.A (forthcoming). (2004). 6. S. Benenti, C. Chanu and G. Rastelli, J . Math. Phys. 43 n.11, 5183-5222 (2002). 7. C. Chanu and G. Rastelli, Eigenvalues of Killing Tensors and separable webs in Riemannian manifolds Quaderni del Dipartimento di Matematica, Universith d i Torino n. 18 (2004). 8. C. Chanu and G. Rastelli, SPT2002, World Sci. Publishing, Singapore, 18-25 (2002). 9. S . Benenti, Rend. Semin. Mat. Univ. Polit. Torino 50, 315-341 (1992). 10. P. Moon and D.E. Spencer, Field Theory Handbook, Springer Verlag (1961).
A REGULARITY THEORY FOR OPTIMAL PARTITION PROBLEMS*
M. CONTI, G. VERZINI Politecnico di Milano Dipartimento d i Matematica via Bonardi 9, 1-20133 Milano, Italy monica. contiomate. polimi. it, gianmaria. uerziniomate.polimi. at S. TERRACINI Uniuersitci degli Studi di Milano-Bicocca Dipartimento d i Matematica e Applicazioni via Bicocca degli Arcimboldi, 8, I-20126 Milano, Italy
[email protected] t
A recent literature 4,’,6,7 shows that a key role in the understanding of different kind of problems in nonlinear analysis, is played by functional class S introduced by the authors in connection with some optimal partition problems. The aim of this paper is to give a presentation of S and an overview of the regularity theory so far developed for its elements.
1. The functional class 5
The purpose of this paper is to present some recent results concerning the main qualitative properties of the solutions of several problems connected with optimal partition questions. The object of the theory is the so called class S, introduced by the authors in in the following manner. First, let N 2 2, R c RN be a connected] open bounded domain with regular boundary dR and let Ic 2 2 be a fixed integer. Then we consider the set of all the possible partition of R in k disjoint subsets which are supports of
*This work is supported by the Italian MIUR Research Projects Metodi Variazionali ed
Equazioni Differenziali Nonlineari
91
92
H’ (R)positive functions with assigned boundary data: (U1,.
. . , uk) E (H1(R))k:
uilan = &, ui 2 0 V i = 1,.. . ,k uj . ui = 0 , i
#j
a.e. on
R
Here we prescribe di 2 0, di E H1/2(dR)and such that di . d j = 0 for i # j , almost everywhere on dR. Roughly speaking, the class S consists in that subset of U in which the densities ui are not general H’ functions but instead obey to a set of 2k differential inequalities. Namely, let us fix fi(z,s ) : R x IK+ -+ EX, for i = 1,.. . , k and define
S =
{
,
( ~ 1 , .. . ~ k E) ZA
:
- A u ~5 f.(z,~
,
i )-AZi
A
2 f ( ~ , Z i ) , V=i 1,.. . , k )
Here the “hat” operation is defined as
i#i
for U E U , and the corresponding procedure on the nonlinearities leads to
i
-fj(z,uj)
if z E { u i > O} if z E {uj > 0 } , j
# i.
Our plan is the following. First, we provide some motivations to a systematic study of the elements of the class s,and we show that it appears in a natural way when studying different problems in nonlinear analysis, from population dynamics to optimal partition problems. Then we focus on the regularity and free boundary theory so far developed by the authors, and we present the main results which are proven in concerning the properties of the elements of S and their supports. To this aim, throughout the paper we always assume that fi(z,s) is Lipschitz continuous in s, uniformly in z and fi(z, 0 ) G 0.
2. The class S in different problems of nonlinear analysis: some examples 2.1. Highly competing diffusion systems Let us consider the system of k reaction-diffusion equations
93 for i = 1,.. . , k (see ’). This system governs the steady-states of k competing species coexisting in the same area 0. Here ui represents the population density of the i-th specie, whose internal dynamics is prescribed by f i ; the positive constants w . aij determine the interaction between the population ui and uj,which is possibly asymmetric. This systems has been introduced in as a model for studying the pattern formation driven by strong competition, see 8,9. It turns out that the presence of large interactions of competitive type produces, to the limit, the spatial segregation of the densities. In other words, in the limiting configurations all the populations survive, but have disjoint habitats. As a matter of fact the limiting configurations as the competition term tends to infinity, belong to S .
’
Theorem 2.1. Let (u+) be a positive solution of the above system. Then, there exists (ui)E S such that, u p to subsequences, I(ui,+- U ~ I I H I 0 as w--+m. Therefore, the class S is the natural framework where investigating the properties of the asymptotic limits of highly competing diffusion systems a. 2.2. A n optimal partition problem fi(z, u)du and consider the following variational problem:
Let F,(q s) =
Problem. Find the minimum of the functional
J(U)=
{ /’(f @(x) 1 V ~ i ( x ) [Fi(z, ~ ui(z)) -
i=l,...,k
where U E ZA. We can think to J ( U ) as the sum of the internal energies of k (see positive densities ui having diffusions di and internal potentials Fi (x,s ) ; a relevant assumption is that they interact solely through the boundaries of their supports. The minimizers of the problems can be viewed as a class of segregation states in population dynamics, which are governed by a minimization principle rather than competition-diffusion. In this perspective, the following result explains the connection between the variational problem and the functional class S:
Theorem 2.2. Let U be a minimum for J in 24. Then U E S ~~
~
aIn the case when aZ3 # a3%the definition of S has t o be changed according to an % asymmetric “hat” operation which takes into account the presence of the ratio a Z 3 / a 3# 1, see for the details
94
Remark 2.1. Under a suitable global Lipschitz continuity of the fi's it is possible to prove that the variational problem does have at least a minimizer U . The question whether the minimizer is unique has in general a negative answer; on the other side, a sufficient condition to prove uniqueness of the minimizer consists in assuming that the Fi's are concave. 3. The regularity theory for the elements of S As a first step in the study of the elements of S, let us list some regularity properties which follow as straightforward consequence of the validity of the Maximum Principle for elliptic equations (here it is crucial the locally Lipschitz continuity of the fi's). We first need a definition:
Definition 3.1. The multiplicity of a point z E R is m ( z ) = # { i : m e a s ( { u>i O } n B ( z , r ) ) > O V r > O }
.
We shall denote by
2 h ( U ) = {Z E R : m ( z ) 2 h} the set of points of multiplicity greater than or equal to h E N.
Lemma 3.1. Let U E S and xo E R: then U satisfies the following properties in a neighborhood of 20: m(z0) = 0 , then 3r > 0 such that ui = 0 on B ( q , r ) ,'di. (2) I f m ( z 0 )= 1, then 32, 3> 0 such that uk = 0 for k # i and
(1) If
-Aui = fi(z,ui)
in B(z0,r ) .
Hence ui > 0 and ui E C'I"(B(z0,r)) (for every 0 < a < 1). (3) If m(z0) = 2, then 3,j, r > 0 such that U h E 0 for h # z , j and setting fi,j(z,s) := fi(z,s+) - fj(z,s-) there holds -A(ui
-
u j ) = fi,j(z,ui - u j )
in B ( x 0 ,r ) .
Hence ui - u j E C'>"(B(zO,r)).
Remark 3.1.
- We can not exclude, at this stage, the occurrence of points of multiplicity zero, although this possibility will be ruled out at the end of Section 4.1, at least in two dimensions, under a weak non degeneracy assumption. Note that a{z E R : m ( x ) = 0) c 2 3 u 6%.
95
- If m ( z )= 2, as a consequence of assertion (3) in the previous proposition, it holds lim
11-S 11€(*,>0>
Fz
Vui(y) = -
Vuj(y)
Y€(U)>O)
If, by the way, the above limit is not zero, it follows that the set {z : m ( z )= 2) is locally a C' manifold of dimension N - 1 '. Let us now come to our main regularity result. (u1,. . . , un) E S, we consider the function
For any Ic-tuple
n
i=l and we wonder which is the best regularity for U . As it is clear by the above discussion, due to the presence of double points, one cannot expect U to be more regular than locally Lipschitz continuous. In fact we can prove that this is always the case, as stated in the following.
Theorem 3.1. Let M > 0 , h > 0 a fixed integer and w bounded domain. Let us define (211,.
. . ,U h ) E
(Hl(W))h
:
c RN
be an open
2 0 , ui ' U j = 0 i f i # j - A u ~5 M , -AGi 2 -M
ua
and let U E S & , k ( w ) . Then U as Lipschitz continuous in w . The next result shows that the regularity of the elements of S can hold up to the boundary, provided that the boundary is regular enough and the boundary data are Lipschitz continuous.
Theorem 3.2. Let dR be of class C 1 , U E S with uilan = W'@(dR) for every i. Then U E W',m(n).
da
and
di
E
We conclude the section by pointing out that the proof of these results is based on a local analysis (blow up procedure) and requires as a key tool a suitable version of the celebrated monotonicity theorem by Caffarelli & al., see and in particular 3. presence of variable diffusions the inequality turns into
lim
11-2
IJEt%>O)
di(z)Vui(y) = -
=i) U€(Yj>O)
dj(z)Vuj(y)
96
4. Further regularity in dimension N=2 Our analysis proceeds in dimension N = 2 with the study of the local behavior of the elements U of the class S around multiple points, that is, z E R with m(z) 2 2. The main goal is to show that, near a multiple point, U and its null set exhibit the same qualitative behavior of harmonic functions and their nodal sets. We refer the readers to the fundamental papers of Alessandrini and Hartman Winter lo for the main results about the zero set of harmonic functions and, more in general, of functions in the kernel of a divergence type operator. From now we will assume the further regularity on f
'
f
E C1(R x
W).
Under this assumption, the first crucial result states the gradient of U is null at points of multiplicity three (or greater) and moreover it vanishes continuously
Theorem 4.1. If ~0 E
2 3
then lVU(z)I
+0
as z
-+50.
We remark that the result is established through the application of a monotonicity formula with three or more phases in dimension N = 2, which is proven by the authors in '. Then the proof follows by a local analysis and exploiting topological arguments bases on the dimension two of the space.
4.1. Local p r o p e r t i e s of the f r e e boundary Let again N = 2 and consider an element U E S with the property that each component has connected support, i.e.
U E S such that {ui > 0} is connected V i .
(1)
Then we can go further with our analysis by providing a complete description of the geometrical properties of the null set of U . We already noticed in Remark 3.1 that the set of double points 22 = {z E R : m(z) = 2}, is locally a regular C' arc around those points where V U does not vanish. In fact this is always the case
Theorem 4.2. Let xo E Z 2 . Then V U ( x 0 ) # 0 and 22 is locally a C1curve through ZO. Let us now consider points of higher multiplicity. Our major result concerns with the topological structure of the set 2 3 . As a matter of fact there holds
Theorem 4.3. The set Z3 consists of a finite number of points.
97 We finally present an asymptotic formula describing the behavior of
C ui in the neighborhood of a multiple point which is isolated in 2 3 . Theorem 4.4. Let xo E Z3 with rn(z0)= h. Then there exists 80 E (-r,7r] such that
as r
--f
0 , where ( r ,8 ) denotes a system of polar coordinates around xo
It is worthwhile at this point to adopt a more geometrical perspective. To this aim, let us associate with the class S a family of partitions of the domain R in k subset, namely the set
P = ( ( ~ 1 , ..., w k )
: wi = {ui> 0 ) for
some U E S, wi connected
}
Then we obtain the following regularity theory for P : each wi is open (Th. 3.1); the k-tuple ( w i ) constitutes a true partition of R,namely
Indeed, having proved that the multiple points are isolated, the existence of points of multiplicity zero can be easily ruled in light of Remark 3.1. Finally, the set i#i
is made up by a finite number of C1 arcs ending either at the boundary of 0 or at a finite number of possible internal points Z h . Each Zh is a multiple intersection point where at least three different subsets wi meet; if the number of these subsets is 1, then the angle around Zh is equally shared in 1 parts by half lines pointed at zh and cutting angles of 2r/l. We wish to remark another peculiar feature of the class of partitions P . If k 2 4, the occurrence of a point of multiplicity four (or greater) cannot be excluded. This shows the different nature of our problem, compared to other singular minimization problems involving discontinuities (in the functions and/or in their gradients) where minimizers are known to possess only triple junctions (e-g. the minimizers of the Mumford-Shah functional in image segmentation). To exhibit a point of multiplicity k = 4, let R be the unitary ball of EX2, and let d(8) = Isin(28)I. Let us now assume the Fi’s to be convex and consider boundary data +i(O) = 4(8) for 8 E [in/4,(i 1)r/4], q5i = 0 otherwise. Then, in light of Remark 2.1, the minimizer U of the internal energy J , which is an element of S, is unique
+
98
and inherits the rotational symmetry of the data. Hence the origin is a point of multiplicity four for U .
Conclusions. The regularity theory we propose applies to the elements of the class S. Hence, it is sufficient to prove that the solution of a certain problem belongs to this class, in order to have immediately a variety of properties for the solution itself. On the other side, our theory also provides a tool for finding changing sign solutions to elliptic partial differential equations. For instance, let N = 1 and take fi = f for all i = 1,.. . ,Ic. Then, in light of Remark 3.1, to each eIement U E S correspond a changing sign solution of -Au
= f(u)
in
a,
namely the function u = ~ ~ = l ( - l ) i - l u ~ . By exploiting this feature, in we obtain a new enlighten on the structure of the h E i k spectrum for the Laplace operator and we prove possible variants and extensions of the already mentioned monotonicity formula by Alt-Caffarelli-Friedman 2 .
References 1. G. Alessandrini, Critical points of solutions of elliptic equations in two uariables, Ann. Scuola Norm. Sup. Pisa C1. Sci. (4) 14 (1987), 229-256 2. H.W. Alt, L.A. Caffarelli, A. Friedman, Variational problems with two phases and their free boundaries, Trans. A.M.S. 282 (1984), 431-461 3. L.A. Caffarelli, D. Jerison, C.E. Kenig, Some new monotonicity theorems with applications to free boundary problems, Ann. of Math. 155 (2002), no. 2, 369-404 4. M. Conti, S. Terracini, G. Verzini, A n optimal partition problem related t o nonlinear eigenualues, J. Funct. Anal. 198 (2003), no. 1, 160-196 5. M. Conti, S. Terracini, G. Verzini, A variational problem for the spatial segregation of reaction-diffusion systems, Indiana Univ. Math. J., to appear 6. M. Conti, S. Terracini, G. Verzini, O n a class of optimal partition problems related t o the FuE& spectrum and to the monotonicity formulae, Calculus of variation, to appear 7. M. Conti, S. Terracini, G. Verzini, Asymptotic estimates f o r the spatial segregation of competitive systems, submitted 8. E.N. Dancer, Competing species systems with diffusion and large interaction, Rend. Sem. Mat. Fis. Milano 65 (1995), 23-33 9. E.N. Dancer, Y.H. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Diff. Eq. 114 (1994), 434-475 10. P. Hartman, A. Winter, O n the local behauiour of solutions of nonpambolic partial differenzial equations (111) Approximation by spherical harmonics, Amer. J. Math. 77 (1955), 329-354
LAMBDA AND MU-SYMMETRIES
GIUSEPPE GAETA Dipartimento d i Matematica, Universitci d i Milano via Saldini 50, I-20133 Milano, Italy
[email protected] t
Lambda-symmetries of ODEs were discussed by C . Muriel in her talk at SPT2001. Here we provide a geometrical characterization of lambda-prolongations, and a generalization of these - and of A-symmetries - t o PDEs and systems thereof.
Introduction Symmetry analysis is a standard and powerful method in the analysis of differential equations, and in the determination of explicit solutions of nonlinear ones. It was remarked by Muriel and Romero” (see also the work by Pucci and Saccomandi14) that for ODEs the notion of symmetry can be somehow relaxed to that of lambda-symmetry (see below), still retaining the relevant properties for symmetry reduction and hence for the construction of explicit solutions. Their work was presented at SPT200111, raising substantial interest among participants. Here I report on some recent work4r6v7 which sheds some light on “lambda-symmetries” , and extends them to PDEs - and systems thereof as well; as the central objects here are not so much the functions A, but some associated one-forms p , these are called “mu-symmetries”. The work reported here was conducted together with Giampaolo Cicogna and Paola Morando; I would like to thank them, as well as other friends (J.F. Cariiiena, G. Marmo, M.A. Rodriguez) with whom I discussed these topics in the near past. It is also a pleasure to thank C. Muriel and G. Saccomandi for privately communicating their work on A-symmetries and raising my interest in the topic.
99
100 1. Standard prolongations Let us consider equations with p independent variables ( X I , ..., zp) E B = Rp and q dependent ones, ( u l , ..., uq)E F = Rq. The corresponding phase space will be M = B x F ; more precisely, this is a trivial bundle ( M ,7r, B ) . With the notation 8, := d / d x i and a, := a Lie-point vector field in M will be written as
x
= tZ(z,u)&
+ Cp”(z,u)&, .
(1)
We also write, with J a multiindex of length IJI = j 1 + ...+ j q , 8,” := a/&:. Then a vector field in the n-th jet bundle J“M will be written (sum over J being limited to 0 5 IJI 4 n ) as
Y
=
tia,
+ dQaJ,”
.
(2)
The jet space J”M is equipped with a contact structure, described by the contact forms
8:
:= duy
-
u”Jidzi
(IJI 5 n - 1) .
(3)
Denote by E the C m ( J n M ) module generated by these 8:. Then we say that Y preserves the contact structure if and only if, for all 29 E E , Cy(8) E E .
(4)
As well known, this is equivalent to the requirement that the coefficients in (2) satisfy the (standard) prolongation formula Qy,i =
DiQy - u”J,,(Dit”).
(5)
We note, for later reference, that for scalar ODES formula (5) is rewritten more simply, with obvious notation, as Qk+l
=
Dz Q k
-
uk+l
( D zt ) .
(6)
We also recall that the vector field Y is the prolongation of X if Y satisfies (4) and coincides with X when restricted to M ; X is a symmetry of a differential equation (or system of differential equations) A of order n in M if its n-th prolongation Y is tangent to the solution manifold SA c P M , see standard references on the subject2i5~8~9~13,16~18. Note that condition (4) is also equivalent to conditions involving the commutator of Y with the total derivative operators Di; in particular, it is equivalent to either one of
[D,,Y]A8 = 0 W E E ;
(7’)
101
with hy E C”(JnM) and V a vertical vector field in J n M seen as a bundle over J ” - ~ M .
2 . Lambda-prolongations 2.1. The work of Muriel and R o m e m In 2001, C . Muriel and J.L. RomerolO, analyzing the case where A is a scalar ODE, noticed a rather puzzling fact. They substitute the standard prolongation formula (6) with a “lambdaprolongation” formula
here X is a real C” function defined on J I M (or on J k M if one is ready to deal with generalized vector fields). Let us now agree to say that X is a “lambda-symmetry” of A if its “lambda-prolongation’’ Y is tangent to the solution manifold SAc JnM. Then, it turns out that “lambda-symmetries” are as good as standard symmetries for what concerns symmetry reduction of the differential equation A and hence determination of its explicit solutions. As pointed out by Muriel and Romero, it is quite possible to have equations which have no standard symmetries, but possess lambda-symmetries and can therefore be integrated by means of their approach; see their works1°-12for examples. 2.2. The work of Pucci and Saccomandi In 2002, Pucci and Saccomandi’* devoted further study to lambdasymmetries, and stressed a very interesting geometrical property of lambdaprolongations: that is, lambda-prolonged vector fields in J n M can be characterized as the only vector fields in J n M which have the same characteristics as some standardly-prolonged vector field. We stress that Y is the lambda-prolongation of a vector field X in M, then the characteristics of Y will not be the same as those of the standard of X ,but as those of the standard prolongation X ( n ) of prolongation X(”) a different (for X nontrivial) vector field 2 in M. This property can also be understood by recalling (4) and making use of a general property of Lie derivatives: indeed, for a any form on J n M ,
LAY(..)
= XYAda
+ d(XY-Ia) =
XLy(a)
+ dXA(YAa).
(9)
102
2.3. The work of Momndo It was noted6-10that lambda-prolongations can be given a characterization similar to the one discussed in remark 1 for standard prolongations; that Y ]1 6 = is, with h r and V as above, (8) is equivalent to either one of [Dz, X(Y 1 6 ) for all 6 E E , and [Dm, Y ]= XY hYD, V . This, as remarked by Morando, also allows to provide a characterization of lambda-prolonged vector fields in terms of their action on the contact forms, analogously to (4).In this context, it is natural to focus on the oneform p := Adz; note this is horizontal for J"M seen as a bundle over B , and obviously satisfies D p = 0, with D the total exterior derivative operator. Then, Y is a lambda-prolongedvector field zf and only zf C y ( 6 ) + (Y _] 6 ) p E E for all 6 E E .
+
+
3. Mu-prolongations; mu-symmetries for PDEs
The result given above immediately opens the way to extend lambdasymmetries to PDEs'. As here the main object will be the one-form p , we prefer to speak of "mu-prolongations" and "mu-symmetries" . Let
(10)
p := X i d ~ ~
be a semibasic one-form on ( J " M , 7rn, B ) , satisfying D p = 0. Then we say that the vector field Y in J"M p-preserves the contact structure if and only if, for all 6 E E ,
CY(6)
+ (YJ6)p
E E .
(11)
Note that Dp = 0 means DiXj = DjXi for all i, j ; hence locally p = D@ for some smooth real function @. With standard computations6, one obtains that (11) implies the scalar p-prolongation formula *~,i =
( D i + k ) * ~- u ~ , , ( D i + x i ) [ " .
(12)
Let Y as in (2) be the p-prolongation of the Lie-point vector field X (l), and write the standard prolongation of the latter as X(")= Fai @ J @ ; = 'p. We can obviously always write 9J = @ J FJ, note that 90= and FO = 0. Then it can be proved6 that the difference terms FJ satisfy the recursion relation
+
F J , ~= (Di where Q := 'p - u& is the
+ Xi)Fj + XiDjQ
character is ti^^^^^^^^ of the vector field X.
+
(13)
103 This shows at once that the p-prolongation of X coincides with its standard prolongation on the X-invariant space I x ; indeed, I x c J"M is the subspace identified by D J Q = 0 for all J of length 0 5 IJI < n. It follows that the standard PDE symmetry reduction ~ n e t h o d ~ ?works ~ ~ , 'equally ~ well when X is a p-symmetry of A as in the case where X is a standard symmetry of A; see our work6 for examples. The concept of p-symmetries is also generalized to an analogue of standard conditional and partial syrnmetries3,l, i.e. partial (conditional) psymmetries4. 4. Mu-symmetries for systems of PDEs The developments described in the previous section do not include the case of (systems of) PDEs for several dependent variables, i.e. the case with q > 1 in our present notation. This was dealt with in a recent work6, to which we refer for details. To deal with this case, it is convenient to see the contact forms 295,see (3), as the components of a vector-valued contact form17 2 9 ~ .We will denote by 0 the module over q-dimensional smooth matrix functions generated by the 295, i.e. the set of vector-valued forms which can be written as 77 = (R~)g29b,with RJ : J n M Mat(q) smooth matrix functions. Correspondingly, the fundamental form p will be a horizontal one-form with values in the Lie algebra g l ( q ) (the algebra of the group G L ( q ) ,consisting of non-singular q-dimensional real matrices) 17. We will thus write --f
p = Aidx'
(14)
where hi are smooth matrix functions satisfying additional compatibility conditions discussed below. We will say that the vector field Y in J"M p-preserves the vector contact structure 0 if, for all 29 E 0 , Cy(29)
+
( Y ~ ( h i ) ; 2 9 'dx' ) E 0.
(15)
In terms of the coefficients of Y , see (a), this is equivalent t o the requirement that the obey the vector p-prolongation formula
where we have introduced the (matrix) differential operators
104
If again we consider a vector field Y as in ( 2 ) which is the p-prolongation of a Lie-point vector field X, and write the standard prolongation of the latter as X(n) = ti& (with $; = @; = cp"), we can write \k; = @; F,", with F$ = 0. Then the difference terms FJ satisfy the recursion relation
+
+
where Q" := p a -@ti, and rJare certain matrices (see ref.6 for the explicit expression). This, as for the scalar case, shows that the p-prolongation of X coincides with its standard prolongation on the X-invariant space I x ; hence, again, the standard PDE symmetry reduction method works equally well for p-symmetries (defined in the obvious way) as for standard ones. See ref6 for examples. 5. Compatibility condition and gauge equivalence
As mentioned above the form p , see (14), is not arbitrary: it must satisfy a compatibility condition (this guarantees the S $ defined by (16) are uniquely determined), expressed by [ V z r V k ] Dihk - Dkhi
+
[hi,&]= 0 .
(19)
It is quite interesting to remark4 that this is nothing but the coordinate expression for the horizontal Maurer-Cartan equation
DP
1
+ 5 [P,PI
= 0.
(20)
Based on this condition, and on classical results of differential geometry15, it follows that locally in any contractible neighbourhood A C_ J n M , there exists Y A : A + GL(q)such that (locally in A) p is the Darboux derivative of Y A . In other words, any p-prolonged vector field is locally gauge-equivalent to a standard prolonged vector field4, the gauge group being G L ( q ) . It should be mentioned that when J n M is topologically nontrivial, or p present singular points, one can have nontrivial p-symmetries; this is shown by means of very concrete examples in our recent work4. Note that when we consider symmetries of a given equation A , the compatibility condition (20) needs to be satisfied only on Sa C J " M . When indeed p is not satisfying everywhere (20), p-symmetries can happen to be gauge-equivalent to standard nonlocal symmetries of exponential form; see again ref.4 for details.
105
References 1. G. Cicogna, “A discussion on the different notions of symmetry of differential equations”, Proc. Inst. Math. N.A.S. Ukr. 50 (2004), 77-84; “Weak symme-
2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
12.
13. 14. 15. 16. 17. 18.
tries and symmetry adapted coordinates in differential problems” Int. J. Geom. Meth. Mod. Phys. 1 (2004), 23-31 G. Cicogna and G. Gaeta, Symmetry and perturbation theory in nonlinear dynamics, Springer 1999 G. Cicogna and G. Gaeta, “Partial Lie-point symmetries of differential equat i o n ~ ~J.’ , Phys. A 34 (2001), 491-512 G. Cicogna, G. Gaeta and P. Morando, “On the relation between standard and p-symmetries for PDEs” , preprint 2004 G. Gaeta, Nonlinear symmetries and nonlinear equations, Kluwer 1994 G. Gaeta and P. Morando, “On the geometry of lambda-symmetries and PDEs reduction”, J. Phys. A 37 (2004), 6955-6975 G. Gaeta and P. Morando, “PDEs reduction and A-symmetries”, to appear in Note d i Matematica N. Kamran, “Selected topics in the geometrical study of differential equations”, A.M.S. 2002 I.S. Krasil’schik and A.M. Vinogradov eds., Symmetries and conservation laws for differential equations of mathematical physics, A.M.S. 1999 C. Muriel and J.L. Romero, “New method of reduction for ordinary differential equations”, I M A Journal of Applied mathematics 66 (2001), 111-125 C. Muriel and J.L. Romero, “Coosymmetries and equations with symmetry , Symmetry and Perturbation Theory (SPT2001), D. algebra SL(2,R ) 7 ’ in: Bambusi, M. Cadoni and G. Gaeta eds., World Scientific 2001 C. Muriel and J.L. Romero, “Coo symmetries and nonsolvable symmetry algebras”, I M A Journal of Applied mathematics 66 (2001), 477-498; “ Integrability of equations admitting the nonsolvable symmetry algebra so(3, r)” symmetries and Studies in Applied Mathematics 109 (2002), 337-352; 11C03 reduction of equations without Lie-point symmetries”, Journal of Lie theory 13 (2003), 167-188; M.L. Gandarias, E. Medina and C. Muriel, “New symmetry reductions for some ordinary differential equations”, J . Nonlin. Math. Phys. 9 (2002) Suppl.1, 47-58 P.J. Olver, Application of Lie groups to differential equations, Springer 1986 E. Pucci and G. Saccomandi, “On the reduction methods for ordinary differential equations”, J. Phys. A 35 (2002), 6145-6155 R.W. Sharpe, Differential Geometry, Springer 1997 H. Stephani, Differential equations. Their solution using symmetries, Cambridge University Press 1989 S. Sternberg, Lectures on differential geometry, Chelsea 1983 P. Winternitz, “Lie groups and solutions of nonlinear PDEs”, in Integrable systems, quantum groups, and quantum field theory (NATO AS1 9009), L.A. Ibort and M.A. Rodriguez eds., Kluwer 1993
POTENTIAL SYMMETRIES AND LINEARIZATION OF SOME EVOLUTION EQUATIONS
M.L. GANDARIAS Departamento de Matematicas, Universidad de Cadiz, PO. BOX 40 11 510 Puerto Real, Cadiz,Spain
[email protected] In this paper we consider a class of (C-integrable) autonomous second-order evolution equations in (1 1) dimensions. These equations have been linearized in via generalized hodograph transformations. We prove that these equations admit an infinite number of potential symmetries. Consequently these equations can be linearized by a non invertible point mapping.
+
1. Introduction In
we observed that the well known nonlinear diffusion equation (1)
U t = [(u-2u&.
can be linearised by using the invertible nonlocal hodograph transformation I ' Z 3 , given by
d X ( 5 ,E) = i i d Z + ii-2ii,dE
r2 :
i
dT(5,E) = dE
U ( X ,T ) = Z
We also observed that the linearization
UT
-
uxx
=0
of (1) can also be achieved by the invertible contact transformation
d x ( X ,T ) = UdX
+ UxdT (3)
u ( x ,t ) = u-l
106
107
Moreover, we define autohodograph transformations, that is nonlocal transformations of hodograph type that transforms a given differential equations into itself. This autohodograph transformation for (1) is given by
A
=
rl rz.
+
In and N. Euler et a1 presented a class of (C-integrable) (1 1)dimensional second order evolution equations which can be linearised using a generalised hodograph transformation as well as the recursion operators for this class of equations. Lie classical symmetries admitted by nonlinear PDE’s are useful for finding invariant solutions, as well as to discover whether or not the equation can be linearized by an invertible mapping and construct an explicit linearization when one exists. Nevertheless an obvious limitation of group-theoretic methods based in local symmetries, is that many PDE’s, do not have local symmetries. It turns out that PDE’s can admit nonlocal symmetries whose infinitesimal generators depend on the integrals of the dependent variables in some specific manner. It also happens that if a nonlinear scalar PDE does not admit an infinite-parameter Lie group of contact transformations is not linearizable by an invertible contact transformation. However most of the interesting linearizations involve non-invertible transformations, such linearizations can be found by embedding given nonlinear PDE’s in auxiliary systems of PDE’s z. For a given PDE one can also find useful nonlocal symmetries by embedding it in an auxiliary ” covering” system with auxiliary dependent variables A point symmetry of the auxiliary system, acting on the space consisting of the independent and dependent variables of the given PDE as well as the auxiliary variables, yields a nonlocal symmetry of the given PDE if it does not project onto a point symmetry acting in its space of the independent and dependent variables. In 1,2 Bluman introduced a method to find a new class of symmetries for a PDE. By writing a given PDE, denoted by R{x,t,u} in a conserved form, a related system denoted by S{x,t,u,v} with potentials as additionals dependent variables is obtained. If u ( z ,t ) ,w(z,t ) satisfies S{x,t,u,v}, then u ( z ,t ) solves R{x,t,u} and w(z, t ) solves an integrated related equation T{x,t,v). Any Lie group of point transformetions admitted by S{x,t,u,v} induces a symmetry for R{x,t,u}; when at least one of the generators of the group depends explicitly of the potential, then the corresponding symmetry is nei-
’.
108
ther a point nor a Lie-Backlund symmetry. These symmetries of R{x,t,u} are called potential symmetries. The nature of potential symmetries allows one t o extend the uses of point symmetries to such nonlocal symmetries. In particular: (1) Invariant solutions of S{x,t,u,v}, respectively T{x,t,v}, yield solutions of R{x,t,u} which are not invariant solutions for any local symmetry admitted by R{x,t,u}. (2) If R{x,t,u} admits a potential symmetry leading to the linearization of S{x,t,u,v}, respectively T{x,t,v}, then R{x,t,u} is linearized by a noninvertible mapping. Suppose S{x,t,u,v} admits a local Lie group of transformations with infinitesimal generator d VS = P ( x , ~ , u , v ) dX
+ q(z,t,u,w)-atd + r ( z , t , u , v )dU-d + s ( s , ~ , u , vdvd) - , (4)
this group maps any solution of S{x,t,u,v} to another solution of S{x,t,u,v} and hence induces a mapping of any solution of R{x,t,u} t o another solution of R{x,t,u}. Thus (15) defines a symmetry group of R{x,t,u}. If
then (15) yields a nonlocal symmetry of R {x,t,u}, such nonlocal symmetry is called a potential symmetry of R{x,t,u}, otherwise Xs projects onto a point symmetry of R{x,t,u}. Potential symmetries were obtained in for the porous medium equation with absorption and convection Ut
+
= [(u”),: f(5)US%
+ g(z)u”,
(6)
when it can be written in a conserved form. These symmetries lead to the linearization of the equation by non-invertible mappings. It is well-known that (1) admits no linearizing point or contact symmetries. However the associated auxiliary system
v, = u U t = u%,,
(7)
admits potential symmetries leading to the linearization of (1) by a change of variables involving a non-invertible mapping. In this paper, we consider the quasilinear autonomous evolution equations reported in 5 , 6 we prove that all these equations that admit recursion
109
operators riot depending on z admit an infinite-parameter group of potential symmetries that allow us to linearize them by a noninvertible mapping
1. P o t e n t i a l symmetries We consider the general second-order linearisable autonomous evolution equation, given by
ut = a(u)u,,
+ b(u)u: + c(u),u, + d ( u )
(8)
We introduce the potential variable v and the equation can be written as
+
v, = f(u) mt vt = g(u)u, k ( U )
+
+ nx,
(9)
with a ( u )=
$$$ b(u)=
c ( u )= f
(21)
d(u) =
5
(10)
By requiring (9) to be invariant under the transformation with infinitesimal generator
v = E(z,t , u , v ) d z +
T ( 5 ,t , U , V ) &
+ 4 ( x , t , U , V ) d u + +(x,t , u,v)dv.
(11)
one obtains an over determined, linear system of equations for the infinites+(z, t , u , v ) , 4(z,t,u , ~ ) When . at least one imals ((5, t , u,v), ~(z,t,u,v), of the generators of the group depend explicitly of the potential , that is if
(2 + ., +” 4: # 0
(12)
then (11) yields a nonlocal symmetry of (1). A potential s y m m e t r y of (13) i s a classical s y m m e t r y of the associated potential s y s t e m (14) that satisfies (12).
In the following we list the quasilinear evolution equations which appear in and as linearizable via the extended hodograph transformation and that admit Lie-Backlund symmetries and recursion operators not depending on x. In this work we are not considering the semilinear equations. We also list the associated potential systems obtained by introducing a potential variable v as well as the corresponding nonlocal potential symmetries leading to the linearization of the potential systems and to the linearization of the corresponding equations by a non-invertible mapping. Following we define
( h ) u :=
1dh 2du
dh + h(-)du
,d2h
-
du2
110
where h E C 2 ( R ) $ , #0
Equation 1 We consider the equation given by Ut
+
= h ( u ) ~ , , (h),uz,
(13)
can be written in a conserved with the associated auxiliary system given by
Applying the classical method to system (14) we get the generators
i) d
v1 = -
v2 =
d
-
~3
d =z-,
dV’
~4 =
dX
a
d
d dt
t- +v-
d h d - -- (15) d v h’du
h1I2 h d -v)dz dv h’ h’ du and an infinite-parameter Lie group of point transformations with infinitesimal generator val, v5 = 0-
+2t-
V a l = a(t,w)-
d
ax
+2(-z+
h1I2 + 2---av(t, h’
d dU
w)-
where a(w,t ) is an arbitrary function satisfying the linear heat equation at
- a,,
=0
(18)
Equation 2 We consider the equation given by Ut
= h(u)u,, f (h),u?
+ Xh(u)ux,
(19)
can be written in a conserved with the associated auxiliary system given by
111
Applying the classical method to system (20)we get, besides some other generators an infinite-parameter Lie group of point transformations with infinitesimal generator v,2,
d
a
exx
v a 2 = Xexxa(w,t)-
dx
+ -(2Xh1/2a,(t,v) + 2A'h)-d U h'
(21)
where a(w,t ) is an arbitrary function satisfying the linear heat equation
at - Xa,, = 0
(22)
Equation 3 We consider the equation given by
+ (h)UUz+ ( X i - Xh(U))Ux,
= h(u)uXx
(23) can be written in a conserved with the associated auxiliary system given by Ut
Applying the classical method to system (24) we get, besides some other generators an infinite-parameter Lie group of point transformations with infinitesimal generator v a 3
v,3
d
= AeAxa(w,t)-
dx
2XeXx + -(XIha(v,t) h'
+ h1/'a,(t,w))-dUd
(25)
with Qt
+ a,, + X X l Q = 0
Equation 4 We consider the equation given by
+
ut = h(u)uxx (h)Uuz-t- XU,,
(26)
can be written in a conserved with the associated auxiliary system given by
v,
= h-1/2
Vt =
fr
-- h-1/2h'ux + Xh-1/2
(27)
112
Applying the classical method to system (27) we get, besides some other generators an infinite-parameter Lie group of point transformations with infinitesimal generator v,1
3. Linearization
All the auxiliary systems written above (14),(20),(24) and (27), admit an infinite-parameter Lie group of point transformations with infinitesimal generator v,1,va2,v,3 and v,4 = v,l, where cy is an arbitrary function satisfying a linear partial differential equation. one can easily obtain the invertible mapping which transforms any solution of the nonlinear auxiliary system to solution of a linear system and hence to a solution o a liner equation
ut = u x x + X I U X + A2U.
(29)
Equation 1
The infinite-parameter Lie symmetry (15) leads to the invertible mapping
X=v
T=t
U = h(u)ll2
V =x
which transforms a solution of the linear system
to a solution of the nonlinear system (14) and hence to a solution of (13).
Equation 2
For equation 2, the infinite-parameter Lie symmetry (21) with X = -1, leads to the invertible mapping
X=v
T=t
u = h(U)%"
V = - ex
(32)
113
which transforms a solution of the linear system
vx
=
u
(33)
VT = U X ,
to a solution of the nonlinear system (20) and hence to a solution of (19).
Equation 3 The infinite-parameter Lie symmetry (25) with X = -1, leads to the invertible mapping
X=v
T=t
u = h(u)i/2ez
V
(34) = -ez
which transforms a solution of the linear system
v, = u v, = -u, + X l v ,
(35)
to a solution of the nonlinear system (20) and hence, any solution of the linear equation UT -k U X X - XiU = 0
to a solution of (23).
Equation 4 The infinite-parameter Lie symmetry (28) with, leads to the invertible mapping
X=v
T=t
u = h(u)1/2
v =5
(36)
which transforms a solution of the linear system
v,
=
u
VT = -uX
+ A,
(37)
to a solution of the nonlinear system (27) and hence, any solution of the linear equation UT = U X X
to a solution of (26).
114
4.
Concluding remarks
In this paper we have considered a class of quasilinear autonomous secondorder evolution equations in (1 1) dimensions. All of these equations have Lie-Backlund symmetries and recursion operators not depending on x and have been linearized in via generalised hodograph transformations. We have constructed nonlocal symmetries (potential symmetries) which are realized as local symmetries of the related auxiliary systems. We exhibit the infinite parameter Lie group of point transformations. Consequently these auxiliary systems can be linearised by an invertible point mapping. In turn these mappings lead to noninvertible linearising mapping of the original PDES.
+
References 1. Bluman G W and Kumei S 1980 J. Math. Phys. 8 1 1019. 2. Bluman G W and Kumei S 1989 Symmetries and Differential Equations (Berlin: Springer) 3. Clarkson P A, Fokas A S . and Ablowitz M J 1989 SIAM J . Appl. Math. 49 (1989) 1188-1209. 4. Euler N., Gandarias M.L., Euler M. and Lindblom O.J. Math. Anal. and Appl. 257(2001) 21-28. 5. Euler N. and Euler M. J. Nonlin. Math. Phys. 3(2001) 342-362. 6. Euler M. and Euler N. Studies in Appl. Math. l l l ( 2 0 0 3 ) 315-337. 7. Gandarias M.L. J . Phys. A : Math. Gen. 29(1996) 5919-5934. 8. Olver P J 1986 Applications of Lie Groups to Differential Equations (Springer, Berlin)
PERIODIC SOLUTIONS FOR ZERO MASS NONLINEAR WAVE EQUATIONS
G. GENTILE Dipartimento d i Matematica Universita da Roma Tre Largo S u n Leonard0 Murialdo 1, 00124 Roma, Italy
[email protected] In a celebrated paper by Craig and Wayne existence of periodic solutions has been proved for nonlinear wave equations with a mass term. Later the result has been extended t o quasi-periodic solutions, but always under the assumption that the mass is strictly positive. So far, in the case of zero mass and Dirichlet boundary conditions, existence of periodic solutions has been proved only for a zero-measure set of values of the frequencies and amplitudes. I shall discuss a recent result, obtained with V. Mastropietro and M. Procesi, in which the same result can be obtained for a set with large relative measure.
1. Introduction 1.1. The model
+
Consider the one-dimensional nonlinear wave equation utt - u,, Mu = p ( u )with Dirichlet boundary conditions (DBC) u(t,0) = u(t,7r) = 0 , where cp(u)= @u3 0 ( u 5 )is an odd analytic function and M is a real parameter (mass). We assume @ # 0, and, without loss of generality, we can set @ = 1 (or Q, = -1). We look for analytic (in 2,t ) small amplitude solutions. This suggests naturally a rescaling u + &u, with E > 0, so that the equation becomes
+
i +
U t t - u,,
+ MU =
E+(u,
u(t,0 ) = u(t,7r) = 0 ,
E),
(1)
with +(u, E ) = u3 0(&u5). If E = 0 any solution of (1) can be written as superposition of linear oscillations with frequencies w, = d m . In particular there are infinitely many periodic solutions. A natural question is under which conditions (if any) periodic solutions can be continued for E # 0.
115
116
1.2. The results Results for this kind of problems read as follows. For concreteness let us consider as unperturbed solution the function uo(t,x ) = w1 cos w l t sin x (more general statements can be obviously made).
Theorem 1.1. One can fix M in a suitable subset M c R so that the following holds. There exist EO > 0 and a Cantor set E E [O,EO] of large relative measurea such that for all E E E there exists a solution u(t,x ;E ) of (1) analytic in ( t ,x ) and 2 r / w ( ~ ) - p e r i o d i cin t , with ~ w ( E-) w11 < CE and Ilu(t,2 ;E ) - uo(w(E)t,x)II < CE,for a suitable constant C . The first result of this kind was by Craig & Wayne', for M a full measure set obtained by requiring that M verifies the Diophantine conditions 1nw1+ wI, > Co/lnl' for all Iml, 1121 > 1 and for suitable positive constants COand T . Later ~ Bowgain* and Kuksin & Poschel'0~'2 extended the result t o all M # 0; in this case one uses part of the nonlinearity t o remove resonances between the frequencies w,. Here we consider the case M = 0, which was left out in literature:
{
Utt
- u z z = c.p(,.
E),
u(t,0 ) = u(t,7 r ) = 0.
Note that for E = 0 one has w, = Iml, so that this is a completely resonant case. Any solution for E = 0 can be written as
The result which can be proved now is as follows."
Theorem 1.2. Set
uo(t,X )
W(E)
= Vm
=
fi. T h e n there i s a n unperturbed
(sn(%(t
+x),m)
-
sn(Rm(t - x ) , m ) )
solution
(4)
with R, = 2K(m)/7r and V, = Om-, where sn(.,m) i s the sineamplitude function with modulus fi,K ( m ) is the elliptic integral of first kind, and m M -0.2554 is uniquely fixed, such that the following holds. There exist EO > 0 and a Cantor set E E [0,E O ] of large relative measure such that f o r all E E & there exists a solution u ( t , x ; E )of (2) analytic in aThis means that lim,,,+ &-'meas(& n [0,€1) = 1, if meas denotes Lebesgue measure. bWith the techniques described here it has been re-obtained by Gentile & Mastropietro8. 'Extensions t o solutions with frequency which is a multiple of W ( E ) are easily obtained.
117
( t ,z) and 2 ~ / w ( ~ ) - p e r i o din i ct , with Ilu(t,5 ;E ) - uO(w(E)t, suitable constant C .
< CE,f o r a
The first result of this kind was by LidskiY & Shul'man'l, for the case with periodic boundary conditions (PBC) and y ( u ) = u3,but only for a set of values of E with zero-measure. Later on Bambusi & Paleari3 found the same result for the equation ( 2 ) , but again for a zero-measure set, and the result was then extended to more general nonlinearities by Berti & Bollel. A large measure set was obtained by Bourgain5 in the case of PBC, which presents some simplifications with respect to the case of DBC, as we shall see. Here we discuss the main ideas of our proof for the case of DBC,d by referring to the original paperg for a more detailed exposition.e
2. General scheme 2.1. Lyapunov-Schmidt decomposition We start by a Lyapunov-Schmidt type decomposition, which consists in writing u ( t ,z) = v ( t ,x) w ( t , x),with
+
(n,m)& lnlflml
where we have set w = W ( E ) = fi. We can assume v, = -v-, and w , > ~= - w , , - ~ by the parity properties of the equation. Note that v ( t ,x) has the same form of the unperturbed solution (3). Then we obtain two sets of equations, usually denoted the Q and P equations,
The new difficulties with respect to the case M
# 0 are:
(1) The Q equation is infinite-dimensional, and this gives rise to two problems: d A similar result has also been announced by Berti & Bolle2. eNon-perturbative smooth solutions have been obtained through variational methods. We mention also that there are examples of nonlinear wave equations in which discontinuous periodic (generalized) solutions can be provided7. In both cases the periods are (sub-multiples of) 2 ~ .
118
(a) Which solution can be continued? (b) How can one avoid loss of regularity in solving the Q and P equations?
(2) The frequency w is 1 plus order E , so that if one wants to impose a 2CO/(~ how ( ~can , one obtain Diophantine condition like (wn&mm( a constant Co which is &-independent? The problem of loss of regularity is the following one. In the case M # 0 the Q equation is one-dimensional: for instance if one looks for a solution with frequency close to w1 one has an equation for the amplitude u1. By considering u1 as a parameter one can solve the P equation and one finds a solution w(u1),which can be inserted into the Q equation and solved by an implicit function argument. In our case the Q equation involves all n E Z, hence it is an equation for a function w (determined by the collection of its Fourier components w,’s): if one takes u as a parameter the solution W(U) of the P equation will be less regular than the function u, so that, when inserting such a w into the Q equation one has the problem to find a solution with the originally assumed regularity of u. Note that the Q and P equation look very different: the latter yields a small divisors problem but can be treated by perturbation theory, while the first one has no small divisor problem, but the right hand side is not of order E . Problem (2) reflects the fact that if Co is of order E then also the Q equation is not a perturbation equation any more. 2.2. Zero-order solution For E = 0 one can take w = 0, so that the P equation disappears and the Q equation reduces to n2w, = [w3],,,. We can write ( 5 ) as w(t,x) = u ( [ ) b(E’), with [ = wt x and [’ = wt - x , and with b ( [ ) = - u ( ( ) (u and b involve the Fourier components with m = n and m = -n, respectively). Then the Q equation gives a = -u3 - (b2) u and b = -b3 - (u2)b, where the dots denote derivatives with respect to the corresponding arguments, and (.) denotes average (that is the Fourier component with zero Fourier label). Because of the relation between the functions a and b, the two equations above correspond to the only equation a = -u3 - ( u 2 )a , which is a closed equation for a. This is an integro-differential equation, because of the average term: the only odd 2r-periodic solution is found to be the function (4) appearing in the statement of Theorem 1.2.
+
+
119
In the case of PBC the equation for a is simpler: it is the ordinary differential equation a = -a3, which admits (trivially) the solution a ( [ ) = sn(O[, 1/2). Such a function solves the full equation utt -uzz = &u3, for [ = w t + z . More generally the same argument shows that one can easily construct periodic solutions of the form u(t,z) = U([) for any nonlinearity cp(u),simply by taking as U the solution of the one-dimensional system with potential energy P(u,E ) du. Then in order to have non-trivial results in the case of PBC one needs corrections to the nonlinear term explicitly depending on z, for instance5 p(u, z) = u3 a(z)u4. Hence the P equa= ~~/~[a(z)u~],,,, and one can impose tion becomes (-d-n2 m’)w,,, a Diophantine condition with CO= O ( E ) ,without destroying the perturbation character of the P equation. In this way both problems l ( a ) and 2 are solved in the case of PBC, while in the case of DBC problem 2 still remains.
+
+
2.3. Avoiding the loss of regularity The idea to solve problem l ( b ) is very simple: we try to solve both equations P and Q simultaneously, by using as zero-order solution (w,zu) = (WO,0), with w o ( t , z) = a ( [ ) b([’), and writing w = wo V = ( a A, b B ) , with obvious meaning of symbols. Then we can write the Q equation as ii A = G(a A ) F ( a ,A, w,E ) , where G(a A ) is the part of [u3],,,depending only on w (what automatically defines also the function F ) , and we are explicitly using that b and B can be expressed in terms of a and A (by the symmetries of the equation). If G denotes the part of G which is at least of order two in A, then the equation Q can be solved formally by variation of constants. This means that we can write A ( [ ) as the first component of the vector
+
+
+
+
+
+
+ +
where W ( t ) is the Wronskian matrix of the linearized system A = dG(a([))Aand the initial condition has to be chosen in such a way that X ( t ) turns out to be periodic: in other words it is used to eliminate the secular terms which in principle can arise from the equation. One has to be careful as G(a+A)depends on ( A 2 ) which , is not known. A classical trick in dealing with problems of this kind is just to replace (A’) with a parameter C, and to work out the solution as a function A(E,C) depending on such a parameter. At the end one has to impose (A’(., C)) =
x
120 C, and in order to solve such an equation one has to check some nondegeneracy condition. In a general setting2 this could be obtained as a genericity condition on the dominant term of the nonlinearity; however in our case the latter is given, so that non-degeneracy has t o be checked by explicit computation.
3. Lindstedt series method
3.1. Power series expansion
So far the analysis is only formal. As a first attempt we try to find a solution in the form of a power series in E , that is
which inserted into the equation (7) and in the second set of equations in (6) give recursive equations for the coefficients V ( k )and w ( ~ )For . instance wn,m = &(-w2n2 m 2 ) - 1 [ ~ 3 ] n.,.m . gives
+
+
where . . . are the higher order terms, and the * denotes the constraints k1 kz + k g = k - 1, nl 122 +ng = n and ml +m2 +m3 = m, and of (k 1 ( k ) , , k% 2 1 in’ Wn,:m,. course ki 2 0 in u ~ , : ~while The quantity g(n,m)= 1/(-w2n2 m2)is called propagator. Then (9) admits a very natural representation in terms of tree graphs (or simply trees), which are given by a collection of points and lines connecting them without forming any cycles. One introduces some rules in order to associate to any tree a number, which is called the value of the tree; for instance to each line one associate two integers (n,m)and a propagator g(n,m). In this way each coefficient u (,k ,) ~is given as a sum over trees of the corresponding values; we refer to the original paperg for details and pictures.
+
+
+
3.2. Counterterms and resummations The problem is that to any order k there are trees whose value grows as a factorial k! to some power, and it is very unlikely that there are cancellations as in the case of maximal KAM tori for systems with finitely many degrees of freedom. Such trees look like self-energy graphs in quantum field theories, that is they are subgraphs G which can be represented as sets of lines
121
and nodes between two external lines such that (i) the external lines have the same propagators and (ii) the latter are larger than the propagators of all the lines of G. Analytically such subgraphs can be associated with numerical values of the form M('))n,m ) ~if ~ g ( n,, m) are the propagators of the form of the external lines, and they give rise to contributions to g ( n , m)M('I)(n,m)wc2,with kl IC2 = k - 1. Then we can try a different approach with respect to naive power series expansion (8). Write the P equation as
witk
+
(-u2n2
+ 5:)
wn,m = pvmwn,m
+~ [ + ( u ' ~ ) I n , m ,
(10)
where we have added and subtracted a term vmwn3,, inserted a factor p = 1, and written GL = w i - v,; we call counterterms the quantities v,. Let us neglect for the moment the constraints p = 1 and LZL I = w; - v,, and look for a periodic solution analytic in p, by considering v, as free parameters. We expand also the parameters v, in powers of p , so writing (10) to order k as
w p k = g ( n ,m)
c
vgl'wiy$
+ .[+(.., &)p;l),
(11)
kl+kz=k-l
and obtaining a tree expansion analogous to the previous ones, with some obvious changes; for instance now the propagators g ( n , m ) are of the form g(n,m) = ( -u2n2+WL)-'. Then we use v g l ) to compensate the dominant contribution of the value M ( ' l ) ( n , m ) of the self-energy graphs: of course To really this fixes the counterterms as functions of p, E and {W,,},,. implement such a construction requires some work, and again we refer to the original paperg for an exhaustive discussion. At the end one finds that the power series expansion in p is convergent, provided the following Diophantine conditions (known respectively as first and second Mel'nikov conditions) are imposed for some positive constants COand 7:
> C O J ~ J -vlnl ~ # J w nf 5 , fGm/J> CoJnl-' V'Jnl# Jmf m'J. Iwn*GrnI
(12)
The radius of convergence is found to be of order of an inverse power of this means that for E small enough one can take p = 1. As we said, the expansion turns out to be well defined only if the coeffiare chosen in a precise way. Hence one has to invert the relation cients 6%= m 2 - V ~ ( { W ~ ~which } ~ ~ is) not , a trivial implicit function problem as the frequencies W, have to be such that the Diophantine conditions (12) are satisfied. The inversion problem can be solved by following an iterative E:
vg'
122
and so scheme, that is by defining (;rE)2 = m2,W g ) 2 = m2 - v,({w~,},~), - (0) on: in general at step p one has W%l2 = m2 - v,({W~~~)},~), so t h a t one has to eliminate some values of E by imposing Diophantine conditions like (12) with W2-l) replacing 5,. Then we have t o show t h a t the procedure converges, and that the Cantor set E of values of E which are left at the end has large relative measure in the interval [0,E O ] ; see the original paperg for a discussion of these issues. Of course the larger is r the larger is the relative measure of admissible values for E (but of course the smaller is the value of E O ) : what we found is that any value of r > 1 is allowed.f Finally, coming back to problem 2, we note t h a t what emerges from the analysis is t h a t the Diophantine conditions (12) have to be imposed only in certain cases. An important issue is that the first and second Mel’nikov conditions have t o be imposed only when 1721 # Iml and 1721 # Im f m‘l, respectively. Furthermore when imposing the first Mel’nikov condition one has t o use that both w and W, differ from integers by terms of order E , so that wnf(;r, can be small for 1721 # Iml only if 1721 and Iml are large enough (of order l / ~ )and , as a consequence one can choose COof order 1 (in E ) . Analogous considerations allow us t o deal with also t h e second Mel’nikov conditions, again with a constant COindependent of E .
References 1. M. Berti and Ph. Bolle, Comm. Math. Phys. 243,315 (2003). 2. M. Berti and Ph. Bolle, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 7, 519 (2004). 3. D. Bambusi and S. Paleari, J. Nonlinear Sci. 11,69 (2001). 4. J. Bourgain, Geom. Funct. Anal. 5 , 629 (1995). 5. 3 . Bourgain, Periodic solutions in nonlinear wave equations, Harmonic analysis and partial differential equations, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 1999. 6. W. Craig and C.E. Wayne, Comm. Pure Appl. Math. 46, 1409 (1993). 7. J.P. Fink, A.R. Hausrath and W.S. Hall, Proc. Roy. Irish Acad. Sect. A 7 5 , 195 (1975). 8. G. Gentile and V. Mastropietro, Construction of periodic solutions of the nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series method, to appear on J . Math. Pures Appl. (9). 9. G. Gentile, V. Mastropietro and M. Procesi, Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions, to appear on Comm. Math. Phys. Berti & Bolle announcement2 one needs 1 < T the proof slightly simplifies.
< 2:
in our case for such values of
T
123 10. S. Kuksin and J. Poschel, Ann. of Math. (2) 143,149 (1996). 11. B.V. Lidski’i and E.I. Shul’man, Funct. Anal. Appl. 22 332 (1988). 12. J. Poschel, Comment. Math. Helv. 71 269 (1996).
FUNDAMENTAL COVARIANTS IN THE INVARIANT THEORY OF KILLING TENSORS
J. T. HORWOOD Department of Applied Mathematics and Theoretical Physics University of Cambridge, Cambridge, United Kingdom C B 3 O W A jh4230cam. ac.uk
R. G. MCLENAGHAN* Department of Applied Mathematics Unaversity of Waterloo, Waterloo, Ontario, Canada NZL 3G1 rgmclena@uwaterloo. ca R. G. SMIRNOV* Department of Mathematics and Statistics Dalhousie University, Halifax, Nova Scotia, Canada B 3 H 355 Roman.Smirnov@dal. ca
D. THE+ Department of Mathematics and Statistics McGill University, Montre'al, Que'bec, Canada H 3 A 2K6
[email protected]. ca
This work is a further development of the invariant theory of Killing tensors (ITKT). We consider the vector space of Killing tensors of valence three defined on the Euclidean plane and present a complete solution to the problem of the determination of a set of fundamental covariants of the vector space under the action of the isometry group.
*Work partially supported by National Sciences and Engineering Research Council of Canada Discovery Grants t Work partially supported by National Sciences and Engineering Research Council of Canada in the form of a Canada Graduate Scholarship
124
125 1. Introduction The invariant theory of Killing tensors (ITKT) is essentially a pseudoRiemannian analogue of the classical invariant theory (CIT) of homogeneous polynomials (see Olver' for more details). Its main goal is t o study vectors spaces of (generalized) Killing tensors defined on a pseudoRiemannian manifold ( M ,g) of constant curvature under the action of the corresponding isometry group. Much work has been done recently in developing ITKT (see Refs. 2-6 and the relevant references therein). For a given vector space of (generalized) Killing tensors the first task in the study is the determination of a set of fundamental invariants (covariants) of the space (or an extension), which have the property that any other invariant (covariant) is an analytic function of them. Various methods can be used to solve the problem of the determination of a set of fundamental invariants (covariants), including the methods of moving frames, infinitesimal generators and undetermined coefficients. In this paper we study from this viewpoint the vector space of Killing tensors of valence three defined in the Euclidean plane lE2. More specifically, we consider the action of the corresponding orientation-preserving isometry group I ( E 2 )on the extension of the vector space by the underlying manifold and solve the problem outlined above. Our notations are compatible with those used in Refs. 3 and 7. 2. Invariant theory of Killing tensors (ITKT) Let ( M ,g) be an m-dimensional pseudo-Riemannian manifold of constant curvature.
Definition 2.1. A symmetric contravariant tensor K of valence p defined on ( M ,g) is said to be a generalized Killing tensor (GKT) of order n iff
[[. . . [K, g],g], . . . , g] = 0 (n+ 1 brackets), where [ , ] denotes the Schouten bracket.
It follows immediately from the R-bilinear properties of the Schouten bracket that GKTs of the same valence and order constitute a vector space. Moreover, the GKTs of order zero are the standard Killing tensors which play an important role in differential geometry as well as various branches of mathematical physics, including general relativity, classical mechanics and field theory. We note that GKTs defined in flat pseudo-Riemannian
126
manifolds were originally introduced by Nikitin and Prilipko' and successfully employed in the study of evolution equations in field theory (see Ref. 9 for more details and references). Let ICP,(M)be the vector space of GKTs of valence p and order n defined on ( M ,g ) . Given that pseudo-Riemannian manifolds of constant curvature are locally flat, the dimension d of ICP,(M), according to the Nzkitin-Prilzpko (NP) formula, is given by d = dim ICg(M) = -
We immediately recognize that for n = 0 the formula (1) reduces to the Delong-Takeuchi-Thompson (DTT) formula (see Refs. 3 and 4 as well as references therein for more details). A general element of ICP,(M)is represented by d arbitrary parameters a l , . . . , ad with respect to an appropriate basis. The isometry group I ( M ) of ( M ,g ) induces a corresponding action on ICP,(M),which in turn induces an action on the parameter space CP, determined by a l , . . . ,ud. The same applies to the extended space ICP,(M)x M and the corresponding extended parameter space CP, x M defined by a l , . . . , a d ,x l , .. . ,x m , where x 1 . . . , x m are local coordinates on M . More specifically, the isometry group action on CP, x M is represented by the transformation laws iil = iil(ul,
. . . ,a d , g1 , . . . ,gr),
iid = iid(al,.. . ,a d ,g1 , . . . ,gT), 31 = 3c'(x', . . . , x m , g l , . . . ,gT),
3" - z-m (XI,.. . ,xm,gl,.. . ,gT),
where g l , .. . ,gr are local coordinates that parametrize the group I ( M ) . Note that T = am(, 1). The formulas (2) can be derived explicitly by employing the standard tensor transformation laws3.
+
Definition 2.2. An I(M)-covariant of the extended space ICK(M) x M , p 2 1, is an analytic function C : CP, x M -+ R satisfying the condition
c = F(a', . . . , ad,xl,.. . , x m ) = ~ ( i i l. ., . , i i d , 2 . . .
(3)
under the transformation laws (2) induced by the isometry group I ( M ) acting on M .
127
Similarly, an I(M)-invariant of ICg(M) is a function defined on Eg that remains unchanged under the action of the group. The following Fundamental Theorem of a regular Lie group action’ is employed to determine the number of fundamental invariants (covariants).
Theorem 2.1. Let G be a Lie group acting regularly on an m-dimensional manifold M with s-dimensional orbits. Then, in a neighbourhood N of each point xo E M, there exist m - s functionally independent G-invariants A,, . . . , Amps. Any other G-invariant Z defined near xo can be locally uniquely expressed as an analytic function of the fundamental invariants through Z = F(A1,. . . , Am-s). Quoting Peter Olver: “While invariants are of fundamental importance ... by themselves they do not paint the entire picture.”l Indeed, if the isometry group in question is not compact it is likely impossible to solve the equivalence problem (i.e., distinguish between the orbits of the group action) by means of (fundamental) I(M)-invariants only. Just like in CIT, in our study one can also employ both I(M)-invariants and covariants t o solve the equivalence problem. The concept of an I(M)-covariant of a vector space of standard Killing tensors was introduced in Ref. 5 and successfully employed to solve the problem of classification of the orthogonal coordinate webs defined in the Minkowski plane. In the following section we present a complete solution t o the problem of the determination of the I(E2)-covariants of the extended space Ki(E2)xE2 for the open submanifold of E: xE2 where the isometry group acts regularly.
3. The main result It is well known that any Killing tensor defined on a pseudo-Riemannian space of constant curvature is expressible as a sum of symmetrized products of Killing vectors. Thus, the three-dimensional vector space of Killing vectors KA(E2) admits a basis generated by the following Killing vectors given in terms of Cartesian coordinates x2,i = 1 , 2
where eij denotes the Levi-Civita symbol. We note the commutation relations
[Xi, Xj]= 0,
[Xi, R]= ~’iXj,
(5)
128
for i , j = 1 , 2 . In view of (4), the elements of the vector space Kb(IE2) have the general form
Ki = U i j k X i @ xj @ Xk + 3 b i j X i @ xj @ R +
@R
@R
where K i and KA are the general forms of the elements of K2(IE2) and K1(IE2), respectively. It is easy to check that the parameters that appear in (6) constitute four algebraic quantities, namely a = u i j k , b = bij, c = ci and e = e, i , j , k = 1,2, whose components represent the dimension of the vector space Kg(IE2). Clearly a, b and c represent S 3 ( N ) ,S 2 ( N ) and S ' ( N ) , where S ' ( N ) denotes the space of symmetric ( ~ ~tensors 0 ) defined over a manifold N of dimension two. It follows immediately from (6) that the parameters bij,ci,e and ci,e represent the dimensions of Kz(lE2) and K,$(IE2)respectively. Hence, we conclude, taking into account the symmetries of the algebraic quantities a and b, i.e. uijk = a ( i j k ) bij , = b(Zj), that dim Kg(IE2) = 10, dim Kg(IE2) = 6 and dim K,$(IE2)= 3 , as expected according to the NP formula (1). Next, we find the infinitesimal action of the isometry group I(IE2) in the space of parameters Eg N RIO determined by uajk,bij, ci and e that appear in (6). In view of the standard parametrization of the isometry group I(IE2), the transformation from one system of Cartesian coordinates xi to another system ?a is given by xi = A i j 2 + bi, where A i j E S 0 ( 2 ) , 6a E R2, i , j = 1,2. Since I(IE2) acts on Kg(IE2) as an automorphism, the induced action of the group on Eg is found by compar- 3 . ing the formula for the generic K i given by (6) with K Ogiven with respect to the coordinates 22: e = e,
where pi = € j k A ' a b k . We observe that the induced action of I(E2) on Eg is a subgroup of GL(lO,R), which is consistent with the result of Theorem 3.5 of Ref. 4. Using (7) and standard techniques from Lie group theory, we
129
derive the infinitesimal generators of the group action in E::
U2 = 3b"-
d dull1
2
d
d
d
+ 2b12-dull2 + b22-da122 + 2c1-abll
d
+
d
c2s + es7
d ac2
8
+ c --cl-. ac1
As expected from Theorem 3.5 of Ref. 4,the generators (8) satisfy the same commutator relations as the Killing vectors (4),namely
[Ui, U j ]= 0 ,
[Ui,V ]= EJiUj,
(9)
for i , j = 1,2. The problem of the determination of the I(E2)-covariants of the extended space Ici(E2) x E2 amounts now to solving the system of
PDEs Ui(F)
+ X i ( F )= 0 ,
+
V ( F ) R ( F )= 0
-+ R is an analytic determined by the vector fields (4)and (8), where F : f ~ n c t i o n l >We ~ . note that the group I(IE2) does not act regularly, but we can investigate the generic case by restricting attention to the open (invariant) submanifold where the group acts regularly and with three-dimensional orbits. Applying Theorem 2.1 on this open submanifold, we expect to derive nine fundamental I(IE2)-covariants that determine the space of all of the covariants of the extended space Ici(E2) x E2. Solving the system of PDEs by the method of undetermined coeficients (see Ref. 6 for more details), we arrive at the following result.
Theorem 3.1. Let Ki(E2) x E2 be the eztended vector space of Ici(IE2). A n y algebraic I(IE2)-covariantC defined over the open submanifold of I$ x E2 where the isometry group I(E2) acts freely and regularly with threedimensional orbits can be locally uniquely expressed as a n analytic function C = F(A1, A2, A3, A4, As, As, A7, As, As), where the fundamental I(lE2)-
130 covariants Ai, i = 1 , . . . , 9 are given by
A1 = e , A2 = c'ci - b'ie, A3 = 6ikEjjebij(2ckCe- bkee), A4
=~
As
= aijk(aijke2- 6bijcke
i k ~ j e { ~- 6(3bkme ~ ~ ~ -[ 2ckcm)ce] 3 ~ ~ ~ ~ e ~ 6bijbmm(bkee- 3ckce)+ 6bke(bijcm% - aimmc?e)},
+
+ 4c2c?ck) + bii[bjj(3ckck- 2bkke)
- 6bjk(c?ck - V ' e ) ] ,
A6
=
+
(uiijajkk - aijka,jk)(cece - beee) 2aiij[ajke(ckce- b"e) (10)
+
- akke(>ce - V e e ) ] &ik€je[aimn~kbmnCe
- aimm(bjkbencn + Vnbknce)]- EikEjeEmp~nqbijbkebmn bpq
1
A7
= ~ik~je{uijm[amnn(ck cb"e) e
+ 3aimmaikn(cecn
+ 3Vkbencn)e
-
-
+ akem(cncn - brine)]
bene)}e - 2EikEjeaimm[(VnbknCe
4V3[k~n1cecn] - 3biib3j(bkeck - bkkCe)Ce 2aiimbnPe)cq biibmmbnne],
+
- 2€ik€jebk e [EmpEnq(biibmnCp
+
+ exzxi,
As
= bii - 2~ijcZxj
Ag
= Eikfjebijbke
+ 8€ikbj[i$1Xk
-
2(cicj - bije)xixj.
Corollary 3.1. Any algebraic I(IE2)-invariantZof the vector space IC2(IE2) defined over the open submanifold of E; where the isometry group I(IE2) acts freely and regularly with three-dimensional orbits can be locally uniquely expressed as an analytic functionZ = F(A1, Az, A3, A4, As, As, A7), where the fundamental I@')-invariants Ai, i = 1,.. . , 7 are given by the formulas (10). Corollary 3.2. Let Ki(IE2) x IE2 be the extended vector space of Kg(IE2). A n y algebraic I(E2)-covariant C defined over the open submanifold of x IE2 where the isometry group I(IE2)acts freely and regularly with threedimensional orbits can be locally uniquely expressed as an analytic function C = F(A1, A2, A3, A,, As), where the fundamental I(IE2)-covariants A,, A,, A3, A, and A9 are given by the formulas (10). Proof. The result follows immediately from Theorem 2.1, the formulas ( 6 ) and (10). Note that here we have assumed the general element of ICg(IE2) 0 is written: @, = 3 b i j X i 0X j +3ciXi 0 R + e R 0R.
131 We note that the result of Corollary 3.2 is compatible with the corresponding result obtained in Ref. 5 by making use of the method of moving frames. 4. Conclusions
T h e results exhibited in the previous section constitute the first step in a comprehensive study of the vector space Icg(lE2), ultimately leading to a complete classification of its elements by I@’)-invariants and covariants. This has been done by determining reduced I@’)-invariants of Icg(lE2), which are the I(lE’)-invariants on the invariant submanifolds where the group orbits degenerate (i.e. are maximal). These results will be published elsewhere.
References 1. P. J. Olver, Classical Invariant Theory, London Mathematical Society, Student Texts 44, (Cambridge University Press, 1999). 2. R. G. McLenaghan, R. G. Smirnov and D. The, J. Math. Phys. 43, 1422 (2002). 3. R. G. McLenaghan, R. G. Smirnov and D. The, J. Math. Phys. 45, 1079 (2004). 4. R. G. McLenaghan, R. Milson and R. G. Smirnov, “Killing tensors as irreducible representations of the general linear group,” to appear in C. R . Acad. Paris Se‘r. I Math. 5. R. G. Smirnov and J. Yue, “Covariants, joint-invariants and the problem of equivalence in the invariant theory of Killing tensors”, to appear in J. Math. Phys., math-ph/0407028. 6. J. T. Horwood, R. G. McLenaghan and R. G. Smirnov, “Invariant theory and the geometry of the orthogonal coordinate webs in Euclidean space,’’ in preparation. 7. B. O’Neil, Semi-Riemannian Geometry. With Applications to Relativity, (Academic Press, 1983). 8. A. G. Nikitin and A. I. Prilipko, “Generalized Killing tensors and the symmetry of the Klein-Gordon-Fock equation”, preprint-90.23, Acad. Sci. Ukr. SSR., Institute of Mathematics, Kiev, 1990, 59 pages. 9. W. I. Fushchich. and A. G. Nikitin, Symmetries of Equations of Quantum Mechanics, (Allerton Press Inc., 1994).
G L O B A L GEOMETRY OF 3-BODY TRAJECTORIES WITH VANISHING A N G U L A R MOMENTUM
WU-YI HSIANG Department of Mathematics University of California Berkeley, C A 94720-3840, USA
Let m = ( m l , m 2 , m 3 ) be the percentages of masses of a given 3-body system (thus having C mi = 1)and {al,a 2 , a3) be the triple of position vectors of its three particles. Fully utilizing the translation symmetry and the conservation of linear momentum, we shall always assume t h a t C mzai = 0 and call such a triple {ai} an m-triangle. The classical 3-body problem of celestial mechanics studies the global geometry of those trajectories of such m-triangles under the gravitation-force, which can be concisely represented by the Newtonian potential function
Such trajectories are locally characterized by the Newton’s equation:
Note that most problems in classical mechanics study the correlations between “initial conditions” and “global geometric behaviors”, while the following “boundary value problem” is certainly one of the simplest and also the most basic type of global geometric problem on trajectories, namely, for a given pair of points { P , Q } in the configuration space M , what are those trajectories going from P t o Q? Inspired by the Fermat’s principle of least time of geometric optics, a kind of least action principle for classical mechanics was proposed by Leibniz, Euler, Maupertuis and Lagrange, t o characterize the above trajectories. The following precise mathematical formulation was due t o Lagrange: Lagrange’s L e a s t A c t i o n Principle: The solutions of the above boundary value problem are characterized by the extremality of the following
132
133 action integral
J1[7]=
s,
Tdt,
T
= kinetic energy
(3)
among those virtual motions with the given pair of terminal points and the same constant of total energy. In 1840, again inspired by results in geometric optics, Hamilton formulated another least action principle, namely
Hamilton’s Least Action Principle: The solutions of the above boundary value problem are characterized by the extremality of & [ y ]=
s,
(T
+ U)dt,
(often written as
I
Ldt)
(4)
among those virtual motions with the given pair of terminal points and the same time interval. 1. Jacobi’s reformulation of Lagrange’s least action principle, a profound geometrization of mechanics Jacobi first took the important step of introducing the concept of kinematic metric on the configuration space, namely, defining the metric ds2 in terms of the kinetic energy T by setting
2T ds2 = -dt2
( p : total mass)
P
(5)
For example, in the case of an n-body system
where mi (resp. (xi,yi, zi))is the “percentage of mass” (resp. the Cartesian coordinates) of the i-th particle. He then proceeded t o reformulate Lagrange’s least action principle which amounts t o a profound geometrization of mechanics. Let U be the potential function. For each given constant of total energy h, set
M(U,h)= {x E M ; h
+ U(Z) > o},
d s i = ( ht U ) d s 2
Then
dSh
=d
m d s =a d s
(7)
134
Therefore, just in one stroke, Jacobi succeeded to transform the Lagrange’s least action principle into a simple geometric statement, namely, trajectories of a given h as its total energy are exactly those geodesic curves in the metric space (M(u,h),d s i ) of ( 7 ) . Jacobi’s remarkable reformulation not only accomplished a complete geometrization of classical mechanics, but also initiated a new topic of geometric study of fundamental importance, namely, the study of global geometry of geodesics in certain metric spaces which is, nowadays, called Riemannian manifolds, while the study of this topic is often called the Morse theory of geodesics. The Jacobi equation along a geodesic curve remains to be the fundamental equation for analyzing the global geometry of geodesics. 2. Conservation of angular momentum and the orbital geometry of ( S 0 ( 3 ) ,M ) The rotation group SO(3) acts isometrically on ( M , d s 2 ) , and also on (M(u,h),d s i ) provided that U is SO(3)-invariant. Therefore, the orbital geometry of (S0(3), M ) plays an important role in both classical and quantum mechanics. (1) Tangential part: Let {e,, a = 1,2,3} be a chosen coordinate frame, S0(2), be the rotation subgroup with e, as the axis and K, be the Killing vector field generated by S0(2),. Then, along a given trajectory y
=
C m i ( e , x ai) . a i = Ernie,. (ai x ai) = e,
.R
(9)
where R = Cmiai x & is the angular momentum. Therefore, the conservation of angular momentum is just the Noether theorem which proves the constancy of < K ( y ) ,+>. In the special case of vanishing angular momentum, the geodesic curve y is always perpendicular to the SO(3)-orbits.
(2) Normal part: The normal part of the SO(3)-orbital geometry can be concisely organized into an induced metrics on the orbit space M / S 0 ( 3 ) ) measures the infinitesimal distances between (resp. M ( u , h ) / S 0 ( 3 )which nearby orbits, called the orbital distance metric. In the case of a 3-body system the orbit space % is exactly the moduli space of congruence classes of m-triangles, while ds2 provides a natural measurement of their differences in sizes and shapes. (3) The phase (i.e. connection) structure: Note that the lifting of a given closed curve in to a virtual motion with R = 0 (i.e. perpendicular to orbits) is, in general, no longer closed. This phenomenon is measured
135 by the kinematic phase (i.e. connection) structure whose understanding will play an important role in fully utilizing the conservation of angular momentum.
3. Kinematic geometry of rn-triangles The following is essentially a summary of some of the results of two unpublished papers [Hsi-11' [H-SI3 on the kinematic geometry of m-triangles: (1) Basic invariants and basic formulas: An m-triangle is a triangle with a center (i.e. of gravity) fixed at the origin, which subdivides it into a triple of subtriangles. Set Ij (resp. Tj, 0,) to be the individual moment of inertia (resp. kinetic energy, angular momentum) and aj (resp. Aj) to be the centrial angles (resp. areas) of the subtriangles. Then, the classical Ceva theorem asserts that Aj = m j A and it is useful to retool the usual trigonometry into the following Ceva-type trigonometry, namely Ceva sine law: sinal --
sinaz
- --
-
sinas --
2A (10)
mlhl m21a21 m31a31 (all.b 2 t . (a31 Ceva cosine law: (z,j,k ) (1,2,3) (i.e. cyclic permutations) N
- 2 d s c o S c Y k = miIi
+ m j I j - mkIk
(= &)
Ceva Heron formula:
L e m m a 3.1. Suppose that x = {aj} is non-degenerate ( i e . Q Set A0 2 A 1 2 A2 t o be the eigenvalues of ( h @ ( x ) ) , ha@(.) K " ( x ) ,K @ ( x ) > .T h e n A0
=I=XIj,
A1
+ A2 = I,
A1
. A2
= 4mlm2m3A2
#
0).
=< (13)
Corollary 3.1. A 1 = A2 when and only when x attains the maximal area among m-triangles with the same I. L e m m a 3.2. Let w j be the angular velocity of aj of a planary virtual motion with 0 = 0 . T h e n 1 w' - -(Ik&j - Ijhk), ('i,j,k ) (1,2,3) (14) a I N
136
Lemma 3.3. Set p = J? and M' to be the subspace of setting I = 1, da2 = ds2 Then
IM*.
dS2 = dp2
+p2da2
(i.e.
defined by
is the Riemannian cone of M ' )
CmiIjIkdI? -
&IidIjdIk 2
C o r o l l a r y 3.2. Set dA to be the area element of ( M ' , d a 2 ) . Then
( 2 ) B a s i c Theorems on kinematic g e o m e r y of m-triangles: T h e o r e m 3.1. (Kinematic Gauss-Bonnet formula) Suppose that the shape curve y* of a virtual motion y with 0 = 0 constitutes the boundary of a domain D in M * . Then
l, IL widt =
2dA = 2 . Area of D
T h e o r e m 3.2. (Sphericality) The moduli space of shapes of oriented m triangles as always isometric to the Euclidean sphere of radius namely
i,
( M * , d o 2 )2 S 2 ( $ ) S 3 ( l ) / U ( l )
(19)
T h e o r e m 3.3. Let bjk be the point representing the shape of ( j ,k)-binary collision and 6* be a given point in M ' . Set oi (resp. si) to be the distance between 6* and bjk in M * (resp. the side-length of 6' opposite of the i-th vertex) and to be the distance between baj and bik in M'. Then
C o r o l l a r y 3.3. The poles relative to the equator of degenerate m-triangles (i.e. with A = 0 ) is given b y ua = 2 (i.e. Ii = f ( l - m i ) ) , i = 1,2,3, which has the maximal area under the constraint of I = 1. It is convenient to identify M' with S2(1) by a magnification of factor 2, thus representing points of M* by unit vectors.
137 Corollary 3.4. Let bi (resp. p) be the unit vector representing a given shape of m-triangle). Then
bjk
(resp.
4. Shape curves, cone surfaces and the geodesic equation of
(M(U,h)ds;) 7
Let y ( t ) be a trajectory with R = 0, y ( t ) be its image curve under the and y * ( t ) be the image curve of y ( t ) under orbital projection p : M -+ the radial projection of %= cone of M* onto M * . Geometrically, ?(t) (resp. y * ( t ) )records the changing of both the sizes and shapes (resp. the shapes only). We shall call the former (resp. latter) the associated moduli (resp. shape) curve of y. It follows readily from the geometric meaning of R = 0 that y ( t ) is uniquely determined by y ( t ) up to a global motion of S 0 ( 3 ) , while y ( t ) is characterized by the geodesic equation of ( % ( ~ , h ) ,ds;). Therefore, the study of global geometry of trajectories with R = 0 can be completely re) , duced to that of the geodesics (namely, the Morse theory) on ( p ( ~ , h ds:), which is a conformal modification of d s 2 ) by the factor of h U . Note that (m,di?) is a Riemannian cone of ( M * , d a 2 ) , while ( M * , d a 2 )2 S2(a) (cf. Theorem 3.2). It is advantageous to introduce the cone-surface construction to facilitate the usage of the above geometric simplicity, thus further reducing the global geometric study to that of the shape curve y*.
m
(x,
+
-
Definition 4.1. The associated cone surface of 7 is defined to be the cone over y* c M * , consisting of all those rays passing through y * ( t ) , namely,
C(Y) = C(Y*)= { O y * ( t ) ) . Set a to be the arclength parameter of y* and .(a) to be the value of U at ?*(a).Then ( p , a ) can be regarded as a polar coordinate system of C(y*),while the restriction metric on C(y*) is given by
( +“”)
dS2 = h
(dp2
+ p2da2)
138 Theorem 4.1. The geodesic equation of ( z ( u , h )dg:) , is given by
where Q is the angle between and the tangent of 7, and u (resp. the unit normal (resp. geodesic curvature) of y* in M'.
K*)
is
Remarks: (i) The first ODE is just the geodesic equation of C(y*), while the second ODE means the normal sectional curvature of 7 c C(y*) is always zero. (ii) Note that both y ( t ) and y * ( t ) are originally given in terms of time, while the above ODE'S are in terms of the arclength parameter s. However, they are simply related by = fl= Moreover
Jm.
2
and it follows readily from (23)2 that
--u*/
p3 = I d 2 du
dy*
IE*
(25)
thus providing a simple formula to reconstruct y ( t ) from the local geometric invariants of y* and &U*.
Theorem 4.2. A 3-body trajectory y ( t ) with Cl = 0 and a given total energy h is uniquely determined up to congruence by the subset { y * ( t ) } (i.e. nonparametrized shape curve) and the initial speed 5. Formula of VU* and a monotonicity theorem Set ki = 2(mjmk)$(l - mi)-i and
Pi
to be given by (20)3,
Then, it follows from (21) that
vu*= UT = u - ( u . p ) p ,
pE
S2(1)
(27)
139 Corollary 5.1. Let po (resp. pb) be a minimal point of U * . T h e n {PO,pb} represent the pair of equilateral m-triangles with I = 1 and opposite orientations. Moreover
Lemma 5.1. For a unit vector p other t h a n {pa,pb}, set V(P) = P x [(P - Po) x (P - Pb)l
(29)
Then
Corollary 5.2. u(p) . (p - pa) i s always strictly positive for a n y p other t h a n {Po, Pb) Note that the pair of minimal points {pa,pb} are no longer situated at the poles { N , S } in the case of non-uniform mass distribution. However, there exists a unique Mobious transformation r which maps po to N , pb to S and the equator of degenerate shapes to itself.
Definition 5.1. For a given m = ( r n l , m n , m s ) , the m-modified latitude of p E S2(1)is defined to be the latitude of r(p) in radian. Theorem 5.1. Let y * ( t ) ,a 5 t 5 b, be a segment of the associated shape curue of a 3-body trajectory with R = 0 , A,*(t) be the f u n c t i o n recording the m-modified latitudes along y*[a,b]. Suppose that a < to < b i s a critical point of A*, (i.e. A$ ( t o ) = 0). T h e n A*, ( t o ) m u s t be a local maximum (resp. minimum) when y*(to) is situated inside of t h e northern (resp. southern) hemisphere. [We refer to [Hsi-2I2 for the proofs of all the above results.]
6. Some problems on global geometry of geodesics in (M(U,h) 7 da;)
The above discussion shows that the study of global geometry of 3-body trajectories with R = 0 can be completely reduced to that of geodesics of ( = ( ~ , h ) ,d S i ) which can be further reduced to that of the shape curves. We mention here some natural problems of such an approach in the following:
140
Problem 1: T h e existence (resp. uniqueness) problem o n t h e shortest path in ( x ( u , h )d,s ; ) linking a given point P t o the base point 0. Problem 2: L e t S be the set of those shortest geodesic segments in (ZTu,o), d s ; ) (i.e. h = 0 ) between p 6 F (eclipses) and 0, and S* be their associated shape curves. Is it true that S* covers MT \ {PO}simply? Problem 3: Let xo be a generic point of b%h, h = -1, and yzo be the geodesic curve starting at XO. H o w t o provide simple and useful lower bound estimate of the distance between xo and the first focal point in t e r m s of the geometric invariants of XO? Problem 4: Is it true that a 3-body trajectory with a closed nonparametrized shape curve necessarily periodic? Is a periodic traject o r y already uniquely determined u p t o congruence by its associated nonparametrized shape curve? Problem 5 : W h a t i s the minimal number of eclipse points o n a closed shape curve (resp. periodic trajectory) ? Moreover, i s a n y even number at least equal t o t h e minimal number can be realized as such a number? Problem 6: W h a t are those homotopy classes of closed curves in M* \ {bl, bz, b3) that can be represented by closed shape curves (resp. periodic trajectories)? Definition 6.1. Let y* be the non-parametrized shape curve of a complete trajectory r[-m,m] with R = 0, and D ( y * ) be the closure of y*. We propose to define the area of D ( y * ) to be the chaoticity of y. Problem 7: W h a t are the non-chaotic (i.e. with zero area of D ( y * ) ) trajectories other t h a n those with closed non-parametrized shape curve? Problem 8: W h a t are the realizable values of chaoticity? References 1. W.Y. Hsiang, Geometric study of the three-body problem, I, PAM-620 (1994) Center for Pure and Applied Math. preprint series, Univ. of Calif., Berkeley. 2. W.Y. Hsiang, O n the global geometry of 3-body trajectories with vanishing
angular momentum, I, preprint (2003). 3. W.Y. Hsiang and E. Straume, Kinematic geometry of triangles with given mass distribution, preprint, May 1995, Univ. of Calif., Berkeley.
THE RELATION BETWEEN THE TOPOLOGICAL STRUCTURE OF THE SET OF CONTROLLABLE AFFINE SYSTEMS AND TOPOLOGICAL STRUCTURES OF THE SET OF CONTROLLABLE HOMOGENUOUS SYSTEMS IN LOW DIMENSION
ABDELOUAHABKADEM Department of Mathematics, Faculty of Science University of Setif, 19000 Setif, Algeria abdelouahabkQyahoo. fr
In this paper, we establish the existence of a relation between the topological structures of the set of controllable affine systems denoted by C, and topological structures of the set of controllable homogeneous systems Ch. We consider the following affine system X = A X u(Dx b) (1) x E R 2 , b E R 2 ,U E R .
+
{
+
where u is a piecewise constant control with values in the subset 0 of R , bounded or not; A and D are two real 2 x 2 matrices. To (1) we associate the homogeneous system X= ( A uD)x
{
+
2
E R2
- (0)
,u E
R.
(2)
It is known that if (2) is controllable in R2 - ( 0 ) and ( 1 ) has not a fixed point, then (1) is completely controllable in R2 z. It turns out that in certain cases (1) can be controllable but (2) is not controllable, but that these cases are marginal. The set of such pairs is the boundary of the set of controllable pairs, the interior being constituted by pairs for which (2) is controllable. It is also known on the one hand that the set C h is connected and its boundary present two types of boundary points, on the other hand the set C, is connected. The question is: can one establish a relation between the boundary of C, and the one of Ch ?
1. Preliminaries and examples In this work we establish a relationship between the topological structures of the set of the controllable afine systems that we will note C, and the set of controllable homogeneous systems noted ch.
141
142
We consider the affine system X= AX + U ( D Z+ b ) XER',~ERU ',ER.
and the homogeneous system
according to the technique of extensions instead of studying (l),we are going to study the finite family ( A , f ( D b ) ) , which is equivalent to (1) from the controllability point of view. We will therefore identify a system by the triplet ( A ,D, b ) element of RIO provided with its usual topology. It is known that if (2) is controllable on R2 - (0) and (1) has not a fixed point, then (1) is completely controllable on R2 2. These systems, which are denoted by 9, form an open set in RIO and form also an open subset of the set of controllable affine systems; let us remind however that this last one is not open, i.e. that generally controllable affine systems do not remain such under the influence of arbitrarily small perturbat ions. It seems that in some given cases (1) can be controllable unless the system (2) be it also, but these cases are marginal. In RlO,the set of such pairs is the border of the set of the controllable pairs, the interior being constituted by pairs for which (2) is controllable. One can surmise that this result is true in general for the affine systems. We know on the one hand that the set of the controllable homogeneous systems is connected and its border presents two types of boundary points such as the border of its outside and the inside of its closing3; on the other hand, the set of affine controllable systems C, is connected '. In this work we establish a topological relationship between and As motivation one is going to give some examples of the set of the controllable affine systems, or not controllable, having their part on the but previously let us give a result of controllability. border of
+
ch
c,.
ch,
1.1. Proposition
+
Let C be a trajectory of &(D b ) ; the affine system (1) is controllable on R2 if and only if the vector field A points to each of the two regions determined by the curve C.
143
1.2. Example: Consider
A=
[ii]
; b=(bl,b2)
,D=
, b2>0.
Let z ( t ) be a point of a trajectory C. The tangent vector to C at z ( t ) is D z ( t ) b; an orthogonal vector to this one is J ( D z b ) , where
+
+
the affine system (1) is controllable on R2 if and only if the expression < A ( z ( t ) J, ( D z ( t ) b ) > changes sign for t E R. In fact the trajectories of (D b) are parabolas, their common axis is a straight line passing via the point ( O , - - b l ) , and parallel to the abscissas axis, their concavity is directed towards positive abscissas. All the trajectories of *(D b ) cross the abscissas axis; let (zo, 0) be a point of the abscissas axis and C is the trajectory of f(D b ) passing via this point. The equations of C are (t E R):
+
+
+
{ qq (( tt ))
+
= (b2/2)t2
+blt +
50
= bat ;
+
an orthogonal vector at s ( t ) E C is D z ( t ) b and has components (bat + b l , b 2 ) An orthogonal vector to this one is T'(t) = J ( D b ) ; it has as components (-b2, b2t b l ) . The scalar product of 2 ( t ) and the vector A z ( t ) is equal to f ( t ) = -$tZ - b l b z t - b 2 z o . The quantity A = bg(b? - 2 b z z o ) can be made negative by a good choice of zo, and f ( t ) doesn't change sign; which means that the affine system (1) is not controllable on R 2 . As for the homogeneous system (2), it is not controllable on R2 - (0): as a matter of fact b = 0 means geometrically that the fields A and D have a proper axis in common; that is one has an invariant zone.
+
+
1.3. Example:
Let consider A =
[ik], D =
, b=
(b1,b2), b 2 f O .
To study the controllability of theaffine system (1) one is going to study the intersection of the trajectories of f(D b ) and the curve defined by < J ( D z b ) , A z >= 0. Actually the trajectories of f(D b ) which have
+
+
+
144
+
+
as equation: X I = x:eXlt and 2 2 = b2t x8. The trajectories of f(D b) consist of (i) the ordinate axis; (ii) logarithmic curves with the ordinate axis as an asymptote. We have A’ = 0 and < J ( D b ) , A x >= -b2x2. As each trajectory of f(D b) cross the abscissa axis, the affine system (1) is controllable on R2. As for the homogeneous system (2), it is not controllable on R2 - (0): in fact bc = 0 means geometrically that the fields A and D have common proper axis; that is one has an invariant zone.
+
+
So, we gave two examples of non controllable homogeneous systems leading to either a not controllable affine system (example l),or to a controllable affine system (example 2). 2. The study of the boundary of the set of the controllable
affine systems C , The border of C, is constituted solely of affine systems having their homogeneous parts at the border of Ch (i.e. the non controllable homogeneous systems). Let’s remind that the border of C, contains as well the controllable affine systems as not controllable affine systems; one can wonder to what type of border these systems belong knowing the type of border on which their homogeneous part is. Contrarily to what one would expect, three types of borders because C, is not an opened it appears in this border only two types of frontier points, which are: bl = FrC, = FrEzt(C,) where contains a controllable affine systems or not controllable; 62 = FrC, n I n t c which contains a not controllable affine systems for which the trajectories of D are cycles. Let us note that in both cases the corresponding homogeneous systems are of types Frl of ch ’. The inside of C, is constituted: (i) on the one hand of affine controllable systems whose homogeneous part is controllable, i.e. systems S ; (ii) on the other hand of affine controllable systems to which correspond noncontrollable homogeneous system characterized by collinear fields (bounded spirals). Outside C, one finds the not controllable affine systems to which correspond homogeneous systems belonging to the outside of ch.
145
Finally, the set of affine controllable systems C, is described as being the disjoint union of (i) 9; (ii) the set of affine controllable systems belonging to the border of C;, and (iii) of system whose homogeneous part is constituted of bounded spirals. These last two parts being of empty interior, one deduces that systems 9 are dense in C., One will find in the table given in appendix all cases of border points of ch,to which correspond different situations of affine systems. In all these cases the homogeneous system is not controllable.
2.1. Commentaries Among the homogeneous systems belonging to the border of ch one observes three possible behaviors with regards to the passage to affine systems. These three behaviors correspond to different properties of the homogeneous system ( A ,fD)of the start. These three types of systems are: - Systems of ”affine controllability ” i.e. for which Vb E R2-{O}, ( A ,f(D+ b ) ) is controllable; these systems are characterized by the adherence of the homogeneous system to the inside of the closing of ch. - ”Ordinary” systems, i.e. for which it 3 b E R2 - ( 0 ) ; these systems are characterized by the adherence of the homogeneous system to the border of the outside of ch and D singular. - Systems of ”affine non controllability” i.e. Vb E R2 - { 0 } , ( A ,f(D b ) ) is not controllable; these systems are characterized by the adherence of the homogeneous system to the border of the outside of ch and regular D.
+
3. Study of border cases
Lemma 3.1. Let be the homogeneous system ( A ,fD)E F?-(EZt(Ch))then V b E R2 - ( 0 ) , the a f i n e system ( A ,f(D b ) ) E Fr(Ezt(Ca)).
+
Proof. We have
Then ( A ,fD)E
61n Cl,where
UI = { ( A ’ ,fD’) / A(D’) > 0, 0} ‘
146
Moreover,
/ A(D’) V1 = { ( A ’ ,fD’)
> 0 , [ ( A ‘ ,D o 2 - A(A’).A(D’) < 0} ‘.
u(1) and v(1)are open subsets, u1 C ch, and v1 C Ext(Ch). Let us denote U i = U1 n { ( A ’ ,fD’) invertible} x R2 - ( 0 ) . Then Ui is C,. In fact {(A’, fD’)invertible} is an open set of controllable systems in open set (dense), the intersection with U1 is so open; furthermore Ui c C,, because any system belonging to Ui has controllable homogeneous part and has no fixed point, {(A’,3 3 ’ ) invertible and b # 0 ) by application of the theorem of Jurdjevic-Sallet. Also V{ = V1 x R2 - ( 0 ) form an open set of not controllable system. -
-
-
+
Finally, by construction Ui = U1 x R2 and Vi = vl x R2,so ( A ,f(D
b)) E
6’1n
f’1,
+
i.e. the affine system ( A ,f(D b ) ) E Fr(Erct(C,)).
Let us remind that so that one has controllability of the homogeneous system ( 2 ) on R2 - ( 0 ) it is necessary that the following 3 conditions are satisfied: (i) rank condition (i.e. that fields A and D are free); (ii) radial controllability (i.e. existence of incoming directions and outgoing directions); (iii) directional controllability. So, ( A ,fD)E Fr(Ezt(Ch))is equivalent to A ( D ) 2 0 and [ ( A ,D ) 2 A(A).A(D) = 0; these is turn are equivalent to stating that system ( A ,H I ) does not verify the directional controllability condition (i.e. (A, * D ) is not controllable in R2 - ( 0 ) ) One arrives by generic perturbations at:
’.
{ ( A ’ ,fD’) / A(D’) 2 0 , [ ( A ’ ,D’)2 - A(A’).A(D’) # 0 } ; let U1 = { ( A ’ ,3 0 ) / A(D’) > 0 , [(A’,D’)2- A(A‘).A(D’) > 0} V1 = { ( A ’ ,+to’) / A(D’) > 0 , [ ( A ’ ,D’)2 - A(A’).A(D’) < 0 } U1 and V1 are open sets, then ( A , f D )
‘
n v1, where
U1 = { ( A ’ ,&to’) -
/ A(D’) 2 0 , [(A’,D’)2 - A(A’).A(D’) 2 0} V1 = {(A’,&D’)/ A(D’) 2 0,[(A’,D’)2- A(A’).A(D’) 5 0 } .
u1 is an open subset of ch , and v1 is an open subset of Ext(Ch) ’. Let us denote Ui = Uln{(A’, fD’) invertible} x R2-{O}. Ui is open set of controllable systems in C, , In fact { ( A ’ ,fD’)invertible} is an open set (dense) the intersection with U1 is so open; furthermore Ui c C,, because any system belonging to U i has the controllable homogeneous part and has
147
no fixed point, (A’,4zD‘)inwertible and b # 0 by application of the theorem of Jurdjevic-Sallet. Also Vi = Vl x R2 - ( 0 ) form one open set of not controllable system. -
-
Finally by construction Ui=fil x R2 and Vi=fl x R2 , SO ( A ,* ( D + b ) ) -
EU‘I n ? I , i.e. the affine system ( A ,f ( D + b ) )
E Fr(Ezt(C,)).
Lemma 3.2. Consider the homogeneous system ( A , f D ) E (Fr(Ch)-
F r ( C h ) ) ;two cases should be distinguished: i) zf A ( D ) < 0, trD = 0, Q(A.D) = 0 (i.e. the homogeneous system is not controllable), then: V b # 0 , ( A ,+(D + b ) ) E (Fr(C,) - Fr(C,)). # 0 , ( A ,D ) bounded (i.e. the homogeneous system is not controllable), then: V b # 0 , ( A ,&(D + b ) ) I n t ( C a ) .
ii) if A ( D ) < 0 , trD
Proof. The homogeneous system ( A , & D ) E (Fr(Ch) - F r ( C h ) ) H A ( D ) < 0 and either A = pD, t r D # 0 , or A ( A . D ) = 0 , t r D = 0. i) ( A , f D ) E (Fr(Ch)- F r ( C h ) ) ,then: ( A , f D ) E {(A’,*D’) / A(D’) < 0 , trD’ # 0 , A’ # pD’ }. Moreover U2 c Ch is open [4]; then let us note:
-
U2
where
U2
=
Ui = U2 n { ( A ,fD)invertible} x R2 - ( 0 ) which is an open of controllable systems in Int(C,), by application of the -
theorem of Jurdjevic-Sallet as first and one has the affine system ( A ,f ( D
+ b ) ) E (Fr(C,)
-
-
Ui = U2 x R2;therefore
Fr(C,)) .
+
ii) Let be ( A , & ( D b ) ) such that A ( D ) < 0, t r D # 0 , and A , D bounded, b # 0. Then 3~> 0 such that V ( A ’ , f D ’ ) E B ( ( A , & D ) , € )the , ball of centre ( A ,&D) and radius E such as one has: A(D’) < 0, trD’ # 0, (A’,D’) bounded or not. - if (A’,D’) is bounded, then (A’,+(D’ b ) ) is controllable V b # 0. - if (A’,D’) is free, then (A’,f ( D ’ b ) ) is controllable V b # 0. So V b # 0 , ( A ,f ( D + b ) ) E Int(C,) the homogeneous system ( A ,f D ) E
+
+
(Fr(Ch)- F r ( C h ) ) A ( D ) < 0 and either A = p D with t r D # 0 or A ( A . D ) = 0 with trD = 0. I7 Let us remind that ( A ( D )< 0 ; A = pD and trD # 0 means that the rank condition is not satisfied and A ( D ) < 0, A ( A . D ) = 0 , t r D = 0 means that the radial controllability condition is not satisfied) so ( A ,f D ) is not controllable in R2 - ( 0 ) ) ’.
148
i) ( A ,fD)E (Fr(Ch)- F r ( C h ) ) ;then one arrives by generic perturbations at: ( A ,fD)€ 6 2 where
U2 = { ( A ’ ,+to’) / A(D’) < 0 , trD’ # 0 , A’ u 2
is an open subset of
# pD’ }
.
chi then let us denote
U i = U2 n { ( A , f D ) invertible} x R2 - (0) which is an open of controllable systems in Int(C,), by application of the -
-
theorem of Jurdjevic-Sallet as first and one has U.=U2 xR2 , therefore the
+ b ) ) E (Fr(C,) - Fr(C,)) . ( A ,+ ( D + b ) ) such that A(D) < 0, t r D
affine system ( A ,f(D
ii) Let be # 0 , and A , D bounded, b # 0, 3~ > 0 such that V (A’,fD’) E B ( ( A ,fD), &)ballof centre ( A , f D ) and radius E such as one has: A(D’) < 0 , trD’ # 0 , (A‘,D’) bounded or not. - if (A’,D’)bounded, then (A’,f(D’ b ) ) controllable V b # 0. - if (A’,D’)free, then (A’,f(D’ b ) ) controllable V b # 0. 2 . SO V b # 0 , ( A ,f(D b ) ) € Int(Ca). We obtain as a consequence of the previous two lemmas the following theorem, which is a description of C,.
+
+
+
4. Main result
Theorem 4.1. The set of controllable afine systems verify the following three properties: i) C, = S U K U K’ (disjoint). ii) I n t ( C a )= S u K‘. zzz)
Fr(C,) = Fr(Ezt(C,)) u (FT(C,) - F r ( G a ) )(disjoint).
Here K denotes the set of controllable affine systems and of not controllable ones (to which corresponds homogeneous systems belonging to border outside ch),and K’ the set of controllable affine systems to which corresponds homogeneous system characterized by the bounded spirals.
Proof. i) Immediate. ii) I n t ( C a )= S U K’; as a matter of fact: a) S U K’ c Int(C,) because S c Int(C,) as S is open; K’ c Int(C,) by lemma 2. This implies that S U K’ c I n t ( C a ) ;b) Int(C,) c S U K’, because according to i), C, =
149
SUKUK'(disjoint). Let x E Int(C,) + x E C,; we know that K c FT(Ca) so x 6K. Therefore K Int(C,), so x E 9U K', and hence the result. 0 5 . Conclusion
Thus, by this study one got a satisfactory description of the set of affine controllable systems C, in dimension 2. It appeared a very strong relations between the topological structures of the set of affine controllable systems C, and of ch. In particular, the properties of FT(C,) and the connectedness of C, results directly from analogous properties for The result that one obtained is that on the one hand C, makes appear two types of border points - while it could have had a priori three types of border points. The inside of its closing contains affine non controllable systems for which trajectories are cycles, the remaining cases are on the border of its outside; on the other hand the inside of C, is constituted solely of systems 9 and affine controllable systems to which correspond the non controllable homogeneous systems characterized by collinear fields (bounded spirals). As far as applications are concerned, the most important result is the fact that 9 is a open and dense in C,. So, when modelling a controllable system by an affine system of type (l),one will be able to confine oneself to a system in 9. One can suppose that as in R2, the controllability of an affine system is nearly equivalent to the controllability of its homogeneous part. The central problem would therefore be the controllability of bilinear systems in R". It would be interesting to know how to establish in R" results analogous to those obtained in dimension 2 between sets C, and ch. Although one doesn't know how to describe ch, one can nevertheless put this problem. It is presumably a problem of method that puts itself since it doesn't seem reasonable to describe of in an exhaustive way Ch and a fortiori C,. The choice of adequate mathematical tools to demonstrate this conjecture remains an open problem.
ch.
References 1. D.E. KODITSCHEK and K.S. NARENDRA, The controllability of planar bilinear systems, IEEE trans. Automat. Control. AC-30 (1985), pp 87-89. 2. V. JURDJEVIC and G. SALLET, Controllability properties of affine systems, SIAM J. Control. Optim. 22 (1984), pp 501-508.
150 3. A. KADEM, Etude de la structure topologique des systQmes homogQnes contrblables, Maghreb Mathematical Review 10 (2001), pp 235-254. 4. A. KADEM, Etude de la connexitk de l’ensemble des syst6mes affines
contrblables; submitted. 5. G. SALLET, Sur la structure de l’ensemble d’accessibilit6de certains systhmes: Applications a l’kquivalence des systbmes, Math-System T h e o w 18 (1985), pp 125-133.
Appendix We give for each case: (i) the canonical form of D , (ii) the algebraic expression of the frontier point of ( A & D ) , and (iii) the resulting properties for ( A ,& ( D b ) ) being controllable on R2.
+
Case I: D =
xoZ],
:[
with XI
#
Xz.
Border points: ( ( A , D ) 2-
A(A).A(D) = 0 H bc = 0; then Vb E I m ( D ) ,( A ,f ( D + b ) ) is not controllable.
Case 11: D =
[::] [::]. [:i] +
with X
# 0.
Border points: ((A,D)’ -
A(A).A(D) = 0 H bc = 0; then 3b / ( A ,f(D+ b ) ) is controllable. Case 111: D =
Border points: ((A,D)’ - A(A).A(D) = 0 H
+
b = 0; then V b E I m ( D ) ,( A ,f(D b)) is not controllable. Case IV: D = b = 0; then 3b
/ ( A ,k(D
Case V: D = is controllable.
Case VI: D = Then Vb, ( A ,f ( D
. Border points: ( ( A ,0)’ - A(A).A(D) = 0 H
b ) ) is controllable.
[ i’]
. Border points: A = p D ; then Vb, ( A ,f(D+b ) )
[ .I’;
Border points: A(A.D) = 0; ( b + ~ ) ~ - 4 a = d 0.
+ b ) ) is controllable.
ON PRESERVATION OF ACTION VARIABLES FOR SATELLITE LIBRATIONS IN ELLIPTIC ORBIT WITH ACCOUNT OF SOLAR LIGHT PRESSURE*
I. I. KOSSENKO Moscow State University of Service, 99 Glaunaya Street, Cherkizovo, Moscow Region 141221, Russia
[email protected] Planar librations of a satellite, with its mass center moving in the elliptic orbit, are considered. T h e force of Solar light pressure is assumed to act on the satellite besides the gravitational one. The rotation of a dynamically symmetrical satellite is examined as an unperturbed one. Direct application of the KAM-theorem is impossible because Hamilton’s function has only two continuous derivatives. T h e existence of invariant tori and the preservation of action variables near their initial values are proven through the reduction of the perturbed Hamiltonian system t o the sequence of symplectic maps, and with the help of Moser’s theorem on invariant curves. Analysis of the limit case when the orbit eccentricity e is close t o one is carried out. T h e order of perturbation in this case is supposed t o be fixed. It turns t h a t in this case the action variables also conserve their values for asymptotically large time intervals.
1. Equations of motion
Librations of an asymmetrical satellite in the plane of the elliptic orbit’ are examined. Orbital motion is considered as a given one. The light source, i. e. the Sun, is considered as a point at infinity, so that the luminous flux has the same direction at all points of the orbit. Using known expression for the torque of the forces of light p r e ~ s u r ethe ~ >equations ~ of planar librations of a satellite can be presented in the form4
6 = c f ( 6 ) - p [ 1 + e cos 4t)13sin(6 - 2 4 t ) + 2 9 1 ,
(1)
*This paper was prepared with partial support of Russian Foundation for Basic Research, grants 02-01-00196, SS-2000.2003.1, and Ministry of Education of Russian Federation, grant T02-14.0-1054.
151
152
Here the angle 6 defines the orientation of the satellite with respect to an inertial coordinate system, v(t).is the true anomaly, a known function of time, c is a constant embodying the reflecting properties of the spacecraft outer surface, 9 is the azimuth of the light source position. The values of e, p, c, are parameters of the problem, and the value of p characterizes the dynamic asymmetry of the satellite. We assume that 0 5 e 5 1. The contribution of the light pressure is given by the function
1 - cosx for sin 2 0 -1 cosz for sing < 0 '
+
f
E CYR)
The obtained mechanical system having 3 / 2 degrees of freedom is defined by the kinetic energy
T ( &= ) &2/2, and by the force function
V(6,t ) = U ( S ,v ( t ) )= CF(S)
+ p [ 1 + e cosv(t)13cos(6 - 2 v ( t ) + 2 9 ) .
Here cF(6) is the force function of the light pressure torque which is simply the antiderivative of cf (6). The nonautonomous second order ODE (1) can be represented in the form of an autonomous system of the third order by adding a differential equation for the true anomaly. Finally Eq. (1) can be represented as an autonomous Hamiltonian system with two degrees of freedom. Indeed, assume q1 = 6, q2 = v, p l = 6. The second generalized impulse p2 is defined as an additional variable, canonically conjugated to 42 = v. In this case Hamilton's function has to take the form
+
H (41, q 2 , ~ 1 , ~ = 2 )~ 1 / 2 u (41, q2) + (1 e c 0 s q 2 ) ~132.
(3)
The system of differential equations of motion takes the canonical form q i = Hp,, pi = -Hpi
(2
=
1,2).
Let us examine the case where the parameter of dynamic asymmetry is small enough, p
7
at left and right points of the librations contour. It is evident from the constructions produced that the function q1 (w1,II) is an analytical one in intervals of the form (kx,(k 1)n) x { I I } . At points of the form (kn,11) analyticity is violated. Here smooth “sewings” are carried out, such that left and right derivatives of q1 ( w l ,11) on w1 coincide up to the second order. Derivatives of the third and higher orders have at these points an ordinary discontinuity. The canonical system with Hamilton’s function H02 (42, p 2 ) is also integrated in quadratures. The action variable 1 2 can be introduced according to the formula
+
where T is the period of a satellite orbital revolution. In the elliptic case using the selected physical units we have T = 27ra3/’ = 27r (1 - e 2 ) - 3 ’ z . Thus I2 = h2 (1 - e2)-3’z, and the generating function & (q2,I Z ) ,used to transform t o the action-angle variables with the aid of the canonical transformation, can be computed in the form
v, (42,121 = (1 - e
2 y 12t (qz) ,
where t ( 4 2 ) is a function which yields the value of physical time, which in turn corresponds t o the value of true anomaly 4 2 , if zero time corresponds to an instant when the pericenter is crossed. Thus, the angular variable w2 is defined according to the formula w2 = (VZ), = (1 - ez)3’2 t ( 4 2 ) = I , where 1 is the mean anomaly, and (1 - e z ) 3 / 2= n is the mean motion.
3. Existence of conditionally-periodic motions The canonical transformation ( q l ,qz,pl,p2) H (w1,W Z ,1 1 , Iz) to actionangle variables allows t o write down the Hamilton function as
+
G ( w ,1)= Go(1) p G i ( w ,1). Here
Go(1) = H0(4(w,I),p(w7~)) = hl(11) + h2 ( 1 2 ) I
(8)
156 Gl(W, I ) = Hl(41 (W1,Il) ,qz (wz, 1 2 ) ) ’ The frequencies of the unperturbed Hamiltonian system read
To apply the results of KAM-theory one can use one of the examples given in the Markeyev paper7 and then carry out an analysis of the sequence of symplectic maps “along” the fast angular variable w1 instead of examination of the continuous, but nonanalytic, dynamic system. We use a standard construction of the isoenergetic reduction’. We will perform the reduction in the vicinity of the unperturbed system torus, defined by the vector of action variables (Il0,Izo) = (11(0),12(0)). For this purpose let us switch from action variables to their deviations (11,Iz) H (J1,Jz) such that 1 i = I;, J z , i = 1 , 2 . The map, which generates cascade, will be constructed in “time” w1 E [0,27r]. In this case by virtue of the foregoing constructions the analyticity of the perturbation G1 (w, I ) is violated on the sets in phase space defined by equations w1 - k 7 ~= 0 , k E Z. Really there are only two of such “vertical” lines. They are defined respectively by the equations
+
w1 = 0
(mod 27r),
w1 = 7r
(mod 27~).
The Hamiltonian system under consideration has an analytical perturbation. We will conduct isoenergetic reduction separately in each of the enumerated phase space “strips”. Thus, within bounds of the analyticity regions the initial canonical system is reduced to the nonautonomous one W;
=Kjz,
L J
=
-K,,,
(9)
with one degree of freedom (indeed, 3/2 DOFs). Here the Hamilton’s function K (w1, w2, Jz) is defined implicitly, via the integral of energy for the source system from the transcendental equation G (w1, W 2 , I l O - K , 120
+ J z ) = h,
where the constant of energy for the perturbed problem is to be calculated using initial conditions according to the formula h = G ( ~ 1 0~, 2 0 , 1 1 0 I, ~ o ) , where, in turn, one should assume w10 = 0, or w10 = 7 ~ . It is clear that the function K depends analytically on the small parameter p inside the region of analyticity. Then the formula of expansion
K
(
,
~ wz, i
Jz)
= KO(Jz)
+ pKi (wi ,wz,Jz) + . . .
157
is valid in view of the implicit function theorem and the appropriate condition a h l / a I l ( 1 1 ~ )# 0 of monotonicity. Here the function KO(&) is implicitly defined by the LLunperturbed” transcendental equation hi (110 - K O )- h i
(110)
+ w2 J2
= 0.
Along the solution of the Hamiltonian system (9) in the segment w1 E [O, TI we obtain an analytic first return map (w2, J 2 ) H (w;, J;). In the segment w1 E [ T , 27r] we have also an analytic map (w;, J . ) H (w;, J - ) of the same type. The resulting map (w2, J2) H (w;, J l ) over the period w1 E [ 0 , 2 ~is ] a composition of analytic maps. Therefore it is also an analytic one. Performing a new expansion on the small parameter p we obtain
c
+
+
w; = WZ 2T Y ( J z ) p f (w2, J2, p ) J; = 5 2 + p g ( w 2 , J 2 , p ) .
7
-
(10)
It is easy to ascertain that Y (J2) = ( K o )( J~z )~. The function
52
y ( J 2 ) is an analytic one and not identically equal to zero in a vicinity
of the unperturbed torus. Indeed, if we represent the expansion of the k3J. ... unperturbed Hamiltonian in the form KO( J z ) = k152 k2J; then it is sufficient for the map (10) to be twisting, if p is sufficient small, that among the coefficients kz, ks, . . . there are some not equal to zero. In this case by virtue of Moser’s theoremg the map (10) has an invariant curve, which corresponds to the torus swept by the trajectory of conditionallyperiodic solution to the perturbed problem. In the simplest, typical case the condition k2 # 0 is equivalent to known isoenergetic nondegeneracy condition of the unperturbed system. After some tedious analytical computations one can verify that this condition is fulfilled in the region of (ql,pl)-phase space at some distance from the separatrix area and also far from stable equilibrium. Summarizing the results, one can say that the majority of the nondegerate tori for the unperturbed problem, which correspond to conditionally-periodic solutions, are preserved in view of the condition of nondegeneracy with sufficiently small p. As it is known”, according to dimensional reasons in this case stability takes place with respect to the action variables. They are preserved secularly, being only a little distorted. This phenomenon is correct in phase space in that measure, in which it is provided by the condition of a nondegeneracy. Since the manifold of constant energy divides phase space, it follows also from the obtained result that with sufficiently small p there exists an invariant vicinity of the equilibrium, stable in the unperturbed motion.
+
+
+
158
Thus all motions which have initial data in this vicinity, remain in it permanently. This means that a satellite will permanently perform bounded librations near the azimuth of the light source. In fact, if the projection of phase point on the plane of variables (41,p 1 ) begins to be moved away from vicinity of the equilibrium cited, then it will be unavoidable t o approach a separatrix. This region, in view of nondegeneracy, is filled with a n invariant two-dimensional tori, which separate the three-dimensional manifold of constant energy and therefore in such a way prevent further diffusion through phase space. In a limit case we fix the value of the perturbation parameter p sufficiently small and obtain estimates for the magnitude of the action variable 11 drift from its initial value 110 in dependence on eccentricity as e + 1. An analysis similar to the one performed by the author earlier5 makes it possible t o obtain the bound IIl(t) - 1101 5 const ‘ p , correct when t E [-const . p - ’ , const . p-’1 uniformly on eccentricity e in a vicinity of the limit case.
Acknowledgments The author would like to thank Prof. A. P. Markeyev for useful discussions.
References 1. V. V. Beletsky, Motion of an Artificial Satellite about Its Center of Mass, Israel Program for Scientific Translation, Jerusalem, 1966. 2. A. A. Karymov, Journal of Applied Mathematics and Mechanics 26, 1310 (1962). 3. R. C. Flanagan and V. J. Modi, The Aeronautical Journal of the Royal Aeronautical Society 74, 835 (1970). 4. I. I. Kosenko, “On Continuous Depedence of Solutions to Satellite Oscillations Equation on Orbit Eccentrity with Account of the Light Pressure”; in Problems of the Investigation of Stability and Stabilization of Motion. Part 2 , Computer Center of the Russian Academy of Sciences, Moscow, 2001. 5. I. I. Kosenko, Cosmic Research 40, 581 (2002). 6. L. A. Pars, A Treatise on Analytical Dynamics, Wiley, 1965. 7. A. P. Markeyev, Journal of Applied Mathematics and Mechanics 53, 685 (1989). 8. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies with a n Introduction t o the Problem of Three Bodies, Cambridge University Press, Cambridge, 1927. 9. C. L. Siege1 and J. K. Moser, Lectures o n Celestial Mechanics, Springer-Verlag, Berlin, Heidelberg, New York, 1971. 10. V. I. Arnold, Russ. Math. Surveys 18, 9 (1963).
AN EXPLICIT SOLUTION OF THE (QUANTUM) ELLIPTIC CALOGERO-SUTHERLAND MODEL
EDWIN LANGMANN Mathematical Physics, Department of Physics, K T H , AlbaNoua, SE-106 91 Stockholm, Sweden
Dedicated to the memory of Ludwig Pittner We present explicit formulas for the eigenvalues and eigenfunctions of the elliptic Calogero-Sutherland (eCS) model as formal power series to all orders in the nome of the elliptic functions, for arbitrary values of the (positive) coupling constant and particle number. Our solution gives explicit formulas for an elliptic deformation of the Jack polynomials.
1. Introduction
The elliptic CalogercFSutherland (eCS) system is a quantum mechanical model of identical particles moving on a circle and interacting via a twobody potential given by the Weierstrass elliptic function It is defined by the 2nd order differential operator p . 1 9 2 1 3
where N = 2 , 3 , .. . is the particle number, the circle,
y = 2 q x - l),
-T
5 x j 5 T are coordinates on
x > 0,
is the coupling constant, and the two-body potential
159
(2)
160
which is essentially equal to the Weierstrass elliptic function 27r and iB."
with periods
The eCS system is known to be integrable in the sense that there exist differential operators of the form N
Hn = X(-i)"j=1
a" 8x7
+ lower order terms
for all n = 1 , 2 , . . . , N , which include the eCS Hamiltonian, H2 = H , and which all mutually commute, [Hn, H,] = 0 for n,m = 1,2,. . . , N . 3 Moreover, in the trigonometric limit p = co where the two-body potential reduces to a trigonometric function, the explicit solution of this model was found by Sutherland more than 30 year ago.2 In the two-particle case, N = 2, the eigenvalue equation of the eCS system is equivalent to the Lam6 equation studied extensively at the end of the 19th century; see Ref. 4 for a review of the classical results. There has been quite some recent interest in the eCS ,ystem.5,6,7,8,9,10,11,12 In this paper we present a generalization of Sutherland's solution to the elliptic case without restrictions on parameters (see the Result in the final section). More specifically, we present explicit formulas for the eigenfunctions $(x;n) and corresponding eigenvalues &(n)of the eCS Hamiltonian,
H$(x;n) =&(n)$(x;n), x = (Xl,...rxN),
(4)
which are labeled by integer quantum numbers
n = (721,. . . , n ~ ) n, 1 2 722 2 . . . 2
nN,
(5)
and which are of the following form,
$(x;n) = @(x;n)@(x) where
@(x)=
u l<j
z r ) , ( ~ r 3 - 1~, r + 2..., , Yn))
such that the x-coordinates are constant on every fiber of the foliation
Bn-,.n U ( p ) , the y-coordinates are constant on every fiber of the foliation B, n U ( p ) , and the metric fj has the block-diagonal form r
n
i,j=l
a ,j=r+1
where the first block depends on the first r coordinates and the second block depends on the remaining n - r coordinates. Theorem 2.1 already gives us the complete list of all geodesically rigid closed surfaces": a closed connected surface is geodesically rigid if and only if its Euler characteristic is negative. More precisely, a closed connected surface of negative Euler characteristic has infinite fundamental group, and admits no one-dimensional foliation, so it is geodesically rigid. A closed connected surface of nonnegative Euler characteristic is not geodesically rigid: it is diffeomorphic to the sphere or the projective plane or the torus or the Klein bottle. Examples of nonproportional geodesically equivalent metrics on the sphere and on the projective plane were essentially constructed by Beltrami2. Since the geodesics of every flat metric are straight lines, every two flat metrics on the torus (or on the Klein bottle) are geodesically related (that is, there exists a diffeomorphism that takes the geodesics of the first metric to the geodesics of the second), so the torus or the Klein bottle are not geodesically rigid as well.
201 For dimension three, a direct corollary24of Theorem 2.1 is
Corollary 2.1. Suppose M 3 is a connected closed manifold. Suppose there exist nonproportional Riemannian metrics on M 3 that are geodesically equivalent. Then, modulo the Poincare' conjecture, M 3 is finitely covered by the sphere S3 or by the product F 2 x S1, where F 2 is a closed surface. It appears that Corollary 2.1 is t r ~ e also ~ ~ without v ~ ~ assuming the Poincark conjecture:
Theorem 2.2. Let nonproportional Riemannian metrics g and be geodesically equivalent on a closed connected three-dimensional manifold M 3 . Then the manifold is homeomorphic either to a lens space or to a Seifert manifold with zero Euler number. Every lens space and every Seafert manifold with zero Euler number admits geodesically equivalent metr i c ~which are nonproportional. Theorems 2.1,2.2 give us a complete list of closed connected manifolds of dimension two and three admitting nontrivial geodesic equivalence. It will be much more complicated to obtain such list in every dimension. But still, in every dimension TI 2 2, there exists infinitely many geodesically rigid r n a n i f ~ l d s ~ ~ , ~ ~ :
Theorem 2.3. Every closed connected manifold admitting a Riemannian metric of negative sectional curvature is geodesically rigid. Note that in view of result of Bore17,in every dimension there exist infinitely many closed manifolds admitting metrics of negative sectional curvature. 3. Methods and ideas
3.1. Integrability fog.the geodesic flows of geodesically equivalent metrics New methods for the global (= on closed or complete manifolds) investigation of geodesically equivalent metrics are based on the following o b s e r ~ a t i o n ~ ' ! ' ~the - ~ ~existence : of g geodesically equivalent to g allows one to construct commuting integrals for the geodesic flow of g . Let g = ( g i j ) and 9 = ( g i j ) be Riemannian metrics on a manifold M". Consider the (l,l)-tensor L given by the formula
202
Then, L determines the family St, t E R, of (1,1)-tensors
St
efdet(L - t Id) ( L - t Id)-'.
(3)
Remark 3.1. Although ( L - t Id)-' is not defined for t lying in the spectrum of L, the tensor St is well-defined for every t. Moreover, St is a polynomial in t of degree n - 1 with coefficients being (1,l)-tensors. We will identify the tangent and cotangent bundles of M" by g. This identification allows us to transfer the natural Poisson structure from T*M" to T M " .
Theorem 3.1. If g , ij are geodesically equivalent, then, f o r every t l , t 2 E R, the functions Iti : T M "
-+
R, Iti (w)
efg ( S t , (w),w)
(4)
are commuting integrals for the geodesic flow of g In other direction these theorem is wrong; a counterexample could be found in19.
Theorem 3.2. l9 Suppose for every t E R the function It given by (4) is an integral for the geodesic flow of g . If the Nijenhuis torsion N L vanishes, the metrics are geodesically equivalent. Theorem 3.3. Let g , tj be geodesically equivalent. Then the Nijenhuis torsion N L vanishes. 3.2. What is special i n these integrals?
Definition 3.1. Two metrics g and 9 are strictly-non-proportionalat 5 E M", if all roots of P ( t ) := det(g - t g ) are simple. Let us assume that there exists a point where the metrics are strictlyat almost every point, and the family It non-proportional. Then, it is contains n integrals that are functionally independent almost everywhere. Hence, the geodesic flow of the metric is Liouville-integrable. Let us note that 0 0
the integrals are quadratic in velocities, at every point 5 E M", the integrals (considered as quadratic forms) can be simultaneously diagonalizable.
203 Integrable systems with such propertieszg are known as Stackel systems. Locally, in a given coordinate system, it can be defined by using a ( n x n)matrix such that its columns depend on the corresponding coordinate only. Not every stackel system can come from geodesically equivalent metrics. The additional assumption is that the stackel matrix can be chosen to be a Vandermonde matrix. Stackel systems with such property were also intensively studied. One of the reasons for it that they satisfy Robertson'sz8 condition, which imply that its quantization is quantum-integrable as well. The second reasons is that all stackel systems coming from physics are of this type (or, a degeneration of systems of this type). It is possible to showlg, that this extra-condition is also a sufficient condition for the existence of geodesically equivalent metrics. The systems with this condition appear independently and under different names (L-systems, Benenti systems, quasi-bihamiltonian systems) in works of different author^^^^^^^.
3.3.
If
the metrics are strictly non-proportional
Theorem 3.1 can be used most efficiently when there exists a point of a manifold where the metrics are strictly-non-proportional. Then, the geodesic flow of g is Liouville-integrable, and we can apply the well-developed machinery of integrable systems. For example, the following theorem follows directly from Theorem 3.1 and Taimanov3'.
Theorem 3.4. Suppose M" is a connected closed manifold. Let the realanalytic Riemannian metrics g and g on M" be geodesically equivalent. Suppose there exists a point of the manifold where the metn'cs are strictlynon-proportional. Then, the following statements hold. (1) The first Betti number b l ( M n ) is not greater than n. (2) The fundamental group n l ( M " ) is virtually Abelian. The integrals are quadratic in velocities. Combining this fact with topological obstructions1* for the existence of quadratically-integrable geodesic flows on closed surfaces, we obtain Theorem 2.3 for dimension two18. Note that, in view of results31, Corollary 2.1 follows from Theorem 3.4 under the additional assumption that the metrics are real-analytic and that there exists a point where the metrics are strictly-non-proportional .
204
3.4. Geodesic equivalence and zero entropy All results of this section are joint with Kruglikov; the proofs will be published elsewhere. It is expected, that an integrable geodesic flow has zero topological entropy. This is not always the case. There are example^^,^ of integrable flows with non-zero entropy. But still it is possible to show that if the integrals come from strictly-non-proportional geodesically equivalent metrics by applying Theorem 3.1, the topological entropy of the geodesic flow must be zero.
Theorem 3.5. Suppose the R i e m a n n i a n metrics g , ij o n a closed connected M" are geodesically equivalent. Suppose there exists a point where the metrics are strictly non-proportional. T h e n , the topological entropy of t h e geodesic Pow of g is zero.
Combining this theorem with the famous Yomdin's Theorem", we obtain
Corollary 3.1. Suppose the R i e m a n n i a n metrics g , g o n a closed connected M" are geodesically equivalent. Suppose there exists a point where the metrics are strictly non-proportional. T h e n , the manifold is finitely covered by the product of a rational-elliptic manifold and the torus. 3.5. General case
We will sketch the proof of Theorem 2.1. Consider the (1,l)-tensor L given by the formula (2). All its eigenvalues are real. At every point x E M", let us denote them by A l ( x ) 5 ... 5 A,(x). It appears that they are globally ordered23,24,25.
Theorem 3.6. Let ( M n , g ) be a connected R i e m a n n i a n manifold. Suppose every t w o points of the manifold can be connected by a geodesic. Let R i e m a n n i a n metric ij o n M" be geodesically equivalent t o g . T h e n , f o r every i E { 1,..., n - l}, f o r all x,y E M", t h e following holds: (1) Xi(.)
I Xi+l(Y).
< Xi+l(x) f o r s o m e x E M", t h e n Xi(.) < Xi+,(z) f o r almost every point z E M". (3) If Xi(.) = A i + l ( y ) , t h e n there exists z E M" such that Xi(.) = Xi+l ( z ) . (2) If Xi(%)
Thus, if g and on closed connected M" are geodesically equivalent, then the following two cases are possible:
205 Case 1: There exists r E (1, ..., n every x E M"
-
1) and a constant X E R such that, for
&(x)
^ is therefore of interest. Rank 2 tensors are can be classified using eigenvalues and by considering the algebraic properties of the associated characteristic polynomial. In this paper we introduce the more general notion of an aligned null direction - a kind of Lorentzian eigenvector concept, but valid for arbitrary tensor ranks. In analogy with the characteristic polynomial, aligned null directions are
215
216
the zeros of certain corresponding alignment polynomials. However , unlike the invariant , univariate characteristic polynomial, the alignment polynomials are multivariate covariants. The classification and equivalence analysis proceeds by considering the alignment polynomials’ invariant algebraic properties. Previously, alignment polynomials in 2 variables were used by Penrose and Rindlerg for the classification of maximally symmetric spinors in four dimensions. The Penrose-Rindler approach can be generalized by noting that the components of a Lorentzian tensor, of any symmetry type and in any dimension, can be naturally ordered according to boost weight. In essence, the n of the N P tetrad is counted with with weight 1, the t with weight -1, and the space-like components with weight 0. We will call a null direction t aligned with the tensor, if the components with the largest weight vanish along that direction. In 4D, the zero set of the PenroseRindler polynomial is just the locus of aligned null directions. We illustrate alignment-based classification by discussing bivectors and Weyl-type, rank 4 tensors. We provide a complete classification of bivectors in all dimensions, give normal forms. For Weyl-type tensors, we show that the 4D PND equation is equivalent to the alignment equations and hence has meaning in higher dimensions. However, we show that for N > 4, generically, these equations have no solutions. For more on the classification of higher-dimensional Weyl tensors see ‘. A discussion of some more theoretical aspects of alignment can be found in7.
2. Alignment
Our setting is the N-dimensional Lorentz-signature inner-product space. We define a null frame to be a basis .f = mo, n = m l , m2,. . . , m ~ - 1 , satisfying eana = 1, miamja = d i j , with all other products vanishing. Throughout, Roman indices a , b, c, A, B, C range from 0 to N - 1. Lower case indices indicate an arbitrary basis, while the upper-case ones indicate a null frame. Space-like indices i, j , k also indicate a null-frame, but vary from 2 to N - 1 only. We raise and lower the space-like indices using d i j , so that mi = mi. The Einstein summation convention is observed throughout. We let Q A B denote a null-frame orthogonal matrix, and characterize a Lorentz transformation as a change of null-frame, riz, = mAQAB. The group of orthochronous’ Lorentz transformations is generated by null rota-
217
tions (l),boosts ( 2 ) , and spins (3), which are transformations of form
2 = e + z j m j - $zjzj n, 2 = ~ ,i i = A - l n ,
&el
n==,
ii = n ,
rizi =
rizi=mi,
mi - z i n ;
(1)
(2)
A#O;
m 3. -- m . X i .3 , X i 3. X kj = h i k.
(3)
'Let T = Tal...ap be a rank p tensor. For a given list of frame indices Al, . . . ,A,, we call the corresponding TA1...Ap a null-frame scalar. A Lorentz transformation QA, transforms the scalars according to
In particular, a boost ( 2 ) transforms the scalars according to: ~ A ~ . . . A= ,
AbA1"'ApTA,,..A~, b A,...A,
= bA, f
. . . + b,,,
(5)
where bo = 1, bi = 0, bl = -1. We will call bA1,..Ap the boost weight of the scalar TAI.,.Ap. Equivalently, the boost weight of T A 1 . , . A p is the difference between the number of subscripts equal to 0 and the number of subscripts equal to 1. Let [k] = spank, kaka = 0 be a null direction and let t , n , m i ,be an arbitrary null-frame such that t is a scalar multiple of k . We define b(k), the boost order along k, to be the maximum of all bA1,,,Ap for which TAl...Ap # 0. A null rotation about L fixes the leading terms of a tensor, while boosts ( 2 ) ( 5 )and spins (3) subject the leading terms to an invertible transformation (4). It follows that the boost order does not depend on a choice of a particular null-frame, but rather on the choice of null direction spanned by k. Therefore, the definition of b(k) is sound; the boost order is the same for all null frames for which e E [k]. Finally, we let b,, denote the maximum value of b(k) taken over all null vectors k, and say that a null vector k is aligned with the tensor T whenever b(k) < b,,,. In other words, an aligned null direction is one for which the leading boost-weight scalars vanish. The value of b, depends on the rank and on the symmetry properties of the tensor T. Generically, = p . However, if the tensor has some index for a rank p tensor, b, skew-symmetry, then b,, will be smaller than p . For example, for a bivector Kab = -Kba, we have b, = 1; the corresponding boost weights are shown below. An aligned null direction corresponds to Koi = 0. 1
0
-1
21 8 3. Alignment polynomials
We now show that the set of aligned directions is a variety, the zero set of a finite number of polynomial equations. The set of all null directions is an N - 2 dimensional variety:
PKN-2 = {[k]: k"k,
=
2k0kl
+ kiki = 0).
Affine coordinates zi = k i / k l are defined for every choice of null-frame. Over the real field, we regard [n]as a point at infinity, and identify IWPIKN-' with real extended space kNP2 = RNP2U {co}, the one point compactification of IWN-2 homeomorphic to the sphere SNP2.Complexified extended space CNP2is the union of CNP2with points a t infinity having, respectively, the form [zzmi n ] ,and [ z i m z ]where , zizi= 0. Let T be a rank p tensor and mA a null-frame. For every choice of indices Al, . . . ,A, we define the polynomial
+
P A ~ . . . A ~ ( z= ~ )T B ~ . . . ABIAl(zi) B~ . . . A B P . , ( z i ) , where
1 A A B ( ~ i )=
( - y z j
t);;
(7)
0 0
(8)
is the matrix corresponding to a null rotation about n (c.f. equation (1)) with the parameters zi considered as complex indeterminates. By definition, a null vector k = t? - :CiCin Cimi,is aligned with T if and only if zi = (i is a solution of the corresponding alignment equations
+
~ A ~ . . . A ~= ( Z 0,~ )
bA l . . . ~ p= b
a x .
(9)
Henceforth, we will refer to the p A l , , , A(zi), p bA1.,,Ap = b,, as the alignment polynomials corresponding to the tensor T. Of course, the alignment polynomials are only defined up to a choice of a null frame, and undergo a certain covariant transformation when the frame is changed. A Lorentz transformation riZB = mAQAB induces a change of afine coordinate, a birational transformation
@(ii) = @ A 0 + @ A i 22 - ; @ A l iiii.
(11)
The form for the transformation (10) follows from the relation
i
-
;iziz.iL
+ 2riZi = q5 (&)a + &i)n + $ h j ( i Z ) r n j . 0
(12)
219
A real transformation of form (10) is a conformal transformation of S", and is known as a Mobius transformation1. In terms of the affine coordinates, null rotations about n correspond to translations; null rotations about l?to inversions; boosts correspond to dilations; spins correspond to rotations. Proposition 3.1. Let mB= mAQAB be two complex null frames related by Lorentz transformation. The corresponding polynomials ( 7 ) are related
by
(C) . . . T B p A p ( & ) ,where,
$ A ~ . . . A ~ ( % )= P B ~ , . , B ~ ( ~ ~ ) ~ ~ ~ A ~
and where
4A,idenotes
the partial derivative of
(13)
with respect to ii.
Note that TAB= 0 for bB < b,, and hence $ A , . . , A p depends only on pBl.,,Bp for which bB1...Bp 2 bAl.,,Ap. Hence, two sets of alignment polynomials, P B ~ . . . B ~ ( Z ~ ) $, A , . . . A ~ ( & ) )
= brnax,
~ A I . . . A ~
defined relative to different frames, are birationally related. Hence, on the open set of finite points, + o ( i i ) # 0, the zeros of and the zeros of the transformed $ A l , , . A p (&) coincide. 4. Bivectors
As per (7), the alignment polynomials for a bivector Kab are given by
where
2, m j
are defined in (1). Expanding these expressions we obtain
poj(Zi) = Koj
+ ziKij
-
zjKol
1 i ziK1j - zizjKi1.
- -2
2
(15)
Thus, the aligned null directions are the solution set of a system of N - 2 quadratic equations in N - 2 variables. Proposition 4.1. The bivector alignment equations (15) admit a real zero.
This follows because a complex null vector ka is aligned with K,b if and only if it is an eigenvector of the transformation Kab. Furthermore, in Lorentzian
220 signature, a skew-symmetric K a b admits at least one real eigenvalue, and hence a real eigenvector. Henceforth, without loss of generality, we take n as an aligned real direction. Hence K 1 j = 0, and the alignment equations assume a linear form: p o j ( ~ i= )
Koj
+ ziKij - ~ j K o 1= 0.
(16)
The attribute of consistency/inconsistency of system (16) partitions the set of bivectors into two classes; we will refer to these, respectively, as type I and type N bivectors,. Type I bivectors admit at least two real, aligned null directions. This is true if and only if equations (16) are consistent, or what is equivalent, Kol # 0, or K [ i j K k ~ O= 0. Taking f2 and n as real, aligned null directions, and after performing a normalizing spin, type I bivectors admit the following canonical form: 1NPI -1
Kab
= Aon[,f!b]
+
(17)
Ap?T12p[am2p+1 bl ' p= 1
Type N bivectors admit only one real, aligned null direction. This is true if and only if Kol = 0, and K [ i j K k p # 0. After a null-rotation about n and a spin, type N bivectors admit the canonical form: w
K a b = n[,mN-'b]
-
3
)
~
A,
f
~
m2p[am2Pf1b].
(18)
p= 1
The real scalar Ao, and the scalars A, up to reordering, are invariants, and can be used to solve the equivalence problem. 5 . Weyl-type tensors
w e define a Weyl-like tensor C a b c d to be a traceless, valence 4 tensor with the well-known index symmetries of the Riemann curvature tensor, i.e., Cabcd
=
-Cbacd
= Ccdab,
Cabcd f C a c d b
+ C a d b c = 0,
C a b c b = 0.
We let W, denote the vector space of N-dimensional Weyl-like tensors. It isn't hard to show that WN has dimension & ( N 2 ) ( N 1 ) N ( N - 3). The maximal boost weight for a Weyl tensor is given by b,, = 2. The Weyl alignment polynomials are given by
+
b ^b POiOj (Zi)= C a b c d f!" f h i ?f fh;,
aThe terminology is borrowed from relativity".
+
(19)
221 2
1
0
-1
-2
COiOj
COlOi, C O i j k
COlOl, COlij,COilj,C i j k l
COlli, C l i j k
clilj
where 2, T?Zi are defined in (1). Since p0ioi = 0, the aligned directions are the solution set of a system of i N ( N - 3) = i ( N - 2 ) ( N - 1 ) - 1, fourth order equations in N - 2 variables. In 4D, the principal null directions of the Weyl-like tensor are defined in terms of the so-called PND equationg: k b k [ , C a ] b c [ d k f ] k c= 0,
k"ka = 0.
(20)
It is easy to establish that the PND equations are just the homogeneous form of the alignment equations'. Proposition 5.1. For every dimension N , a null vector k" satisfies the PND equation (20) i f and only i f it i s aligned with C a b c d .
For N 2 4 we have $ N ( N - 3 ) 2 N - 2 , with equality if and only if N = 4. Thus, a four-dimensional Weyl-like tensor always possesses at least one aligned direction (see below). For N > 4, the number of equations is greater than the number of variables, and hence, generically, the alignment equations are inconsistent.
Theorem 5.1. If N 2 5 , then the subset of Weyl-type tensors possessing no complex aligned directions is a dense, open subset of W,. I n other words, the generic Weyl-like tensor in higher dimensions does not possess any aligned null directions, not even complex ones. Let us now re-derive the well-known Petrov-Penrose classification of 4-dimensional Weyl-like tensors using alignment. To facilitated the calcum2, with lations we switch to the NP tetrad: e = mo, n = ml m2, m21= fana= 1, rn;m2la = -1. A null rotation about n now takes the form
i = t+z'
m2+z rnz,+zz' n,
ii = n,
riz2 = m2+z n ,
h2, = m2,+zfn,
where zf = Z for a real transformation. Equation (19) gives the alignment polynomials, with p0202l = 0 because C a b & is trace-free. w e also note that ai/az' = T?Zz, and aT?Z2/az' = 0. It follows immediately that dpo202/dz' = 0. Hence, P0202(z,
.') =P0202(z) = c0202(z - e l ) ( . - c2)(z - c 3 ) ( z - c4)
is a fourth degree polynomial of one complex variable. Furthermore, p02t02, is the complex conjugate of ~ 0 2 0 2and , we deduce that, generically, there are
222
16 complex aligned null directions: z = &,, z’ = , with p , q = 1 , 2 , 3 , 4 . The real aligned null directions correspond to z’ = 2. w e also see that a real Cabcd is completely determined by the polynomial p0202(z),and t h a t a change of null-frame transforms the latter by a Mobius transformation. The usual classification of the 4-dimensional Weyl tensor now follows by considering the root multiplicities of this polynomial. We have derived all this directly by means of the alignment paradigm; there was no need t o invoke spinors. Acknowledgements The author was partially supported by a n NSERC discovery grant. Discussions with A. Coley, N. Pelavas, V. Pravda, A. PravdovA are gratefully acknowledged.
References 1. A. F. Beardon, The Geometry of Discrete Groups Springer-Verlag, (1983). 2. M. Carmeli, Group theory and general relativity. McGraw Hill, (1977). 3. A. Coley, R. Milson, N . Pelavas, V. Pravda, A. Pravdovb, and R. Zalaletdinov, Phys. Rev. D 67 104020, (2003). 4. A. Coley, R. Milson, V. Pravda, and A. Pravdovb, Class. Quantum Grav. 21 L35, (2004). 5. P. DeSmet, Class. Quantum Grav. 19 4877-4895, (2002). 6. D. Kramer, H. Stephani, M. MacCallum, C. Hoenselaers, and E Herlt, Exact solutions of Einstein’s field equations Cambridge University Press, (2003). 7. R. Milson, A. Coley, V. Pravda, and A. Pravdovb, gr-qc/0401010. 8. J. M. Overduin and P. S. Wesson, Phys. Rep. 283 303-378, (1997). 9. R. Penrose and W. Rindler, Spinors and Space-time, vol. II. Cambridge Uni-
versity Press, (1986). 10. J. Stewart, Advanced general relativity, Cambridge University Press, (1990).
RENORMALIZATION GROUP SYMMETRY AND GAS DYNAMICS
SOUICHI MURATA Department of Physics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan
[email protected] We present a new scenario in the Renormalization Group Symmetry method, where we allow a Lie point symmetry operator which does not contain arbitrary functions. As a specific example, we give an exact solution for the adiabatic perfect gas dynamics.
1. Introduction The explicit construction of exact solutions to a system of partial differential equations (PDEs) of physical relevance is of great interest. Especially, when exact solutions contain arbitrary functions, these solutions may be used for solving general initial value problems. One of the most important tool to find an exact solution of a system of PDEs is the Lie group theory, which has been applied t o solve a wide rage of problems in physics. If a system of PDEs is invariant under the action of an infinitesimal group of transformations, we can construct special solutions called similarity solutions or invariant solutions, which are invariant under the transformation admitted by the system. Invariant solutions can also be constructed for specific initial value problems. The renormalization group symmetry (RGS) method is introduced t o solve an initial value problem for a system of PDEs. In this approach, constructing a special type of the Lie point symmetry called an RGS is the main ingredient. Initial value problems for some PDEs of physical interest were analyzed by means of the RGS method. All RGS operators obtained in the above studies depend on arbitrary functions, which support some arbitrariness of initial values through the (FSS) conditions (see below). However, except for relatively simple systems with good symmetries, it is rare to find an RGS operator with arbitrary function for the real system of physical
223
224
interest. In this paper, we present another type of mechanism, which allows an RGS operator without arbitrary functions to support some arbitrariness of initial conditions. In our treatment, if possible, an RGS operator is determined so that the conditions of functional self-similarity become degenerate. Then, arbitrariness of initial conditions is ensured even by an RGS operator without arbitrary functions by virtue of the degeneracy condition. This scenario is illustrated for the dynamics of adiabatic perfect gas.
2. RGS method to initial value problem The RGS method with Lie point symmetries consists of three steps. The first step is to calculate the Lie symmetries V admitted by the given partial differential equation. This step is carried out by “pr~longation”~ , where we consider a prolonged vector field on the manifold in the jet space that the differential equation refers to. In the second step, we restrict V on the solution of an initial value problem by imposing the invariant conditions called the FSS conditions. They are written as
1
V ( U i - 02)
-
u;=ui
= 0,
where Uiis the j-th dependent variable and 0 iis the solution of an initial value problem. The final step is to evaluate Eq.(1) at the initial time t = 0 and to find the desired RG-symmetries. If the Lie symmetries admitted by the given partial differential equation are expressed by an operator including arbi-
2 trary functions, the RGS naturally contain arbitrary initial values 0
’.
ILO
Here, we consider the case that the Lie symmetries obtained in the first step do not include any arbitrary functions. The final step gives the following relations among initial values 0 i
:
ILO
1
V ( U i - 02)
= 0.
(2)
u,=i7i
If we can choose a Lie symmetry V so that Eqs. (2) are degenerate, we introduce some arbitrariness to the initial values and construct an exact solution corresponding to the initial values. In the next section, an explicit example of this approach is given for the gas dynamics.
225 3. Perfect gas dynamics We take the initial value problem for the equations that govern the motion of an adiabatic perfect gas. The equations of an adiabatic perfect gas and initial conditions are written in the following form
aP + v ( p v ) = 0 , at
(3)
'
dV
1
at
P
- + (v.V) . v + - V P
= 0,
(4)
where p(t, x) is density, P(t,x) the pressure, v ( t ,x) = ( u ,w,w )the velocity vector, t the time, x = ( r ,8, z ) the cylindrical coordinates, y the adiabatic index. It is k n o ~ n ~that y ~ the , ~ Lie group of point transformation, that leaves system(3)-(5) invariant, constitutes a seven dimensional Lie algebra generated by the following infinitesimal operators:
if y = then we have also the invariance with respect to the so-called projective group that is characterized by the infinitesimal operator:
V7 = t2at
+ tra, + t z d , + ( r - ut)& - wt8, + ( z - wt)aw - 5 t ~ a -p stpa,,.
The restriction of the group admitted by system (3)-(5) on the solution of the initial value problem u = E ( t , r , z ) , w = E ( t , r , z ) , w = W ( t , r , z ) , P = p ( t , r , z ) , p = ; 6 ( t , r , z ) leads to the conditions of functional self-
226 similarity (1):
au - (Ti,
-
VE,
+r
(-
uu,
- ViEZ
+
-
5 +jF, -2
->
7
( (
= 0,
(-
+r TiiP,+mFz+yP
> >> >>
+ ww, + p, = 0 ,
ZLW,
ap - (F, - V H Z i,+gtm, (i v-ci,
- E-cir +7
1
TEr+Ez+ = o~,
W-@,-~'ijz+T
QW -
+ ZOU,
(Ep,+urp,+;iSfor (z,+g+m,oft
=o,
=o.
(9) (10)
(11)
(12)
(13)
These equalities should be valid any values and certainly for t = 0, when the dependencies of Ti, g , m,P and ;iS upon r and z are given by the initial conditions (6) and (7). This yields eight relations between Ciand initial values: (-C5uO
+ ~ 7 )-r~1
+(CiUo - C4)Uor
+
+ 3% = 0 ,
(C1Uo - c5)V (ClUo - C 4 ) r S +(Cl Woz - c 2 - CqZ)%
(14)
= 0,
(15)
= 0,
(16)
C3R+ (C1Uo - C 4 ) r g +(CiWoZ - C2 - c 4 Z ) g zCiR(2Uo + Wo) = 0 ,
(17)
(-c5wOZ
+
c7Z
f
c6)
3%
t ( C 1 ~ oz ~2 - C ~ Z ) W +~
+
+
+
(C3 2C5)Q (ClUo - C 4 ) r g +(CiWoZ - C2 - C 4 ~ ) g CiQ(2Uo Wo) = 0.
+
+
(18)
When the arbitrary constants Ci,Uo and WOare chosen as c 4 =c 5
= ClUO,
c 2
=c 6 = 0,
-5C1u0, uo = wo, (19) the system of Eqs.(l4) -(18) becomes degenerate, that is, Eqs.(l5), (17) and (18) are automatically satisfied while Eq.(14) and (16) yield 1 Q = -R, v 2= a(-r RR , - 27-7. U c 3=
227 Here, R = a R / d i where i = z2 and C7 5 C1 (U," - 2a) where a is a negative constant so that 7 # 0 for any time ( see Eqs.(22) ). Then, R is an arbitrary function of r and z except for the condition rRr -
R
2r2(= .'/a)
< 0.
(20)
Substituting (19) in (8) gives an RGS-operator
-
3-
v = - 2- w .
- P = - -,?tP, 5
We see that the RGS operator for the system ( 3 ) - ( 7 )is presented as a symmetry of one-dimensional algebra with the infinitesimal operator (8). The invariant condition for the initial value problem with respect to the RG with this operator is presented in the form of five partial differential equations. Solving the Lie equations which correspond to the RGS-operator (21) enables us to construct the desired exact solution of the system ( 3 ) - ( 7 ) 7t
u=-r, 27
Tt
w=-z,
27 (23)
R p = -72.5 ' where
1 R a 71.5 '
p = --
228 4. Summary
A new scenario in the RGS method is introduced to solve initial value problems for a system of partial differential equations. The crucial step of our approach is to find a Lie symmetry admitted by the system, which satisfies a degeneracy condition for functional self similarity relations evaluated at the initial time. When we succeed in finding such Lie symmetry, an exact solution is constructed for an initial value problem including arbitrary functions even if the Lie symmetry itself does not contain arbitrary functions. As a specific example, we consider an initial value problem for the adiabatic perfect gas dynamics in the cylindrical geometry and obtain a solution containing an arbitrary function. This exact solution describes a contracting and expanding localized mass of gas.
References 1. V.F. Kovalev, V.V. Pustovalov and D.V. Shirkov, “Group analysis and renormgroup symmetries”, J. Math. Phys. 39, 1170 (1998). 2. V.F. Kovalev and D.V. Shirkov, “Functional self-similarity and renormalization group symmetry in mathematical physics”, Theor. Math. Phys. 121,1315 (1999). 3. D.V. Shirkov, “Renormalization group in modern physics”, Internat. J. Modern Phys. A 3, 1321 (1988). 4. Dmitrij.V. Shirkov and V1adimir.F. Kovalev, “The Bogoliubov renormalization group and solution symmetry in mathematical physics”, Phys. Rep. 352, 219 (2001). 5. P.J Olver, “Application of Lie Groups to Differential Equations,” SpringerVerlag (1986). 6. Francesco. Oliveri and Maria.Paola. Speciale, “Exact solutions to the equations of perfect gas through Lie group analysis and substitution principles,” Internat. J. Non-Linear Mech. 34 1077 (1999). 7. Francesco. Oliveri and Maria.Paola. Speciale, ”Exact solutions to the unsteady equations of perfect gases through Lie group analysis and substitution principles,” Internat. J. Non-linear Mech. 37 257 (2002). 8. N.H Ibragimov, “CRC Handbook of Lie group analysis of differential equations”, vol l CRC Press (1994). \
,
REFINED COMPUTATION OF HYPERNORMAL FORMS
JAMES MURDOCK Department of Mathematics Iowa State University Ames, I A 50011, USA
This article is a short overview of 2 . We first review the main ideas of hypernormal forms (also called unique normal forms and simplest normal forms), due primarily to Baider. Then we outline some new ideas based on ”refinement.” Refinement does not result in normal forms that are different from Baider’s, but affects the method of calculation. For a thorough introduction to normal forms, see ’;section 4.10 concerns hypernormal forms. According to the Borel-Ritt theorem, a formal power series is the same thing as an equivalence class of smooth functions modulo flat functions. This means that “formal” calculations are actually “rigorous,” without becoming involved with asymptotic series. A formal vector field on Rn with rest point at the origin is given by
v(.)
= 110(.)
+
+ v2(x) + . . .
111 (x)
where v, belongs to the space Vi of homogeneous vector fields of grade i (degree i 1). This grading is used because it adds under Lie bracket. A generator is a formal vector field starting with quadratic terms:
+
g(.)
=g 1 ( . )
+.‘.
(For the refined theory, we will want to classify generators by the grade j of their leading term, so that g = g3 . . . with j 2 1.) The time-one map of a generator is a near-identity transformation that fixes the origin and has the identity as its linear term. The group structure on the near-identity transformations can be transferred to the generators in the form
+
where [,] is the Lie bracket. This group of generators is called the CampbellHausdorfl group. Note that * is nonabelian but approximately equal to +: g * h =g
+ h + higher order terms.
229
230 A consequence of this is that the group of generators has many abelian subquotients (that is, there are many pairs of subgroups, one a normal subgroup of the other, such that the quotient is abelian). One of the themes of normal and hypernormal form theory is to use actions of these subquotients whenever possible. The group of generators acts on the space of vector fields. We want to select a unique “simplest” element from each orbit and find the generator that moves a given vector field to the simplest one. But there is not always an obvious choice of what is simplest. A rule for picking the “simplest” is called a hypernormal form style. A vector field is normalized one degree at a time (or faster if quadratically convergent algorithms are used). Therefore hypernormal form styles are presented as a family of subspaces of the vector spaces V,. A generator
+ gj+1 +
g = gj
’
.‘
starting with grade j acting on a vector field
v = V O + v1 + . . . makes no change below grade j. The increment produced in grade j is
If this is zero, the increment produced in grade j [Sj, v11
+ 1 is
+ [Sj+l7 2101
The same pattern continues up to the target t e r m , the first term with a nonzero increment; if the target grade is k , the increment is [ S j ,Vk-j]
+ . . + [gk,2/01. ’
After the target term the increment becomes nonlinear in g. Since terms in g beyond grade k cannot affect the target term, the spaces rjk(v) of generators g i . . . gk that target the grade k in v are natural ones to consider. (In the Baider theory only j = 1 is considered.) Although the increment produced by a generator is always linear in v, and is linear in g up to the target term, the action of the group of generators on vector fields is a nonlinear action. The simplified action of generators on the target term is abelian and can be viewed as an action of an abelian group (and vector space) that is a subquotient of the group of generators (as discussed above). The term full
+ +
231 action will be used to distinguish the orginal nonabelian action from this simplified one. Generators that target grade k produce a vector space A k C v k of achievable increments (called the “removable space” by Baider). These can be used to move v k into any subspace of v k complementary to A k . such’ a subspace is called a style in grade k and is a realization of the quotient Space V k / A k . The standard procedure for computing hypernormal forms involves two steps for each grade. In the first step, the simplified action is used to move the term of grade k into the desired style; the generator that achieves this is found by solving the linear system of equations [g1,vo]=0
[ a 0,1 1 + [gzvo] = 0 [gl,gk-Z]++.”+ [ g l ,v k - l ]
f .’ ’
[gk-l,vO]=O
+
[ g k , W O ] = old v k
- new v k .
In the second step, the full action is used to compute the effect of this generator on terms of grade higher than k . The refined process breaks step one into a sequence of smaller substeps. First, a generator g k , homogeneous of grade k , is used to move Wk into its classical (not hyper) normal form (of some chosen style); the generator is selected via the classical homological equation of normal form theory. Next, the classical normal form is refined by using a generator g k - 1 - t g k of “lag gk-1 g k of lag two is one” that targets grade k . Next a generator g k - 2 used, and so on. At each substep the style space is refined, so that the new style space is a subspace of the previous one, and the space of generators to be used at the next stage is restricted so that (in addition to having the required target grade and lag) the admissible generators map the current style space into itself. This guarantees that what is accomplished in one substep is not undone later. It is clear that for this to be workable, there must be a way to compute the required spaces of generators for each substep. The generators g j . . . g k needed for grade k with lag k - j lie in a realization of a quotient space
+
+
+
+
232 (Here ?rjk : r j , k + l ( v ) + rjk(v) is the projection that drops the term of grade lc 1.) The double quotient removes generators that have already been used, and ones that will not be used until later. The particular style choices that are made will affect the correct choice of realization of Fjk(w). (Recall that a realization is a complement of the denominator in the numerator .) Now the implementation of the method requires nothing but linear algebra. At first sight it would seem to require a great deal more linear algebra than the traditional method, which would be a step in the wrong direction, because there is already too much linear algebra to do effectively. This leads to the second new idea, the special row echelon form. It turns out that a modification of the standard row echelon form reduces the amount of linear algebra that must be done to determine the spaces GI!, used in the traditional method, and at the same time permits determination of the Fjk(w) needed for the refined method without any additional row operations. The linear system to compute r j k ( 2 r ) is
+
Using acadmic -LKK L K - ~ , K L2K LIK LK-I,K-I ' " L2,K-1 LI,K-I " '
L22
-
Ll2 L11 -
Using acadmic Using acadmic
233 1. Put each band into reduced row echelon form.
2. Working from the bottom, clear the column AbOVE each Pivot element (but not below). 3. Repeat steps 1 and 2 until nothing further can be done. Each lower segment fof the mtrix is now row equivalent to the same can from be determined. It can the same same matrix. matrix. the segment of the reduced matrix, can and be all determined also be shown shownthat thatall all The following numerical example (which does not come from a normal form problem) illustrates the notion of special row echelon form. The matrix 10000040 00100026 100353 001219 000000 100 000 000 already has its bands in reduced row echelon form, so step 1 has been done.
Step two results in
-
100000 4 0 0 0 0 0 0 -3 1000 5 0010 1 0000 0 1 0 0 0 0 0
and then a repetition of the steps produces
-
0 3 3 9 0 0 0 0
234
This matrix is in special row echelon form; the pivot elements are marked by parentheses. In contrast to this example, an actual normalization problem is best done symbolically. A typical problem divides into many “cases,” characterized by the requirement that certain matrix entries are zero, others are nonzero, and the rest are unrestricted. (This sort of thing is familiar in bifurcation theory, where it is assumed that certain low order terms vanish, certain intermediate order terms that play a critical role are nonzero, and the remaining high order terms do not matter for the final result.) Once these assumptions are specified, it is possible to compute the special row echelon form, in particular (and most importantly) the positions of the pivot elements. From here the Fjh(w) and the hypernormal form can be determined. The familiar example of the anharmonic oscillator (or single center) is worked out this way in 2 ; this is one of the few examples that has been fully solved, and has repeatedly been used as a test case when new methods are introduced. It is hoped that the present methods will be helpful for problems not yet solved, and will be implementable in symbolic processing systems. Jan Sanders has shown, using homological algebra, that the computations of hypernormal forms can be arranged into a spectral sequence. The ideas developed here allow a derivation of the Sanders spectral sequence without homological algebra (beyond a few definitions). It turns out that the Sanders spectral sequence involves some, but not all, of the refinement ideas described above. (It does not compute the smallest groups F’k(v)of generators.) There is also an easier (less detailed) spectral sequence governing the original approach to hypernormal forms (without refinement). It is given in ’. Often the Lie algebra of vector fields is a direct sum of Lie algebras. In this case the notion of target t e r n can be sharpened: a generator can have different target terms in different summands. This should lead to a “refined” theory of refinement; the way this works for the anharmonic oscillator is discussed in 2 , but no general theory has been attempted. Alternative gradings of the Lie algebra have been used in this situation in the nilpotent case, but they do not work for the anharmonic oscillator. The relationship between alternative gradings and “refined refinement” should be investigated further.
235 References 1. James Murdock. Normal Forms and Unfoldings for Local Dynamical Systems. Springer, New York, 2003. 2. James Murdock. Hypernormal form theory: foundations and algorithms. Jovrnal of Differential Equations, 205:424-465, 2004.
NEW ORDER REDUCTlONS FOR EULER-LAGRANGE EQUATIONS
C. MURIEL, J. L. ROMERO Dpto. de Matema'ticas. Facultad de Ciencias. Universidad de Ca'diz Poligono Rio San Pedro s/n, 11510 Puerto Real, Ca'diz, Spain concepcion.
[email protected] We present a generalization of the concept of variational symmetry, based on the new prolongations called A-prolongations. This will lead to new methods of reduction for Euler-Lagrange equations. Some results related to the conservation of variational symmetries through successive order reductions are also presented.
1. Introduction Most of well-known methods for obtaining exact solutions of differential equations are based on the existence of Lie symmetries. The knowledge of a Lie symmetry X of an ordinary differential equation
A(x,u'"')
=0
(1)
implies that the order can be reduced by one, and we can recover the solutions of (1) from the solutions of the reduced equation by a quadrature. For special types of ordinary differential equations and symmetries (variational symmetries) this reduction adopts a particularly simple form by associating with the symmetry a conservation law. This is done through Noether's theorem for differential equations that can be derived from a variational principle: L[U]=
s
L(z,u("))dz.
(2)
It is well-known that there exists ordinary differential equations without Lie symmetries that can be reduced or integrated by using different methods. One of them, that explains a large variety of these processes, is based on the existence of C"-symmetries1-2. This concept arises from a new way of prolonging vectors fields. For a given vector field
236
237
+
X = ((x,u)& q(x,u)d, defined on M c X x U and for an arbitrary function A E C " ( M ( l ) ) , the A-prolongation of order n of X, denoted by X [ ' I ( ~ is ) ]the ~ vector field defined on by
For this kind of prolongations it is possible to calculate a complete system of invariants by derivation of lower order invariants3. This is the key t o construct new methods of order reduction, based on the existence of C" -symmetries'. The objective of this paper is to investigate the concept of variational symmetries when A-prolongations are considered. This leads us to introduce the concept of variational C" -symmetry and create new methods of reduction for Euler-Lagrange equations. In particular] we prove that a variational C" -symmetry of a given Euler-Lagrange equation provides a reduction by two; this is a "partial" reduction, meaning that a one-parameter family of solutions of the original equation is lost when the reduced problem is considered. Whereas a one-parameter variational symmetry group will, in general, alow one to reduce the order of the Euler-Lagrange equations by two, it is not true that a two-parameter variational symmetry group allow one to reduce the order by four4. In this paper we also show some applications of C"-variational symmetries to this problem. 2. The concept of variational C" -symmetry
Let us consider a variational problem (2) where the Lagrangian L ( x ,~ ( ~ 1 ) is defined on M ( " ) ,for some open set M of the space of the variables X x U. Let n
E[L] C ( - D ) ' ( & , L ) = 0
(5)
i=O
be the associated Euler-Lagrange equation] where D stands for the total derivative operator with respect t o x. To simplify the notation, we will denote by A the space of smooth functions depending on x,u and derivatives
238 of u up t o some finite, but unspecified, order and we write P[u]= P ( z ,u ( ~ ) ) if we do not need to precise the order of derivatives that P depends on. Roughly speaking, a variational symmetry group of the functional ( 2 ) is a local group of transformations that leaves the variational integral C unchanged when u = f(z)is transformed by the action of the group. The infinitesimal criterion of invariance*, characterizes the infinitesimal generators of connected groups of variational symmetries. They are the vector q ( z ,.)au such that fields v = ((z, u)&
+
v(~)(+ L )LD(C)= 0. The hypothesis that the vector field X generate a group of variational symmetries is overly restrictive t o deduce the existence of a conservation law. This motivates a generalization of a variational symmetry: the infinitesimal divergence symmetries are the vector fields 21 such that
v ( ~ ) (+ L )L D ( 0 = D ( B )
for some
B E A.
This concept and the A-prolongation formula inspire the following generalization of the definition of variational symmetry:
+
Definition 2.1. A vector field X = ((x,u)& q ( z ,u)& is a variational z exists C"-symmetry of the functional C[u] = S L ( ~ , u ( ~ ) )ifd there B[u]E A such that
+
x [ x ! ( n ) I ( L )L ( D
+ A)(()
=(D
+ A)(B),
for some A E C"(M(l)). We also say that X is a variational A-symmetry t o precise the function A for which (6) is satisfied. 3. Order reduction through variational Cm-symmetries
It is well-known that one-parameter symmetry groups of ordinary differential equations allow us t o reduce the order of the equation by one. Due to the special structure of an Euler-Lagrange equation, the knowledge of a variational symmetry allows us to reduce the order by two4. as well as Lie symmetries, have associated a Since C"-symmetries, method of reduction, we can expect new order reductions procedures for Euler-Lagrange equations arising from variational C" -symmetries.
Theorem 3.1. Reduction of order Let C[u]= S L ( z ,~ ( ~ ) ) dbe a :an n-th order variational problem with Euler-Lagrange equation Eu[L]= 0 , of order 2n. Let X be a variational
239 A-symmetry, where A E C"(M(l)). Then there exists a variational problem z [ w ] = z ( y , w("-l))dy of order n - 1, with Euler-Lagrange equation of order 2 n - 2, E, [L]= 0 , such that a ( 2 n - 1)-parameter family of solutions of EzL[L] = 0 can be found by solving a first order equation from the solutions of the Euler-Lagrange reduced equation E,[L] = 0.
s
A
Proof. Let us introduce a change of variables y = y ( x , u ) ,a = a(%, u)such that X takes the form 2 = a, in the new coordinates. Let C [ a ]= ?;(y,a("))dy be the corresponding variational problem in remains as a coordinates ( y , a ) . It can be checked' that the vector field variational X-symmetry, where A is the function D,a.A in new coordinates, and the next relation holds:
s
I
---
Let us observe that [2fi>(")l,Dy] = AX[xi(")I,which applied t o any function g provides:
Let us consider any function A such that grangian
B = -&(A).
Then the La-
2= Z ( y , a("))+ D,(A)
(9)
and have the same Euler-Lagrange expression, E,[z], and by (7) and (8) we get
jp?(")l(z) = 0, Let w = w ( y ,a , 0 1 1 ) be a first order invariant for a,(w)
+ xa,,
( w ) = 0.
(10)
2[',(')1, that
is
(11)
A very important property of A-prolongations is that a complete system of invariants for the n-th order A-prolongation can be constructed by successive derivations of lower order invariants3. In this case, by successive derivations of w with respect- to y we get a system of coordinates { y , a , w , . . . , wn-l}, such that X[xr(")I = a,. Let us also denote by z(y, W J(~-' )) the Lagrangian in the (y, a, w("-')) variables, that by (10) does not depend on a. By means of the transformation { y = y , w = w ( y , a , a l ) } we get the following relation between E,[z] and the Euler-Lagrange equation of
2
240
z ( y , w ( ~ - ~(see ) ) Ref. 4,Exercise 5.49):
Therefore, by (11):
Ea[Q = ( D y +X)[-dal,(w)Ew[ZlI.
(12)
Let us denote by w = H ( y , C1, . . . ,&-2) the general solution of the = 0. When w is written in terms reduced Euler-Lagrange equation Ew[z] of { y , a , a1} we get the first order ordinary differential equation for a : w ( y , a , a 1 ) = H(Y,Cl,"'
,C2n--2).
(13)
From its general solution a = G(y, C1,. . . , Czn-l) we obtain a (2n 1)-parameter family of solutions
a ( x ,U ) = G ( Y ( xu, ) ,C I ,
' ' '
,G n - 1 )
to the original Euler-Lagrange equation EU[L]= 0.
0
4. Conservation of variational symmetries by order reductions When an ordinary differential equation admits some r-dimensional symmetry solvable algebra S , the order of the equation can be reduced through r successive one-order reductions. Recently, the C" -symmetries theory has successfully been applied to carry out step by step methods of reduction that are even valid for non-solvable symmetry Since a variational symmetry allows us to reduce the order of EulerLagrange equation by two, one could think that two-parameter variational symmetry algebras (always solvable) allow us to reduce the order by four. This assumption is not true in general (see Ref. 4, Exercise 4.11). Motivated by the power of the Coo-symmetries theory in step by step methods of reduction, we can expect variational C" -symmetries become an important tool to solve this problem. This is proved in next theorem.
Theorem 4.1. Conservation of variational symmetries Let C [ u ] = L C ( ~ , u ( ~ ) )be d xa variational problem that admits a two dimensional non-abelian algebra G of (strict) variational symmetries. Let X 1 and X2 be two generators of 6 such that [Xi, X,] = cX1, c E R. For
s
241
v E R, let z v [ w ] be the one-parameter family of reduced variational problems obtained by using X2. Any of them inherits from Xi a variational C" - symmetry.
Proof. Let {y,a} be a system of local coordinates where X2 takes the form 8, and let Z(y, a l , . . . , a,) be a Xz-invariant Lagrangian equivalent to L (see Ref. 4, pag. 257 for details). Since X I remains as a variational symmetry under changes of variables, we get
xp(z)+ ZD,(Xl(Y)) = 0.
(14)
The one-parameter family of reduced variational problems obtained by v E R, where = s(Z(y, d n - l ) )-vw)dy, using X2 can be written as Lv[w] w = a1. For v E R and by (14), we can write
Xi"'(Z - va1) + (Z- ~/ai)D, (Xi (y))
= -vX~")(a1) - valD,
(Xi (y)). (15)
When both members of [Xp),D,] = -Dy(X1(y))Dy (see Ref. 7) are applied to a , we get Xl"'(a1) - DY(Xjn)(a)) = -Dy(X1(y))al. Therefore, (15) becomes
Xi"'(z
+ (Z
-V ~ I )
-
Val)Dy(X1(y))= DY(Xin'(-va)).
(16)
\
Let f be any function such that X2(f) = cf. In coordinates y, a } , we can choose, for example, f(y, a ) = eca. Then we get Xi" ] = 0 for
[fx,'"),
m E N. By Lemma 5.1 in Ref. 1, we also have fXi"' = (fX1)['?(")1, for the function X = D f = -ca1. Relation (16) can be written as follows:
-+
(fXl)[',(,)l(Z
+ (Z
- Val)
-
+
val)(Dy X)(fXl(Y)) X)(fXi"'(-va)).
= (D,
+
(17)
Since [fXi"',Xi"'] = 0 and fXi"' = (fX1)['~(")1,for m E N, the infinitesimals of the vector field (fX1)['~(")l do not depend on a. Let us denote f X 1 = C ( Y ) ~ , + v(Y)&. Let 7r, be the projection r , ( y , a , . . . ,a,) = ( y , a l ; . . ,a,), for m E N.For any m E W, the vector field (fX1)['-(")1 is 7r,-projectable. ) ] )is. also clear that Yi-cal'(m)l = Let us denote Y1 = ( ~ 1 ) * ( ( f X 1 ) [ ' ~ ( ~It (7rm)*((fX1)[',(")]). If we set w = a1, (17) becomes Y;-cw,(n)l(Z
-
vw)+
(Z- v w ) ( D , - cw)(((y)) =(
4 4
- CW)(-V77(Y)).
This proves that Y1 is a variational (-cw)-symmetry reduced problem.
(18)
of the corresponding 0
242 4.1. An example
2)
admits XI = 8, and X2 = .a, as (strict) variational symmetries (see Ref. 4, Exercise 4.11). It is clear that [Xl,Xz] = XI. The one-parameter family of X2-reduced problems are defined by the Lagrangians
It is well-known that any Lagrangian of the form L ( z ,
W1
+
L(Y, ; w) - vw,
v E
R,
(19)
where y = x and w = a.According to Theorem 4.1 the inherited Cco-symmetry Y1 = -waw is a variational Cm-symmetry, for A = -w :
Y p ) ](L(y,
+ w) - vw)
= vw =
( D - W)(-v).
Now we use Theorem 3.1 to reduce again the order of the Euler-Lagrange equations associated to (19). A complete system of invariants of Yp,(’)l= -wa, (w2 - wl)dw,is given by { z = y , p = w %,PI} and let us consider p = -ln(w). In terms of coordinates { z , P , p , p l , p 2 } the vector field Y/[x’(l)lbecomes 8,. To construct the corresponding Y,[x’(l)l -invariant Lagrangians given by (9) we consider A = -up. In this case, (9) becomes L ( z , p ) - v p , for v E R,in coordinates { z , p , p , p l , p 2 } . The associated reduced Euler-Lagrange equation is given by
+
+
L(OJ)( z ,p ) - v = 0.
(20)
When this (algebraic) equation is solved, we get p = H ( z , v). By setting z = y and p = w+% we obtain the first order ordinary differential equation w 2 = H ( y , v). Let us denote by w = G(y, v,Cl), C1 E R, its general solution. Since w = %, by solving the first order ordinary differential equation u, = u.G ( x ,v,C1) we obtain a three parameter family of solutions to the original Euler-Lagrange equations.
+
5 . Conclusions
In this paper we have obtained a generalization of the concept of variational symmetries for Euler-Lagrange equations based on A-prolongations. As well as standard variational symmetries, we have proved that variational Cm-symmetries allows us to reduce by two the order of the EulerLagrange equations. This is a “partial” reduction, meaning that, in general, a one-parameter family of solutions can not be derived from the solutions of the corresponding reduced equation. It corresponds to solutions of the Euler-Lagrange equation for which the expression in the brackets of second member of (12) is neither null nor constant. In other words, in general, that
243 expression is not a first integral of the Euler-Lagrange equations. This can also be interpreted in terms of the formulation of Noether's theorem when A-prolongations are considered, and we will be dealt with in a separate paper'. Finally, we have shown how the new theory provides a method to reduce by four the order of equations with 2-parameter non abelian variational symmetries.
References 1. C. Muriel and J. L. Romero. New methods of reduction for ordinary differential equations. I M A J. Appl. Math., 66(2):111-125, 2001. 2. C. Muriel and J. L. Romero. Cm-symmetries and reduction of equations without Lie point symmetries. J . Lie Theory, 13(1):167-188, 2003. 3. C. Muriel and J. L. Romero. Prolongations of vector fields and the invariants by derivation property. Theoretical and Mathematical Physics, 133(2):289-300, 2002. 4. P.J. Olver. Applications of Lie Groups t o Differential Equations. SringerVerlag, New-York, 1993. 5 . C. Muriel and J. L. Romero. Cm-symmetries and equations with symmetry algebra sl(2, W). In S y m m e t r y and perturbation theory (Cala Gonone, 2001), pages 128-136. World Sci. Publishing, River Edge, NJ, 2001. 6. C. Muriel and J. L. Romero. Integrability of equations admitting the nonsolvable symmetry algebra so(3, R). Stud. Appl. Math., 109(4):337-352, 2002. 7. H. Stephani. Differential Equations, Their Solutions Using Symmetries. Cambridge, 1989. 8. C. Muriel, J.L. Romero and P.J. Olver. Variational Cm-symmetries and Euler-Lagrange equations. Forthcoming paper, 2004.
REGULARITY OF PSEUDOGROUP ORBITS
PETER J. OLVER" School of Mathematics University of Minnesota Minneapolis, M N 55455, U S A
[email protected]. edu
JUHA POH JANPELTO Department of Mathematics, Oregon State University, Corvallis, OR 97331, U S A
[email protected]. edu
Let G be a Lie pseudogroup acting on a manifold M . In this paper we show that under a mild regularity condition the orbits of the induced action of G on the bundle J " ( M , p ) of nth order jets of p-dimensional submanifolds of hl are immersed submanifolds of J n ( M ,p ) .
1. Introduction
Lie pseudogroups, roughly speaking] are infinite dimensional counterparts of local Lie groups of transformations. The first systematic study of pseudogroups was carried out at the end of the 19th century by Lie, whose great insight in the subject was t o place the additional condition on the local transformations in a pseudogroup that they form the general solution of a system of partial differential equations] the determining equations for the pseudogroup. Nowadays these Lie or continuous pseudogroups play an important role in various problems arising in geometry and mathematical physics including symmetries of differential equations] gauge theories] Hamiltonian mechanics] symplectic and Poisson geometry, conformal geometry of surfaces] conformal field theory and the theory of foliations. Since their introduction a considerable effort has been spent on develop*Work partially supported by grant DMS 01-03944 of the National Science Foundation.
244
245
ing a rigorous foundation for the theory of Lie pseudogroups and the invariants of their action, and on their classification problem, see e.g. Refs. [ 5 ] , [6], [7], [ 8 ] , [ Q ] , [ll] and the references therein. More recently, the authors of the paper at hand have employed a moving frames construction [3], [4] to establish a concrete theory for Lie pseudogroups amenable to practical computations. As applications, a direct method for uncovering the structure equations for Lie pseudogroups from the determining equations for the infinitesimal generators of the pseudogroup action is obtained (see, in particular, the work [l]on the structure equations for the KdVand KP-equations) and systematic methods for constructing complete systems of differential invariants and invariant forms for pseudogroup actions are developed. Moreover, the new methods immediately yield syzygies and recurrence relations amongst the various invariant quantities which are instrumental in uncovering their structure, the knowledge of which is pivotal e.g. in the implementation of Vessiot's method of group splitting for obtaining explicit noninvariant solutions for systems of partial differential equations. Let 8 be a Lie pseudogroup (a precise definition will be given in Sec. 2) acting of a manifold M . The action of 8 on M naturally induces an action of 4 on the extended jet bundle J " ( M , p ) of nth order jets of submanifolds of M by the usual prolongation process. Our goal in this paper is to prove that under a mild regularity condition on the action of the pseudogroup 6 on M the orbits of 6 in J " ( M , p ) are immersed submanifolds for n sufficiently large. We were originally lead to the problem in connection of the research reported in Ref. [4] and the result is of importance in the theoretical constructs therein. Interestingly, as we will see, the submanifold property of 8 orbits in J " ( M , p ) is closely related to local solvability of the determining equations for the infinitesimal generators of the pseudogroup action on M . The proof of our main result relies on classical work [la] on the structure of the orbits of a set of vector fields originally arising in the study of the accessibility question in the context of control theory. In Sec. 2 we cover some background material on Lie pseudogroups and discuss the regularity condition for pseudogroup actions needed in our main result. Sec. 3 is dedicated to the proof of the submanifold property of orbits of the action of 6 on the extended jet bundles J " ( M , p ) .
246
2. Tameness of Lie Pseudogroups Let D = D ( M ) denote the pseudogroup of all local diffeomorphisms of a for the nth order jet of 'p E V at z and dn): manifold. We write D(") + M for the associated jet bundle, where d")(j,"'p) = z stands for the ) ~ ( z=) Z source map. We furthermore write d"): 2%") -+ M , ~ ( " ) ( j ; ' p= for the target map.
jy'p
Definition 2.1. A subset Q c V is called a pseudogroup3 acting on M if (1) the restriction idlu of the identity mapping to any open 0 c M belongs to Q; (2) if (p, $J E Q , then also the composition 'p o $J E Q where defined; (3) if 'p E G, then also the inverse mapping 'p-' E Q.
A pseudogroup G is called a Lie pseudogroup if, in addition, there exists N 2 1 so that the following conditions are satisfied for all n 2 N : (4) G(") c D(")is a smooth, embedded subbundle; ( 5 ) n:+' : G("+') + Q(") is a bundle map; (6) a local diffeomorphism 'p of M belongs to G if and only if z is a local section of a(") : @") + M ; (7) Q(") = pr"-NQ(N) is obtained by prolongation.
-+
ji") 'p
We call the smallest N satisfying the above conditions the order of the pseudogroup, and unless otherwise specified, we will assume that n 2 N in what follows. Note that by (1) and (2), the restriction 'pp of a transformation 'p E Q to any open subset 0 of the domain of 'p is again a member of the pseudogroup. Fix local coordinates (zl,. . . ,z m ) about some p E M , and let ( z , Z )= ( z ' , . . . , zm, Z',. . . ,Z")denote the induced product coordinates about ( p , p ) E D(O) = M x M . Due to conditions (4)and (6) above, pseudogroup transformations are locally determined by a system
Fa(z,Z(n)) = 0,
a = 1,.. . , k,
(1)
of partial differential equations, the determining equations for Q. Here n 2 N is fixed and ( z ,Z(")) stands collectively for the coordinates of D(n) induced by ( z ,2).By Definition 2.1 the above equations are locally solvable, that is, given a jet g?) = (zo,Zp') satisfying (l),then there is a solution 'p E 4 of the equations so that j!y"p = g:"). Let X = X ( M ) denote the space of locally defined vector fields on M . Thus the domain U ( v )c M of v E X is an open subset of M . The vertical
247 lift V(") of a vector field v E X to D(") is the infinitesimal generator of the local one-parameter group a:") of transformations acting on D(")defined by @i"'(jp'cp) = j?)(at o cp), where at stands for the flow map of v. Note that the domain of V(") is d")-'(U(v)). Pick v = C~="=,u(z)&a E X and write V = Cr=lVa(2)&a for Then V(") is simply given by the usual the vertical counterpart of v. prolongation formula2, m
a=l l J l < n
where J = (jl,. . . , j p ) is stands for a multi-index of integers, DJ = D 31. . . . Djp for the product of the total derivative operators Dj = a,, CT=lCIJllo Z y j d p , and where the 2: denote the components of the fiber
+
coordinates on D(")induced by ( z ,2 ) .In particular, at the n t h jet the identity mapping, Eq. (2) becomes
vg
=
cc
~2Jv"(z)dz;,
IF)of (3)
a=l I J l < n
where we have again used the obvious multi-index notation. We denote the space of n jets of local diffeomorphisms with source at a fixed z E M by D(")12.It is easy to see that D ( n ) ~is2a regular submanifold Write Rh(,) for the right action of a jet h(") E D(")on the source of D("). fiber D(")IT(n)(h(n)) = ~ ( " ) - ~ ( d " ) ( h (by " ) Rhcn,g(") )) = jz(n)(h(n))(cpo +), where h(") = j,"c,,(hc,,)+, g(") = jm(n,(hcn))cp. Then, by differentiating the identity
R,(,) @.t")(g("))= a p (R,c,,g(")), it is easy to see that V(") is Rh(,)-invariant, that is,
Rh(,)*(V(")(g(")))= V(")(Rh(,)g(")),
(4)
whenever d")(g(")) = d " ) ( h ( " ) ) . Note that the action of R,(,) on V(")(g(")) is well defined since V(") is a vertical vector field. Next let G be a Lie pseudogroup. A local vector field v E X on M is a G vector field if its flow map is a member of G for all fixed t on some interval about 0. We denote the space of G vector fields by X,. The infinitesimal determining equations La(z,jp)v)= 0 for G vector fields v can be obtained by linearizing the determining equations (1) for G
at
248
at
I?),
that is, d ~~(z,j?)v) = - - ~ ~ ( z , j ! n ) a ~= ) Io~ =for ~ all z E ~ ( v ) . dt
(5)
By (2), this is equivalent to the condition
( v ( " ) F ~ ) ( ~ , I ?=) )o for all z E ~ ( v ) . As a consequence of our definition of a Lie pseudogroup, the infinitesimal determining equations completely characterize 9 vector fields.
X is a G vector field zf
and only
f o r all z E U ( v ) with s o m e n >_ N .
(6)
Proposition 2.1. A local vector field v E if
L,(z,j?)v)
=0
Proof. We only need to show that a vector field v satisfying ( 6 ) is a G vector field. First note that equation (6) implies that V ( " )is tangent to G(") at I?) for all z E U ( v ) . Thus, by the right invariance (4),V(")is tangent to G(") at any g(") E G(") n ~ l ( U ( v ) )and, , consequently,
Q ~ ) ( I ? ) >E @")
for all t .
(7)
But (7) implies that
'(")atE @") for all t and z E U(v), and consequently, at E for all t sufficiently small and thus v is a G vector 32
field.
!$in)
Recall that the lift of the flow of v E X to D(")is tangent to the fibers D(")12. We thus obtain canonical mappings" A)(, from the space of n-jets X," ( ', 2 = d " ) ( g ( " ) ) , of local vector fields into the tangent space Tg(n)D(n)12 of D(")12at g(") by A g ( 4 34") Z v ) == v ( " ) ( g ( " ) ) .
Proposition 2.2. T h e mappings A)(,
:
xP)
are well defined.
+T
~(~)D(~)I~,
where z = a ( " ) ( g ( n ) ) ,z = d " ) ( g ( " ) ) ,
(8)
249
Proof. We only need to show that if vl, v2 are two local vector fields so that j g ) v l = jg)v2, then V p ' ( g ( " ) ) = VP'(g(")) for all g(") with d " ) ( g ( " ) ) = 2.
(9)
By (3), Eq. (9) holds for g(") = IF). But then, due to the invariance (4) of Vp', V P )under the right translations Rg(n),Eq. (9) must hold for all g(") with d " ) ( g ( " ) ) = 2. 0 Let G be a Lie pseudogroup and write G(7'),z = G(") n 2)(")1,. Note is a regular submanifold of G("). In fact, the source mapping that zl)"@ dn)restricted to ($")I, is of maximal rank as is seen by observing that for 'p E 6, the local section jp'y of @"I -+ M yields a local right inverse for
d"). By the proof of Proposition 2.2, the mappings A),(
for g(") E G(")
restrict to mappings from the n-jets of G vector fields Xg' at 2 = d " ) ( g ( " ) ) into the tangent space Tg(n)G(n)~z of the source fiber Q(")I, at g(").
Definition 2.2. A Lie pseudogroup G is called t a m e at order n provided that the mappings A,(,
:
are isomorphisms for all g(") E
X;;L
--f
T,(,,G("),,
@").
Remark 2.1. By Eqs. (2) and (3), the mappings A)(, are automatically monomorphisms. Thus, by the right invariance (4), 6 is a tame Lie pseudogroup provided that ),A!, maps XG;i ( onto T,i,)G("$, for all z E M . Thus in particular, when G is tame, the dimension of the space of n jets of 4 vector fields at z E M is constant in z . Proposition 2.3. A L i e pseudogroup G is t a m e at order n i f and only if the nth order infinitesimal determining equations f o r 6 vector fields are locally solvable. Proof. We can use ( 5 ) to identify the solution manifold of the nth order infinitesimal determining equations with tangent vectors w E TI!,)@"). ,1 By Remark 2.1, a Lie pseudogroup is tame if and only if any w E T,!,,)G(")lz can be represented by a solution of the equations, that is, provided that the infinitesimal determining equations are locally solvable. 0 Remark 2.2. While the local solvability of the determining equations for
G is built into Definition 2.1 of a Lie pseudogroup, as far as we know,
250 the infinitesimal version of the equations does not necessarily possess this property. However, as of yet, we have been unable to construct an example of a Lie pseudogroup with infinitesimal determining equations that are not locally solvable.
3. Regularity of Orbits
Definition 3.1. Call g(") E G(") reachable by G vector fields if there are ~ 1 ,. .. , v, E X, with flow maps a:', . . . , so that g(") = jsi)(g(,,)) (a;: o . . o a:;) for some t l , . . . , t,. Write I(") = U z E ~ I [ p E) G(") for the image of the section generated by the identity mapping and denote the connected component of G(") containing I[(n) by G?).
Theorem 3.1. Let 6 be a t a m e L i e pseudogroup at order n. T h e n a n y g(") E
Gp) is reachable by 6 vector fields.
Proof. Denote by RlZoc G~)I,, the set of jets with source at z, that are reachable by G vector fields. Clearly R is non-empty. Our goal is to prove that Rlz0is both open and closed in G?'I,,. First, let g?) E Rlzo and write 2, = d " ) ( g p ' ) . Choose a basis w 1 , . . . ,w, for Tg~n,G(n),zo. By tameness, there are vector fields vj E X, defined in a neighborhood Nz, of 2, in M so that j = 1,.. . , r .
wj = v p ' ( g p ) ) ,
Due to linearity, any w = C;==, aiwi E tion
w=
c
Tg~,,~(")lZu is a linear combina-
aiVjn)(
gp)
i=l
of the vectors V ! " ) ( g p ) ) , Moreover, by shrinking Nz,, if necessary, it is easy to verify that there is E > 0 so that for any a = ( u l , .. . , a r ) with 11 a 11 < 1, the flow map of v = U ' V ~ . . . arvr is defined on (-el E ) x Nz,. By assumption,
+ +
g p
=jL;)(a;;
0 . .
.o
a;;)
for some 9 vector fields y l , .. , , y, and for some tl, . . . , t,. It is clear that there is a neighborhood Pzo of z, so that the composition
a& a; 0
0
'
' . 0 a; ( z )
251
is defined for all v = a l v l + . . . Next consider the mapping
+ arvr with 11 a 11 < 1 and z E Pz,.
k : { ( a ' , . . . ,a r ) } + ~ ( ~ ) l ~ ~ , k ( & . . . , a P ) = j ! ,") (@a'vl+-.+a"vT €12 where 1) a 1)
;@ ;
a;**),
o , ,,
< 1. Then at a1 = . . . = ar = 0, E
9,(a,;) = -wa.
(10)
2
Hence the Jacobian of k at u1 = . . . = ar = 0 is non-degenerate, and so, in particular, the image of iP contains an open neighborhood Qgin)of gp) Now by the definition of k , any g(") E
in Q(")l, .
is reachable by Q
Qp)lzo
vector fields and consequently, RlZoc is open. In order to show that Rlzois also closed in
Qp)lzo, pick a sequence RlZoof jets in Rlzoconverging to gp) E Q ~ ) l z o . By the first part of the proof, there is a neighborhood Qgin) of g?) in Sp)lI,,so that any {gi("'}
C
g(") E Q'"'
9P'
can be expressed as g(n) =
@p)(gp)
for some Q vector field v. Choose i so large that 9:"' virtue of (ll),we see that 90(n)=
(11) E
Qgp'.Then, by
@;'"'(gp)
for some 4 vector field v . Consequently, gp) E Rlzoand Rlzoc closed. This completes the proof of the Theorem.
Qp'l,, is 0
Next let V be a set of locally defined vector fields on a manifold M . An orbit 0, of V through p E M is the set of points that can be reached from p by a composition of flow maps of vector fields in V , 0, = ( 4 =
q
0 .
., o@;;(p) Itl,. . . ,t, E
IW,Vl,.
. . ,v, E V }
We equip 0, with the strongest topology that makes all the maps ( t l , .. . , t r ) E R'
-+
;@ ;
0 . .
. o q - ( p ) E 0,
(12)
continuous. One can easily verify that this topology is independent of the point p on the orbit. We call V everywhere defined if every p E M is contained in the domain of at least one v E V .
252 Theorem 3.2. Let V be an everywhere defined set of local, smooth vector fields on M and let Op be an orbit of V . Then Op admits a differentiable structure compatible with the topology defined by (12) in which it becomes an immersed submanifold of M . Proof. A proof can be found in Ref. [12].
0
We write J" = J " ( M , p ) for the nth order extended jet bundle consisting of equivalence classes of p dimensional submanifolds of M under nth order contact and IP : J" -+ M for the canonical projection. We denote the action of a local diffeomorphism 'p E V on a jet z(") E J" by 'p(") . z("), where z = ~ " ( z ' " ) )is contained in the domain of 'p. This action factors on J" in an obvious into an action of the diffeomorphism jet bundle V(") fashion. The infinitesimal generators of the action of V on J" are, by definition, the prolongations2 t o J" of local vector fields on M obtained in local coordinates by the usual prolongation formula. Similarly, for a Lie pseudogroup G the infinitesimal generators of the action of G on J" are, by definition, prolongations of G vector fields to J". We let g(") c X ( J n ) stand for the Lie algebra of infinitesimal generators of 9. We denote the g(") orbit of a point z(") E J" by Oz(n)and the orbit of z(") under the action of G on J" by Oi(n). Then Oz(n)consists of the points
where each vi is a G vector field. So obviously Oz(n)c OL(,,). Moreover, the orbit O;,,, agrees with the orbit of z(") under the induced action of the pseudogroup jet bundle G("), specifically,
We call a Lie pseudogroup G connected if the subbundle connected.
G(") c V(")is
Theorem 3.3. Let 6 be a tame Lie pseudogroup at order n acting on M . Then the orbits of the action of Q on J" are immersed submanifolds of J". Proof. First assume that is connected. By virtue of tameness of 4 and the prolongation formula, the Lie algebra g(") of infinitesimal generators is everywhere defined on J". Thus by Theorem 3.2, the g(") orbits Oz(n)are
253 immersed submanifolds of J". Hence, to conclude the proof of the Theorem for connected pseudogroups, we only need to show that OL,,, c Oz(,). For this, note that by Theorem 3.1, any jet g(") E G(") can be expressed as
where the vi are
G vector
which shows that O,:
fields and z = d")(g(")). Consequently,
c Oz(n).
Gp)
Next assume that G is not connected and let be a connected component of G(") distinct from the connected component GL") containing I("). Write = Gin) nG(n)lz and pick E @")~z. Since G is tame, we can
Gp)lz
gp)
proceed as in the proof of Theorem 3.1 t o show that any g(") E G$")iz can be expressed as
for some G vector fields v1,. . . ,v,. This implies that the orbit of z(") E J" under Gin) coincides with the orbit of . z(") under This completes the proof of the Theorem. 0
gp)
Gp).
References 1. J. Cheh, P. J. Olver, J. Pohjanpelto, Maurer-Cartan equations for Lie symmetry pseudo-groups of differential equations, preprint , University of Minnesota (2004). 2. P. J. Olver, Applications of Lie Groups to Differential Equations, Second Ed., Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993. 3. P. J. Olver, J. Pohjanpelto, Moving frames for pseudogroups. I. Invariant differential forms, preprint, University of Minnesota (2003). 4. P. J. Olver, J. Pohjanpelto, Moving frames for pseudogroups. 11. Differential invariants for submanifolds, preprint, University of Minnesota (2003). 5. N. Kamran, T. Robart, A manifold structure for analytic isotropy Lie pseudogroups of infinite type, J . Lie Theor?/ 11,57-80 (2001). 6. A. Kumpera, Invariants diffkrentiels d'un pseudogroupe de Lie. I, 11, J . Diff. Geom. 10, 289416 (1975). 7. M. Kuranishi, On the local theory of continuous infinite pseudo groups I, Nagoya Math. J . 15,225-260 (1959). 8. M. Kuranishi, On the local theory of continuous infinite pseudo groups 11, Nagoya Math. J. 19, 55-91 (1961). 9. J. F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach Science Publishers (1978).
254 10. A. Rodrigues, The first and second fundamental theorems of Lie for Lie pseudogroups, Amer. J . Math. 84, 265-282 (1962). 11. I. M. Singer, S. Sternberg, The infinite groups of Lie and Cartan. Part I (the transitive groups), J . Analyse Math. 15,1-114 (1965). 12. H. J. Sussman, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. SOC.180,171-188 (1973).
RELAXATION TIMES TO EQUILIBRIUM IN FERMI-PASTA-ULAM SYSTEM
S. PALEARI, T. PENATI Universita’ degli Studi d i Milano Bicocca, Dipartimento d i Matematica e Applicazioni, via R. Cozzi, 53, 20125 - Milano, Italy
[email protected],
[email protected]. it In this paper we report and discuss some recent results obtained investigating with numerical methods the celebrated Fermi-Pasta-Ulam model, a chain of nonlinearly coupled oscillators with identical masses. We are interested in the evolution towards equipartition when energy is initially given t o one or a few modes. Using the spectral entropy as a numerical indicator there is a strong suggestion that the relaxation time t o equipartition increases exponentially with an inverse power of the specific energy. Such a scaling appears t o remain valid in the thermodynamic limit.
1. Introduction In the last 50 years many studies have been devoted to the so called “FPU paradox”, which arose from the report of Fermi, Pasta and Ulam [8]. In one of the first numerical works ever made they studied a chain of non-linearly coupled oscillators with identical masses (see appendix for details). Quoting from the original words of their paper: “The ergodic behavior of such systems was studied with the primary aim of establishing, experimentally, the rate of approach to equipartition of energy among the various degrees of freedom of the system”. They were interested in looking how long does it take for the following limit to be actually reached:
where Ej is the harmonic energy of the normal modes and E is the total energy, starting with an initial condition such that, e.g., El = E . Surprisingly, in their experiments they could not see equipartition at
255
256 N=31
0=0.250
8=0.250
E-0.050
K-cyc I es
Pades 1
-
8
heraged energies
Mode
Figure 1. FPU chain (parameters shown in figure) with condition close to those used in the original experiments by Fermi, Pasta and Ulam. Left panel: Time evolution of the harmonic energy of the first eight normal modes. It is possible to see a sort of quasi periodic motion, with energy almost completely regained by the first mode at the end of the simulation. Right panel: Initial conditions (empty circles) showing all energy given at time zero to the first mode, and averages over the whole simulation (full circles) showing the spread of energy limited t o a few low frequency modes. Time in natural units of the fastest oscillator.
all. Quoting again from their work: '(Let us say here that the results of our computations show features which were, from the beginning, surprising to us. Instead of a gradual, continuous flow of energy from the first mode to higher modes, all of the problems show an entirely different behavior. (. . .) Instead of a gradual increase of all the higher modes, the energy is exchanged, essentially, among only a certain few. It is, therefore, very hard to observe the rate of 'thermalization' or mixing in our problem, and this was the initial purpose of the calculation". In Fig. 1 it is possible to see a simulation comparable with those explored by Fermi and collaborators. Further investigation has shown that the sharing of energy, if it happens, takes a very long time that, at low energies, becomes unobservable even with the most powerful computers. In the attempt of understanding the mechanisms leading to the phenomena briefly described above, also in order to check their possible persistence in the thermodynamic limit, different conjectures have been proposed. Izrailev and Chirikov [14] suggested that the practically ordered motion observed by Fermi could be explained in view of the KAM theorem.
257 Precisely they conjectured the existence of a threshold in energy, below which there is no equipartition: FPU original experiments were performed, according to these authors, below the threshold. But they claimed that such a phenomenon should be irrelevant in the thermodynamic limit. For, the energy threshold must fall to zero due to the increasing number of resonances created among normal modes when the number of particles N grows. It should be stressed that the applicability of the KAM theory has been proved only recently by Rink [15], in the case of the periodic FPU @-model. A different conjecture was proposed by Bocchieri, Scotti, Bearzi and Loinger [3]. They conjectured the existence of a threshold in specific energy E = E I N , whose relevance is discussed in [ll,12, 6, 41. Such a threshold could be connected with the more recent interpretation of the FPU’s phenomena based on Nekhoroshev exponential stability. Below the threshold, the equipartition might be reached, but only in times that increase as exp(Eca), with some positive a, i.e. times which could become longer than the lifetime of physical system under analysis (see, e.g., [lo, 91). After the formulation of the model, we will discuss some recent result about the rate of thermalization of FPU systems and its dependence on N , concentrating in particular on papers [7, 1, 21.
2. Natural packets: Two time scales
A very direct way to observe the approach to equipartition is to plot the distribution of energy among the modes, like in the right panel of Fig. 1, at increasing times. For sufficiently low energy one clearly observes that the energy, initially given only to the first mode, spreads quite rapidly to a few low frequency modes. This is the first time scale. Then the system enters a kind of metastable state, characterized by an extremely slow flow of energy towards the highest modes, until equipartition is possibly reached on a second and much longer time scale. In the paper by Berchialla, Galgani and Giorgilli [l]the idea of considering packets of modes is introduced. A packet of lenght s is defined as the set of the first s modes, with harmonic energy Es = El . . . E,. Starting with all the energy on the first mode means that all packets have initial energy E . If the system relaxes to equipartition, then the time average of E, tends to s E / N , i.e., the packet s has lost a fraction ( N - s ) E / N of energy. A critical time for a packet may be defined as the instant at which the packet s has lost an energy y ( N - s ) E / N . A plot of the critical time
+ +
258
10'
1o3
1o2
o4
1
Relaxat i o n t
o5
1
o6
1
irne
Figure 2. Natural packets phenomenon. N = 15, energy initially on the first normal mode. It is clear the presence of two time scale: The first one concerning the rapid formation of the natural packet, and the second and longer one, related to the time needed for its destruction.
for different energies is reported in Fig. 2, where different symbols refer to different packets. From such a picture it is clearly possible to distinguish the two time scales: The first one, which appears to be of power law type, related to the creation of the natural packet, and the second one related to its destruction and thus to equipartition of energy among all modes. In paper [l]it is also shown that the maximal frequency involved in the natural packet scale with no dependence from N in the range with specific energy as w 7 - 1023 (see [l],Fig. 7 ) . N
3. Equipartition times An attempt to give a quantitative estimate of the second time scale related to the equipartition is given, e.g., in the paper by De Luca, Lichtenberg and Ruffo [ 7 ] . They find a power law dependence of the kind T E - ~ . Similar considerations are contained also in a paper by Casetti, CerrutiSola, Pettini and Cohen [5]. N
259
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r
-
c90
al
-
.
=0 v.-
*
-
..
*
# ..
.
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I
I
,.=* *
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-
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-
-
em*..
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.#
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-
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Using acadmic 3.1. Numerical indicators The numerical indicators used in the papers cited above were respectively the spectral entropy in [7] and Lyapounov exponents in [5]. We briefly recall the relevant definitions relative to the first one: Introduce the quantities N e j := E j / C jEj and define S := - Cj=l ej In e j , and neff := e S / N . The latter quantity varies in [0,1] and can be interpreted as an effective number of modes involved in the dynamics. Fig. 3 represents the typical time evolution of such an indicator, for different specific energies in a @-chain. 3.2. Exponentially long t i m e s t o equipartition An estimate of the relaxation time to eqipartition in the light of Nekhoroshev theory has been performed in a subsequent work by Berchialla, Giorgilli and Paleari [2]. In order to compare the results with those in Ref. [7], the FPU @-model has been considered, with @ = 0.1, initial data i.e., with energy equally distributed on a packet of modes with fixed frequency range. The specific energy has been varied in the range [0.0089,7.7] for N E (255,511,1023). For every one of these conditions 25
[v, w],
260
.-I
c
m
P
Figure 4. Exponentially long times. Time at which n,ff = 0.4 vs. the fourth root of the inverse of specific energy, in semi-log scale. The exponential behavior clearly fits the data very well.
different orbits have been integrated, changing randomly the phases of the oscillators: The indicator used is the average over these different orbits. In order to see the scaling of the relaxation time with the specific energy, we plotted, for every E the time at which n,ff overcomes a fixed threshold, that we choose equal to 0.4. The results are illustrated in Fig. 4. There is a numerical evidence supporting the exponential scaling of relaxation times to equipartition with respect to specific energy, with a possible law of the type T exp(e-lI4). N
3.3. Thermodynamic limit In order to check the persistence of these phenomena in the thermodynamic limit we repeated the same calculation as before varying N E [63,327671 for a couple of fixed values of specific energies E E {0.052,0.14}. In Fig. 5 one clearly sees that, after an initial decrease of the times for the first low values of N , the subsequent points reported in the plot relative to values of N greater than 1023 remains practically constant, thus showing a numerical indication of the possible persistence in the thermodynamic limit of the phenomena presented in paper [2].
261 0
~=1.37e-01
~=5.16e-02
A
1 ul
o
N
z
111111
U
4
I
lo2
z
. I
.
o
I I l l l l l
.
o
I
I
o
o
1 1 1 1 1 1 1
z
4 lo3 4 loL Number o f oarticles
I
z
I
4
Figure 5. Thermodynamic limit. Time needed by n,tf to overcome the threshold 0.4 vs. the number of particles N , for a couple of specific energies. After an initial transient, there is clearly no dependence on N .
4. Further explorations
We did not mention up to now the content of the paper [5]. They study the problem of the relaxation times to equilibrium using the maximal Lyapunov exponent, since they consider it a more reliable indicator. They explot the different behaviour of such an indicator for the FPU system and for the Toda chain, which is an integrable system “close” in some suitable sense to the FPU one. In such a way they are able to introduce an estimate of the relaxation times, which they find to scale as a power law. They also claim that for increasing N any stochasticity threshold vanish, thus conjecturing that the FPU phenomena are not relevant in the thermodynamic limit. In a forthcoming paper [13] we will investigate the dynamics of the FPU system through the use of Lyapounov indicators in order to better compare and discuss the results presented here with those of [5]. Appendix: The model The FPU a , P-model as a Hamiltonian system is given as follows:
H(Z>Y)=
c“p; + j=1
1
1 2
V(Zj+l - Zj) , V ( s )= -s 2
P4 + -s3 + -s4 , 3
262
+
describing the onedimensional chain of N 2 particles with fixed ends, which are represented by xo = xN+1 = 0. For the free particles, X I ,. . . ,X N are the displacements with respect to the equilibrium positions (that obviously exist). The normal modes are obtained by
(qk,pk) being the new coordinates and momenta. The quadratic part of the Hamiltonian in the normal coordinates is given the form N
HZ
=
C E~ ,
j=1
E~ = 1 (p;
+ wjqj)
,
w j = 2 sin
jn
2(N
+ 1)
E j being the harmonic energies and w j the harmonic frequencies. Acknowledgments The authors wish to thank Antonio Giorgilli and Luisa Berchialla for useful discussion and for the proof reading of the manuscript.
References 1. L. Berchialla, L. Galgani, and A. Giorgilli. Discrete Contin. Dyn. Syst. Ser. B, (2004). (to appear). 2. L. Berchialla, A. Giorgilli, and S. Paleari. Phys. Lett. A , 321,167 (2004). 3. P. Bocchieri, A. Scotti, B. Bearzi, and A. Loinger. Phys. Rev. A , 2, 2013 (1970). 4. A. Carati, L. Galgani, A. Ponno, and A. Giorgilli. Nuovo Cimento B ( l l ) , 117, 1017 (2002). 5. L. Casetti, M. Cerruti-Sola, M. Pettini, and E. G. D. Cohen. Phys. Rev. E (3), 55, 6566 (1997). 6. C. Cercignani, L. Galgani, and A. Scotti. Phys. Lett. A , 38,403 (1972). 7. J. De Luca, A. Lichtenberg, and S. Ruffo. Phys. Rev. E (3), 60, 3781 (1999). 8. E. Fermi, J. Pasta, and S. Ulam. In Collected papers (Notes and memories). Vol. 11: United States, 1939-1954, (1955). 9. F. Fucito, F. Marchesoni, E. Marinari, G. Parisi, L. Peliti, S. Ruffo, and A. Vulpiani. J. Physique, 43,707 (1982). 10. L. Galgani, A. Giorgilli, A. Martinoli, and S. Vanzini. Phys. D, 59, 334 (1992). 11. L. Galgani and A. Scotti. Phys. Rev. Lett., 28, 1173 (1972). 12. L. Galgani and A. Scotti. Raw. Nuovo Cimento (2), 2, 189 (1972). 13. A. Giorgilli, S. Paleari, and T. Penati. preprint (2004). 14. F. M. Izrailev and B. V. Chirikov. Sou. Phys. Dokl., 11, 30 (1966). 15. B. Rink. Comm. Math. Phys., 218, 665 (2001).
ENERGY CASCADE IN FERMI-PASTA-ULAM MODELS
A. PONNO, D. BAMBUSI Universitci degli Studi di Milano, Dipartimento d i Matematica ‘%. Enrigues”, via Saldini 50, 20133 Milano, Italy
[email protected];
[email protected] We show that, for long-wavelength initial conditions, the FPU dynamics is described, up t o a certain time, by two KdV-like equations, which represent the resonant Hamiltonian normal form of the system. The energy cascade taking place in the system is then quantitatively characterized by arguments of dimensional analysis based on such equations.
1. Introduction The problem posed by Fermi, Pasta and Ulam (FPU) [l]concerns “in large” the dynamical characterization of the approach t o equilibrium of nearlyintegrable Hamiltonian systems with many degrees of freedom, which is obviously relevant to build up a meaningful statistical mechanics. FPU considered weakly nonlinear oscillator chains, for initial conditions with energy in the lowest Fourier mode (the longest wavelength mode) and numerically integrated the equations of motion paying special attention to the evolution in time of the modal energies. As is well known, the expected fast trend t o energy equipartition among the Fourier modes was not observed, which is what is known since then as the FPU paradox. For references on history, consequencies and relevant results in the field see [2, 3, 41. The aim of the present contribution is t o look at the FPU problem from a somehow new point of view, which allows us to give some quantitative estimate of physically relevant quantities characterizing the transfer of energy from large spatial scales, where it is put initially, t o small ones or, in other words, from low Fourier modes (large wavelength) t o high (short wavelength) Fourier modes. We will refer t o such a process as to the energy cascade, or simply the cascade. The term is borrowed from the theory of hydrodynamic turbulence, which actually displays a phenomenology similar to that of the FPU problem.
263
264
In the present paper, in order to avoid technical difficulties of minor importance, in place of considering the FPU model itself, we study the simpler problem of a class of PDEs which are naturally related t o it. The structure of the paper is the following. First of all it is recalled how certain PDEs arise in the study of the FPU problem. Then we endow such PDEs with a proper Hamiltonian structure and suitably simplify them by performing one step of averaging. Finally, through dimensional analysis we give an estimate of the effective number n,ff of degrees of freedom sharing the energy and thus actually involved in the dynamics. We also estimate the time 7 needed to reach this state of partial equipartition. As will be shown, such estimates turn out to be in agreement with some recent numerical results available in the literature. The presentation is quite informal; the material consists essentially of “snapshots” taken from a quite longer work in progress by the present authors.
2. Boussinesq equations modeling FPU chains The equations of motion of a weakly nonlinear oscillator chain are
+
?, = [A(T grP-’)],
,
n = 0 , . . . ,L .
(1)
Here T, = qn+l - q,, where q, is the displacement of the n-th particle from its equilibrium position on the chain; A is the usual discrete laplacian ([Af], = fn+l fn-l - 2fn), while g > 0 is the coupling constant and the integer p 2 3 is the degree of nonlinearity (in the potential). We will suppose the chain to be periodic of period L , i.e. ~ ( t=) T L ( t ) , for any t 20. Now, for long-wavelength initial excitations, which is the problem of interest in the present work, finite differences, such as T,+I - T,, are small; one can then formally expand the operator A = 4sinh2(d,/2) appearing in (1) in powers of 8, and retain the first few terms in the r.h.s. of the equation. Renaming the spatial independent variable n as 2,we get a PDE for the continuous field T ( Z , t ) , namely
+
rtt
=
[. + (1/12)Tzz + gTp-l]zz ,
r(0,t)= T(L,t)
.
Such a PDE is a generalized Boussinesq (gB) equation, and was considered as a starting point in approaching the FPU problem e.g. in [5]and [6]. In introducing a PDE, we pass from a system with a finite number (precisely L ) of degrees of freedom to a system possessing infinitely many
265 degrees of freedom. But of course the gB system is meaningful, i.e. it represents a good approximation of the original system, only if finite differences remain small, that is to say only if long-wavelength Fourier modes take part in the evolution in a significant way. The consistency of the approximation breaks down when modes of wavelength of order one (the size of the lattice step of the original chain model) receive a significant amount of energy or, in other words, when a number of degrees of freedom of order L is excited. From now on, we will focus on the gB equation (2).
3. Hamiltonian structure of the g B equation
To our knowledge, the Hamiltonian structure of the gB equation (2) was pointed out first by Zakharov [6], in a famous paper where he showed that the (properly said) Boussinesq equation, corresponding to the case p = 3 is in fact integrable (in the Lax and in the Hamiltonian sense). Zakharov introduced an auxiliary field (periodic on [O,L])thought of as the coordinate, while r was thought of as the corresponding conjugate momentum. Then, if one defines the Hamiltonian
the corresponding pair of Hamilton equations associated to H , namely
at = 6H/6r = r + ( 1/12)rxx + grp-' rt =
-6H/6@
=
,
aXx
(4)
turns out to be equivalent to the second order gB equation. Notice that the L r dx,which has to be set to zero, since flow of equations (4) preserves 0. in the original periodic lattice one always has CtZi r, Now, for our purposes, it is quite convenient to perform a noncanonical change of variables which is analogous to that used e.g. by Craig and Groves [7] in approaching the water wave problem. Let us introduce the change of where variables ( r ,a) H ([,
so
v),
Then, after substitution, the Hamiltonian (3) reads
=
266 while the equations of motion (4) transform into
[E + (1/24)([
Et
=
77t
= -177
+ ~),x + (g/2””)(E +
~)”-‘]x
(7)
+ (1/24)(E + 77)xx + (9/2”’”(E + 77)”-1]z
The latter equations can be quite conveniently rewritten in Hamiltonian form, namely
where C denotes the diagonal Pauli matrix diag(1,-1). It can be easily checked that 28, is a degenerate Poisson operator. The corresponding Casimir invariants of the system are the linear functionals of the form Jt(cl[+c277) dx, with arbitrary constants c1 and c2. By the definition of the L variables E and 77 given in ( 5 ) and by the geometric condition r da: = 0 one deduces that the physically meaningful Casimir leaf is the one defined L L 77 da: = 0. by Jo E dx =
so
so
4. Averaging
One has to keep in mind that a typical long-wavelength initial datum for equations (7) is EO(Z)
= VO(a:) = &cos(2rx/L)
I
(9)
where E plays the role of the specific energy (energy per degree of freedom) in the original FPU system. Indeed, substituting (9) in the expression of the Hamiltonian (6) yields If[&,,7701 = E = EL+o(EL),the leading contribution EL coming from the first two terms of the Hamiltonian. Moreover, if one evaluates the r.h.s. of equations (7) on the initial datum (9), one realizes that the leading terms in the evolution equations for E and 77 are Ex and -qx, respectively. The other terms turn out to be small because both E and 1/L are supposed to be small quantities. As a consequence, one can regard the Hamiltonian (6) as being a perturbation of dx . According to a standard technique in perturbation theory [8], one can then average the whole Hamiltonian (6) over the flow generated by the unperturbed Hamiltonian HO (lo), and thus simplify the dynamics. Such a flow,
267 acting on vector-valued fucntions periodic on [0,L],is given by
at = , p a , and the averaged Hamiltonian
(&=@=I), 3
(11)
JtH [ W ( E q, ) ] d s / Lturns out to be
In the above expression, Cg = p ! / ( n ! ( p- n ) ! )while ( f j ) = f j d x / L ; we will refer to the latter as to the moment o f f of order j, or simply the j-th are moment of f . The equations of motion associated to
SoL
{
Et
= Ex
+ (1/24)Exxx + (g/2"/2) c",;C;-'(qp-"-1
Vt = -qx - (W4)VXXX - (9/2"/2)
c:;
)(t")X '
(13)
c:-1(Ep-n-1)(7")x
These are generalized Korteweg-de Vries (gKdV) equations. One can easily check that for p = 3 and p = 4 they yield, respectively, the KdV and the modified KdV equation, which are both integrable. Notice that now, as a consequence of averaging, the two moments of second order of E and q are constants of motion for system (13). Moments of order greater than two will be time-dependent, and as a consequence the above equations are actually coupled and of integro-differential type for P26. Up to this point we have shown that, for long-wavelength initial data, the gKdV equations constitute the resonant hamiltonian normal form of the FPU system (actually represented by the gB equation). We recall that in the case of short-wavelength initial conditions the resonant normal form has been shown to be constituted by nonlinear Schroedinger equations [9]. 5 . Dimensional analysis
The fundamental role played by the KdV equation in the FPU problem (in the case p = 3) was pointed out first by Zabusky and Kruskal [lo]. In such a work they pointed out that the flow of such an equation displays two distinct regimes, which we recall here. Keeping in mind once again that initial data for the problem in study have the form ( 9 ) , at very short times the derivatives will be small, and the dispersive terms txXx and qxxx can be neglected in the equations of motion (7); as a consequence one gets two generalized Hopf (or inviscid Burgers) equations whose flow would display singularities in a finite time. Anyway, in going towards the singularity
268
the derivatives increase and at a certain time the terms txxxand q,,, can no longer be neglected. Then dispersjon becomes important and a sort of balance between dispersion and nonlinearity prevents the shock formation, thus giving rise to solitons (for a recent aprroach to the FPU problem strongly based on solitons see [ll]). At this stage one expects a state of partial equilibrium to have been reached. So, there remains only to translate in quantitative terms what has been just said, and this is easily achieved through dimensional analysis. An enlightening treatment of the mat hematical foundations of dimensional analysis can be found in the book of Gallavotti [12]. If one denotes by M the typical value of the fields t and q at time t and by e the typical length-scale of variation of the same fields, one then MI[. Analogously, if T denotes the typical time-scale estimates e.g. Ex of variation of the fields, then one has e.g. & M / r . Now, in the equations of motion (7), following what pointed out by Kruskal, one can neglect the dispersive term if the ratio of the dispersive terms to the nonlinear ones appearing on the r.h.s. of the equations is less than 1, namely if
-
-
One the other hand, if such an inequality holds, the dynamics is ruled by the a generalized Hopf equation, and this has the property that the maximum of its solution is a constant of motion. Thus, recalling (9), one can set M = &,which inserted in (14) yields
emin
The latter inequality must be interpreted as follows: gives the order of magnitude of the smallest spatial scale to which energy flows; at smaller spatial scales the dynamics is essentially dispersive and the corresponding Fourier modes of the system are almost frozen. The relaxation time ~~~l needed for the energy to flow up to the spatial scale emin is the one for namely which dispersion becomes important and & txXx,
-
It must also be stressed that if the minimal wavelength to which energy is transfered is X then the highest Fourier mode involved in the l/emin. Such a value of dynamics is the one corresponding to kmax/L kmax/L of coincides with the fraction n,ff of degrees of freedom of the system actually sharing the energy.
- emin,
-
269
6. Comments First of all it has to be pointed out that from the estimates (15) and (16) one realizes that the quantities characterizing the cascade are intensive: they depend only on the specific energy E = E / L . Thus, at least at a formal level, such estimates hold in the thermodynanlic limit E + 03, L -+ 00 at E / L fixed. Of particular significance is the fact that n , f f E ~ ( P - ’ ) / ’ , which is a small number if E is small, so that the system does not reach equipartition at least on time scales of order r,,~. The numerical results available in the literature are mostly concerned with the case p = 3. In such a case, the scaling k,, was observed both by Berchialla et al. [13] and by Biello et al. [14], while in reference ~ was measured. The agreement with the [14] the scaling law r,,~ E - ~ /too simple predictions given in the present paper seems thus to be promising. N
N
N
Acknowledgments The authors thank G. Benettin, A. Carati, L. Galgani, A. Giorgilli and S. Paleari for the many enlightening discussions that “randomly” take place on the subject. References 1. E. Fermi, J. Pasta and S. Ulam, in E. Fermi “Collected Papers”, University of Chicago Press (1965). 2. G. Benettin, in “Molecular dynamics simulation in classical statistical mechanical systems”, Proceedings of the E. Fermi Summer School of Varenna (1986); ed. by G. Ciccotti and W. G. Hoover. 3. J. Ford, Phys. Rep. 213,271-310 (1992). 4. A. Carati, L. Galgani, A. Ponno and A. Giorgilli I1 Nuovo Cimento B 117, 1017-1026 (2002). 5. N. J. Zabusky, in “Nonlinear Partial Differential Equations”, Academic Press, New York (1967); ed. by W. F. Ames. 6. V. E. Zakharov, Sow. Phys. JETP 38, 108-110 (1974). 7. W. Craig and M. D. Groves, Wave Motion 19,367-389 (1994). 8. J. A. Sanders and F. Verhulst, “Averaging Methods in Nonlinear Dynamical Systems”, Springer-Verlag, New York (1985). 9. D. Bambusi, A. Carati and A. Ponno, Discrete and Continuous Dynamical Systems-Series B 2, 109-128 (2002). 10. N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15,240-243 (1965). 11. A. Ponno, Europhysics Letters 64, 606-612 (2003). 12. G. Gallavotti, “Foundations of Fluid Dynamics”, Springer-Verlag, Berlin (2001).
270 13. L. Berchialla, L. Galgani and A. Giorgilli, Discrete and Continuous Dynamical Systems-Series B, (2004) in print; available on the web page of A. Giorgilli: www.matapp.unimib. it/-antonio. 14. J. A. Biello, P. R. Kramer and Y. V. Lvov, in “Proceedings of the fourth in-
ternational conference on dynamical systems and differential equations, Wilmington, NC, USA (2002)”; available on the web: arXiv: nlin. CD/0210007 vl and arXiv:nlin.CD/0210008 vl.
ON BIRKHOFF METHOD FOR INTEGRABLE LAGRANGIAN SYSTEMS
GIUSEPPE PUCACCO Dapartimento d i Fisica - Universith d i Roma “Tor Vergata” INFN - Sezione d i Roma 11
[email protected] We review Birkhoff method based on conditional integrals and apply the approach to find integrable two-dimensional Lagrangian or Hamiltonian systems with scalar and vector potentials.
1. Introduction
The configurational approach to the direct determination of additional integrals in autonomous Hamiltonian systems has been shown to be quite effective in looking for integrable system^.^,^,^ Configurational (or “conditional” or even “weak”) integrals hold for a specified particular value of the energy constant and were firstly investigated by B i r k h ~ f f .Although ~ they have been discussed several t i m e ~ ,their ~ ~ use ~ > is~ still not very common, probably for an understatement attitude. In this note we provide additional arguments to clarify matters, using the weak invariance condition to explore conditional integrability and showing how the approach is a shortcut also towards standard ( “strong”) integrability, namely the case in which integrals hold for arbitrary values of the energy constant. 2. Lagrangian approach
To appreciate all the features of the generic behavior of dynamical systems it is enough to limit the attention to two-dimensional autonomous Hamiltonian systems. These systems are in general non-integrable but it is enough to check the existence of one additional independent integral of the motion to determine a completely integrable system. In the case of a weak integral, this condition is satisfied on a fixed energy hypersurface.6 Many investigations have been devoted to time-reversible systems; much less to irreversible systems, due t o to the complexity of the set of equations ensuing from the
271
272 invariance condition. Here we want to show how the case with a linear dependence on momenta, very interesting for its physical applications, can also be addressed with the configurational approach. In presence of gyroscopic terms, it is simpler to work in the Lagrangian framework. Moreover, since in two dimensions it turns out t o be very useful to exploit conformal coordinate transformations, we will use complex coordinates and velocities defined by z=x+iy,
u=i, U = Z ,
Z=x-iy,
(1)
with dot denoting differentiation with respect to ordinary time. Consider a Lagrangian of the form
L
= iuii - 28{@u}-
w,
(2)
where W ( z ,5) is the real scalar potential and @ ( z ,2 ) is a complex function. The equations of motion are ?i
+ 2(iRU + Wz)= 0
(3)
and its complex conjugate, where the real function
R = 23{@z}, (4) will be referred to as the vector potential. A real phase-space function F ( z ,2 , u,6) is an integral of motion if it is preserved along the flow, namely if dF - = 2!J2{uFz - 2(Wr ZRU)F!} = 0 , (5) dt along the solution of equations (3). A first integral of the system always exists, the Jacobi function,
+
J ( z , Z , U , 2L)
= +uZl+
W ( ZZ, ) .
(6)
With a slight abuse of language, referring to the constant value C of the function J on a given invariant hypersurface, we can speak of the ‘energy’ of the system: actually, only in the case of a constant R playing the role of an angular velocity, C is the energy in the rotating frame. The direct approach to find a second integral consists in making a suitable ansatz on its form involving arbitrary functions, plugging it into the invariance condition ( 5 ) and solving for the unknown functions, including the potentials. The most common and useful ansatz is that of a polynomial in the velocities. Only the case of the linear integral
F ( l ) = AU + A2L
+ B,
(7)
273 can be solved in full generality4 and several examples of weakly-integrable systems can be found.' The case of quadratic integrals is instead still lacking of a general solution.9~10~11~12~13 Recently, a general procedure concerning strong quadratic integrals14 has been presented and some new results on cubic integrals have also been p r 0 ~ i d e d . lMoreover, ~ the challenging issue of the separability of the Hamilton-Jacobi equation has been attempted.16 Let us examine the quadratic case:
F(') = +(Su2+ Su2)+ +Quu+ +(Ru+ Ru)
+ P,
(8) where P and Q are real functions and R and S are complex functions. Plugging F ( 2 ) inside eq.(5) provides a cubic polynomial in the velocities which must identically vanish:
+ 4u3Sz+ +u2uQz+ ~ ( U U R+, u2&) + UP,+ + iRu)(+Su+ +uQ + + R ) }= 0.
dtF ( 2 ) = 8{ 'u.ci2 S,
-2(Wz
(9)
The procedure to solve for the unknown functions is dictated by a possible constraint on phase-space variables. In practice there are two possibilities: no constraint which implies that each coefficient of the polynomial (9) must separately vanish; fixed 'energy' constraint, namely
c,
J ( z ,z,u,u)= (10) and a corresponding extra link between the coefficients. We remark that the constraint-free approach must be reobtained from the constrained one by a suitable relaxation step. Comparing (10) with (6), the simplest way to impose the constraint above is that of writing uU = 2G, (11) where the 'Jacobi potential', G = C - W , has been introduced. We observe that this condition implies the disappearance of any cross terms, namely terms with powers of uu both in the polynomial (9) and in the integral itself. The only coefficients which must separately vanish are those of u k ,0 5 k 5 3 and their complex conjugate, so that
sz = 0, + 2iRS = 0 , Kz + SG, + +S'G + iRR = 0 , Rz
(12) (13) (14)
q ( R G ) , ) = o., (15) where, in (14) we have already exploited ( 1 2 ) , that expresses the fact that S = S ( z ) is an arbitrary analytic function, and defined the function K
=P
+ + GQ,
(16)
274
which accounts for the disappearance of the uii term in the integral (8). Equations (12-15) express the conditions in order t o F ( 2 )of the form (8) be an integral at some given fixed C: if we are interested in a stronglyintegrable system, in the end we must get a solution independent of C. To this purpose, we can write the integrability condition for K which, computing K,, = KErfrom (14), is
S{S”G
+ 3SG, + 2SG,, + 2i(RR),} = 0.
(17)
Now, using the definition G = C - W , we see that if (17) must be satisfied for every C, it turns out that
S { S ” ( z ) }= 0.
(18)
This result, valid even in the purely scalar case R = 0, is related with classical separating coordinates systems. In fact, the function S can be used to ‘standardize’ the integral expression through the conformal coordinate iY defined by transformation w = w ( z ) = X
+
so that
+
F ( 2 )= R{ ( w ’ ) + ~ Ed} +K.
(20)
It can be shown’ that the coordinates systems compatible with condition (18), that can also be expressed as IS1 = A ( X )+ BW)’
(21)
with A and B determined by the specific form of S , are the Cartesian, polar, parabolic and elliptical systems, so that we may grasp a first fundamental result connected with weak integrability: any conformal coordinate system in the plan allows for a family of weakly-integrable systems.
3. Hamiltonian approach In the Hamiltonian framework, the Legendre transformation gives p = +u- a, H = 2(p+
a)@+6)+ w,
(22)
so that
d2) = s ( p+
+ s ( p + 6)2+ R ( p + a) + R ( p + 6)+ K .
(23)
275 The Hamiltonian formulation of the weak invariance is the following ‘weak involution’ condition
{ F ( M ) , H }= f ( M - I ) ( H
-
C),
(24)
where f ( M - l ) is a homogeneous polynomial of degree M - 1 in the momenta. In the present case, M = 2, we may assume f ( l ) = %{+p} and follow a line of reasoning as above. The result is again given by the set (12-15). 4. Strong integrability Recently,14 we have obtained a general procedure to solve this system in the arbitrary energy case. Under the conformal transformation, eq.( 13) transforms into
R, + 2ifi = 0.
(25)
Since the conformal field is real, the solution of (25) is
-
R = -4Ztw,
(26)
where t is an arbitrary real function. In this way, the conformal vector field is given by
R
= 2tWW.
At the same time, eq.(15) at arbitrary C implies
R =-4i~z and
w = W(V),
(29)
with v another arbitrary real function. Comparing (28) with (26) and taking into account of ( 2 1 ) , we get the following differential equations for the functions q and E
A’VY + B’VX - 2 ( A A’ty B’Ex 2 ( A
+
+
+ B)VXY= 0 , + B)&yy = 0.
(30) (31)
Finally, putting all these results into the integrability condition ( 1 7 ) , we obtain VXY W”(q)+ 3-W’(V) V X W
=
2 fly - flx -
A+.(,
V X )
Therefore, the strategy to find strongly integrable systems with scalar and vector potentials supporting quadratic invariants is: to chose a suitable S
276
in the class determined by condition (18) (and therefore A and B ) and solve (30-31) to find q(X, Y) and [(X, Y); to solve (27) to find
ti = 3"xx + t Y Y )
(33)
and use the conformal transformation to find
c?
O(X,Y) = -
A+B and, finally, try to solve (32) for W = W ( q ( X ,Y ) )
(34)
5. Examples
In the general treatment14 we have obtained several solutions of this problem. In order to solve eq.(32), it is natural to assume that in general the condition
is satisfied, where Q, is arbitrary. Actually, it turns out that this condition is too restrictive in the particular case in which one can assume a linear dependence of the scalar potential on its argument, say
w = iaq,
(36)
with a real constant. In this case we see that (32) can be written in the following form
a ( A + B ) V X Y= V
X ~ Y V Y ~ X ,
(37)
which can be satisfied using for example polynomial solutions of (30). We remark that in the cases we obtained so far,14 these solutions are irrational functions. In the following subsections we provide other solutions based on the linear ansatz (36).
5.1. Polar coordinates
In this case the coordinates are r = ex,O = Y and we have A = e2x r 2 ,B = 0, so that eq.(30) is solved by 77 = eXg(Y)
+ f(X).
=
(38)
Using (33) we get, 2 0 = e-4x
(f'- 2f')
- e-3x (9''
+ 9).
(39)
277
In order to solve (37) we can make the choice g = bcosY
+ csinY,
Putting all together and using standard polar coordinates we get the fields a a = 4p - -r2, 2 24
(43)
5 . 2 . Parabolic coordinates
In this case the coordinates are such that x = X2 - Y2,y = 2XY and we have A = 4X2,B = 4Y2. A polynomial solution of eq.(30) is
q = a(X2+ Y2)2(5(X4+ Y4) - 6X2Y2)
(44)
so that
E
= -a
[+(y6 - x 6 )+ x2y2(x2 - Y')]
(45)
and
R = 6a(X2 - Y2). Putting all together and using standard Cartesian coordinates we get the fields
R
= 6ax,
(47)
w = -a2r2(5z2 + y2).
(48)
5.3. Elliptical coordinates
In this case the coordinates are such that
+ J(r2 + A2)2- 4A2y2, 2A2 sin2 Y = r2 + A2 d(r2 + A2)2 - 4A2y2,
2A2 sinh2X = r2 - A2
-
and we have
A
= A2 sinh2 X,
B
= A2 cos2Y.
A polynomial solution of eq.(30) is q = aA6(sinh2X - cos2Y)(sinh2X
+ cos2Y)2
278
so that
A'
[ = -[[3(4cos2Y+cos4Y-3cosh2X+cosh4X)-16cos2
16
Y sinh2X] (53)
and
f2 = 4A2[3(sinh2X
+
- cos2 Y) 21.
(54)
Putting all together and using standard Cartesian coordinates we get the fields
f2 = 4a(3r2 - A2), W = -16a2(r2 - A2)[r4+ 2A2(z2- y2) + A'].
(55)
(56)
Acknowledgments
It is a pleasure to thank Vladimir Matveev, Claudia C h a m and Giovanni Rastelli for very fruitful discussions.
References 1. K. Rosquist and G. Pucacco, J . Phys. A 28, 3235 (1995). 2. M. Karlovini and K. Rosquist, J . Math. Phys. 41,370 (2000). 3. M. Karlovini, G. Pucacco, K. Rosquist and L. Samuelsson, J . Math. Phys. 43, 4041 (2002). 4. G. D. Birkhoff, Dynamical Systems (Amer. Math. SOC.Coll. Publ., 9, New York, 1927). 5. L. S. Hall, Physica 8D, 90 (1983) 6. W. Sarlet, P. G. L. Leach and F. Cantrijn, Physica 17D,87 (1985) 7. J. Hietarinta, Physics Report 147,87 (1987). 8. G. Pucacco, Cel. Mech. and Dyn. Astr., in press (2004). 9. B. Dorizzi, B. Grammaticos, L. Ramani, and P. Winternitz, J. Math. Phys. 26,3070 (1985). 10. E. McSween and P. Winternitz, J . Math. Phys. 41,2957 (2000). 11. J. BQrubQand P. Winternitz, preprint arXiv:rnath-ph/0311051 (2003). 12. H. M. Yehia, J. Phys. A 25, 197 (1992). 13. H. M. Yehia, J . Phys. A 32, 859 (1999). 14. G. Pucacco and K. Rosquist, J . Math. Phys., in press (2004). 15. H. M. Yehia, J . Phys. A 35, 9469 (2002). 16. S. Benenti, C. Chanu and G. Rastelli, J. Math. Phys. 42,2065 (2001).
SYMMETRY OF SINGULARITIES AND ORBIT SPACES OF COMPACT LINEAR GROUPS
G. SARTORI, G. VALENTE Dipartimento d i Fisica, Universiti d i Padova and INFN, Sezione d i Padova via Marzolo 8, I-35131 Padova, Italy
[email protected],
[email protected] After the seminal paper by Arnold', it is well known that the bifurcation diagrams of the simple singularities of type Ak (for k 2 I), Dk (for k 2 4), &, E7 and E8 are diffeomorphic to the manifolds of the non-regular orbits of the homonymous Coxeter reflection group acting on the complex space. The extension of this connection to discover the rest of the finite reflection groups has stimulated deep research in singularity theory. We propose a simple method to identify the singularities which may be associated to general compact linear groups (including finite reflection groups). It is based on a particular equivalence relation which naturally appeared in the classification of real orbit spaces of compact (coregular) linear groups2.
1. Introduction The behaviour of a smooth function in a neighbourhood U of the origin 0 of Rn may be studied through the notion of germ, i. e. through the equivalence classes of functions whose restrictions to U coincide. The natural equivalence relation t o be imposed on the space of smooth germs of real smooth functions f a t 0 is R-equivalence. Two germs of functions f l and f2 are R-equivalent ( f l f2) when there exists an invertible germ g : R" + R" with g ( 0 ) = 0 such that f2(z) = ( f l o g ) ( z ) , for z belonging to the domain of f2. If a germ f is non singular at 0, that is V f ( 0 ) # 0, it is equivalent to the germ IT : (XI,. . . ,zn)---t 21. If a germ f has a non degenerate critical point at 0 and zero target, (2.e. f(0) = 0 ) , Morse's lemma ensures its equivalence with the Morsian form -z1~--22~--. . . - X ~ ~ + Z ~ + .~.+zn2, ~ + . where v is the index of inertia of the real 2-form. A function in general position has only non-degenerate critical points. Degenerate critical points appear naturally when considering family of functions, depending on some control parameters.
-
279
280
In this note, we intend to show that the bifurcation diagrams (BD) of the smooth real functions (including boundary ~ingularities~) can be retrieved from the classification of orbit spaces of compact linear groups through geometric invariant theory2i4. We recall that it was checked that the orbit spaces of all compact coregular finite groups5 ( i e . , the finite groups generated by reflections) and all compact coregular simple Lie groups6 with less than 5 basic polynomial invariants are equivalent to a member of the classes reported in Ref. '. In this way, a natural correspondence between singularity theory and group representation theory is proposed.
2. Classification of singularities In this section, we briefly expose the main ideas behind the classification of singularities under R-equivalence of smooth real functions. We essentially follow the line of Gibson7 and refer to it for omissions and details. Let us first examine the action C of a Lie group Q on a smooth finitedimensional manifold M . In fact, even if neither the manifold of smooth functions nor the group of R-equivalence are finite dimensional, the essential results continue to hold true for them. Given y E Q and f E M , we shall denote by 7 .f the image of the action of y on f . For a point f E M , we consider the &orbit Rf = { h E M : h = y f , y E 8 ) . The tangent space at f to the orbit Rf is the image of the tangent space at the unit element of the Lie group Q, Tn(Q),under the differential D of the map [f : Q -+ M defined by o ( y ) = y'f : T f ( R f ) = Dn[f(Tn(Q)). Thinking of the tangent space at f to the variety M , Tf(M), we can define the codimension o f f as the codimension c of the vector subspace T f ( 0 j )of Tf(M). If c = 0, in finite dimensions, the inverse function theorem guarantees that there exists a neighborhood Uf of f in M such that g E Of, for all g E Uf. In that case, we shall say that g is equivalent to f and write g f . An element f of M is called stable if it is equivalent to every g in a suitable neighborhood Uf of f. In general, if c > 0, a perturbation of f with a y E B changes the equivalence class of f only if it lies in the transversal section to Of. Such a section (a c-dimensional submanifold of M ) can be parameterized by a deformation F o f f , that is by a map F : R" -+ M such that F ( 0 ) = f ; such a map is also called unfolding of f. The deformation directions of F '
N
are the elements
belonging to T j ( R f ) .
Particular relevance have the transversal deformations of f , that is the ones whose deformation directions span all the cotangent space to the orbit Of: T f ( 0 f )@ DoF(Rd) = Tf(M). In fact, it can be shown that they
281 coincide with the class of versal deformations, i.e. with the deformations such that any other deformation H of f can be induced by F through a suitable re-parameterization. Sometimes, a versal deformation involving the minimal number of parameters is called universal. We note that the local data of the tangent space Tf are sufficient to write down versal deformations, which capture all possible behaviors of the singularity of f in a full neighborhood of it, according to the equivalence relation determined by the group G. Coming back to our problem, the manifold M is the set of C" (R", R) diffeomorphisms. The group G is the group of C" mappings on R",acting on elements of M by composition on the right. Owing to the cardinality of the spaces involved, the inverse function theorem cannot be applied in this case. Notwithstanding, the general attitude to reduce to finite dimensions is maintained, at least for the complement of the tangent space to the orbit of the infinite-dimensional group action. In this context, the notion of germ function and the determinacy problem, that is the possibility of using the jet-map, play a fundamental role. The space of (indefinitely) differentiable germs at the origin, denoted by En, is endowed with a real algebra structure. It is also possible to introduce the ideal of the germs with target 0 and the ideal of the (k - 1)-flat germs (i.e. the one composed of the germs whose derivatives at 0 vanish up to order k - l), denoted by M and M k , respectively. It can be proved that M is the unique maximal ideal of En. Moreover, Mk coincides with the k-th ideal power of M : Mk = M k . In other words, Mk is generated by the monomials (in the coordinate) of degree k. By Taylor's theorem, the quotient algebra J," = &,/Mk+l is naturally isomorphic to the polynomial algebra in n variables whose degree is at most k. In the same way, the quotient Mk/Mk+l can be identified with the real vector space of homogeneous polynomials in n variables of degree k. We shall call k-jet of f at the origin 0, denoted by j k f ( x ) ,the k-th order Taylor polynomial of f at x = 0. Loosely speaking, a germ function is k-determined when cutting away the contributions to the Taylor series expansions from the (k 1)-th terms upwards does not alter its qualitative behavior. In this case, one is again dealing with a finite dimensional object. The generalization of the analysis of the action ( B , M ) to the infinite dimensional case proceeds through the individuation of the tangent space at f to the orbit under R- diffeomorphisms, represented by the Jacobian
+
ideal,
Jf.
The ideal
Jf
3.f and its is generated by the partial derivatives -
3Xi
282
codimension in En is identified with the codimension of f and denoted by cod f . A germ is said t o be of finite codimension if cod f < 00. Orbits of positive codimension can form discrete stratifications or continuous families. In the former case the orbits are called simple. To be specific, a germ of a function f with critical point 0 E R” is simple if for sufficiently large k its orbit in the space of k-jets is simple and the number of confining orbits in the space of k-jets is bounded as k 4 00. Table 1. Normal forms of a real function in the neighborhood of a simple critical point with critical value 50. In the second column one can read the codimension of the germ with respect to the space En.
I
Name
1
Germ
Perturbation
I
4
fx14 x15 fx16 21’
x12x2f x~~ x12x2 x~~ x12x2& xZ5 x13 f x~~ x13 ~ 1 x 2 ~
+
+
We cannot enter here the algebraic issues concerning ideal theory and the details of the classification of germ functions. We just recall that determinacy and codimension are intimately related. In fact, it can be proved7 that a germ f E En is finitely determined if and only i f the codimension of f is finite. Moreover, there is a criterium t o test if a function is determined with respect t o R-equivalence: near a @-isolated critical point, even i f it is
degenerate, a g e m can always be reduced to polynomial normal f o r m . 3. The Orbit Space approach
In this section, we give an overview of the study of Orbit Spaces (OS’s) from the point of view of Geometric Invariant Theory’. Invariant functions under the transformations of a Compact Linear Group (CLG) acting in R”, lK = R or C , can be expressed in terms of functions of a finite set of basic real invariant polynomials p ( z ) = ( p l ( z ) ., . . , p q ( z ) ) ,z E K”,which may be chosen t o form a Minimal Integrity Basis (MIB) for the group G.
283 In the coregular case (which is the only case which is considered in this paper), i.e. when there are no syzygies among the elements of the MIB's, the 0s can be identified with the image p ( K n ) of K" through the orbit map p . For K = C, p ( C n ) fills the whole of Cq, while for K = R the orbit space turns out to be a disjoint union of connected semi-algebraic varieties, whose defining relations can be expressed from (semi)positivity and rank apa apb conditions of a matrix pab(p(x)) = Cy=,-(x) -(x), which, through ax2 axa the Hilbert theorem, depends only on the MIB { p i , . . . , p 4 } . In order t o classify the OS's of CCLG's, an axiomatic approach has been proposed which aims at determining the matrices p ( p ) exploiting only their general properties2. It has been shown that the p-matrices can be determined as solutions of a system of differential equations obeying convenient initial conditions. The equations can be derived in the following way. Let a be a general primary stratum of = p ( R n ) , and Z(o) the ideal formed by all the polynomials in p E RQ vanishing on g. Every f(p) E Z(a) defines in R" an invariant polynomial function f(x) = f(p(x)), and f(x) = 0 for all x E Cf = p - l ( a ) . The gradient af(x) is obviously orthogonal to C f at every x E C f , but, it must also be tangent to C f since f(x) is a G-invariant function. Consequently, it has to vanish on C f : 4
0
=
af(.)
,
,abf(p) apb(x)l
= 1
VXECf.
(1)
P=P(x)
By taking the scalar product of (1) with dpa(x), we end up with the following boundary conditions:
2
bpab(P) abf^@)
1
Relation (2) can be re-proposed in the form of a differential relation involving only polynomial functions of p . According to the Hilbert basis theorem, the ideal Z(u) is finitely generated. If a is a codimension one primary stratum of p@") c Rq, the ideal Z(a) has a unique irreducible generator, a ( p ) , and (2) reduces to 4
bpab(p) abak") = xa(p) a ( p ) >
a = 1,.. . , q ,
(3)
1
where the Xa's, (like the Pab'S and a) are w-homogeneous unknown polynomial functions of p . Eq. (3) will be quoted as Muster Equation (ME). The
284
ME holds true also if a ( p ) is the unique generator of the ideal I ( d S ) of the functions vanishing on the whole boundary dS of the orbit space. In this case, a ( p ) will be denoted by A ( p ) and called a complete active factor. 4. Group theoretical interpretation of singularities Arnold established’ the connection between simple singularities and the manifold of non regular orbits of finite groups generated by reflections acting That is quite clear for the singularity series Ak. Consider the on P. action of the permutation group s k + 1 on the (eventually complex) roots of the polynomial equation L ( z ) = zk+l Csso(jzj = 0. Expanding the
n,”z:(z
+
identity L ( z ) = - zj), it is easy to realize that the coefficients may be viewed as functions of the elementary symmetric polynomials s,(z1,. . . ,zk+l) of the roots:
E
jFrom the normal form of the singularity L ( z ) , it is evident that the action of Sk+l is restricted to the hyperplane V = {z E Wk+’ I &(Z) = ~j = o}. The polynomial invariants forming a it is MIB can be chosen to be p j ( z ) = (jIv for 1 5 j 5 k. In Ref. shown how to get an orthogonal representation of the real action of sk+l on V , i.e. of the group G = Ak, and to get the orbit space p(Rk) of the action of G on Rk as a subset of Rk. It is also possible to think of the boundary dS of the real orbit space as a subset of C k . It is worth noting that dS is, in general, just a subset of the real polynomial discriminant C c Ck,as stressed in Remark 4.2 of Ref.g. In this spirit, in what follows we shall consider the notion of extended MIB transformations. It is just a in which we allow the generalization of the notion introduced in Refs. coefficients of the transformation between MIB’s to be complex numbers (see Eq. 14 below). 214
4.1. Bifurcation diagmms and classification of orbit spaces In this section, we show that the BDs of the singularities of the smooth real functions correspond to orbit space solution^^!^ of the ME. For each normal form N ( z ) of the simple singularities appearing in Table 1, we shall construct the bifurcation diagram (BD), and we shall demonstrate that this equation can be viewed as a solution of the ME. In fact, we shall write down the correspondent p-matrix and XA term. Then, we shall give the
285 MIB (extended) transformation which transforms each solution in an 0s one. We just recall that, to get the BD’s (2.e. the discriminant varieties) it is sufficient to eliminate the zi in the system of equations N ( z ) = 0 and d - N ( z ) = 0. Then it is possible to identify the degree of the polynomial dXi
invariants from some dimensional analysis. Think of N ( z ) as an homogeneous polynomial and assign weight di to the coefficients ti,in such a way that any monomial containing tishares the same total weight as the other ones. The values of the di’s are then to be compared to the ones obtained as solutions of the ME. To save space, in what follows we shall associate the polynomial invariants pi to the coefficients & of the normal forms, in such a way as to obtain the minimum number of different BD’s A ( p ) , and, when possible, that A ( p ) is related to the complete active factor solution of the ME by a real MIB transformat ion. Singularity A2: From the normal form of Table 1, after identifying t o = p l and = - p z , we obtain the BD A ( p ) = 27p: - 4 p : = 0. Thinking of A ( p ) as a complete active factor, it is possible to find the other ME terms, i.e.:
It is well known that the orbit spaces of all CCLG’s with MIB formed by two invariants are isomorphic ’. With the MIB transformation given by 3 g1 = - 4 p l , gz = p z , we get the ME solution for ( d l , d 2 ) = ( 3 , 2 ) . 2 Singularity A s : From the normal form appearing in Table 1 , after identiG -p2 and & = p3 for A-3 or &, = -pl, El E -p2 and fying t o = p l , & = -p3 for A3 , we obtain the BD A ( p ) = 0, where
+
+
A ( p ) = 2 5 6 ~ : 27p; - 144plpip3 1 2 8 ~ : ~ 4; p i p : Thinking of A ( p ) as a complete active factor, we find:
&)
=
(
~ P - 4ZP i P 3
PzP3
8pi
8Pi + 2 p z 6 p z ) ’ 6P2 4P3 With the following MIB transformation:
+l6plpi.
’= ( i4)
(6)
-4 P3
’
(7)
286
we get the the solution 111.1 ( m = 1) for (dl1d2,d3)= (4,3,2). Singularity A4: From the normal form appearing in Table 1, after identifying t o = -PI, & = p2, (2 = p3, (3 E -p4, we obtain A(p) = 0, where:
+
+
+ 225OpIp2pz - 27p;p: + 375oP;P3P4 - 144PiPzp4 - 630Pipzp:p4 +
A(p) = 3 1 2 5 ~ ; 2 5 6 ~ ; 16OOplpip3
-2oooP?P;P4 - 1 2 8 ~ ; ~ :- 5 6 0 ~ l ~ ~ ~ 3 ~2 ~2 +2 8 2 5 ~ , ~ 3 I~~4P +~ P 9 ;0+0 ~ ~ ~
+ 72 Pi PZPSP:
f4P2 P3 P4 + 16Pl 2
2
3
- 1 0 8 ~ P: : - 1 0 8 ~pg. 1
(9)
Thinking of A(p) as a complete active factor, we get: ! P ~ ~ + ~ P I - P$ ~ P ~ P B + ~ P I P ~2 ~ 2 ~ 4 - 5 P24p3 6Pl P4 P32 f4PzP4 lop1 - EP3P4 8P2 5 PZ P4 lopi - gp3p4 -8p2 2
y
+
RP)=
+
8 P2
AT
= (4p3, 12p4, 0 , 40)
6 P3
4 P4
.
(10)
With the following MIB transformation: 41 = 15 43 =
fi
(lop1 -P3P4)
-3 @P3
1
7
42 = -60132
+ 9P4'
, (11)
94 = p4
we get the solution E l (s = 1) for (dl, d2, d3, d4) = (5,4,3,2). Singularity D4: From the normal form appearing in Table 1, after identi= p2, (2 = p4, t 3 E p3, we obtain the BD's fying t o = p l ,
A(p)
= 432p':+64~p:pi+576p?p2pz+64~pip2,+128p;pi+
+64 EP! - 288 EP; p2 p4 - 320 ~ pp;l pi p4 + 1 9 2 ~ P; 1
~4
-144 ~ p 2
-16
2
2
2 ~P,1 ~4
2 2 2 - 24 € P I ~3 ~4
+ 3 2 2
- 1 6 ~~3 2 ~4
+ 6 4 ~ : ~+: 72p1p2 pgpi + 27p:p;
+
=0;
(12)
the parameter E equals -1 for singularity 0 - 4 and +1 for singularity D4, respectively. Thinking of A ( p ) as a complete active factor, with the same meaning for E , we get:
RP) =
AT
E(4PlP2-3PzP4) 4 (-Pi+2EPlP4) 4 (-Pz+2ePiP4) 4(3Pi+EP2P4) 4EP2P3 6EP3P4 8PZ 12Pl
= ( 8 ~ ~24Ep4, 2 , 0,48)
.
4EP2P3 12Pl 6EP3P4 8P2 8P3
4P4 (13)
287 Consider the following (extended) MIB transformations:
q3 =
(36Pi-6EP2PdPi), Q2=3d3(4P2-477P3--Ep2) -3E (4P2 f 1277P3 - € P i ) , 44 = P 4 ,
,
(14)
where a new parameter 77 has been introduced, such that 17 = i for singularity 0 4 and 7 = -1 for 0 - 4 , respectively. Through them we get the solution E2 (s = 2) for (dl, d2, d3,d4) = ( 6 , 4 , 4 , 2 ) . 4.2. B o u n d a r y singularities
The classification of simple singularities do not exhaust the list of all the groups generated by reflections. Arnold extended the classification considering the class of the functions defined on a manifold with boundary, i.e. a smooth manifold with a fixed smooth hypersurface. Two functions on a manifold with boundary are equivalent if they are R-equivalent under a diffeomorphism of the manifold which leaves the boundary invariant. Let us identify the boundary of the manifold with the hyperplane x1 = 0, and set F ( x ) = f(x) t 2 1 , with t E R. It is easy to realize that the condition determining the singular points of a function f ( q 2,2 , . . . ,x,) given at page 101 of is equivalent to finding the solution of the following equations:
+
d --F(z)
8x2
= f(2) = t 2 1 = 0 ,
i = l ,...,n .
(15)
Eliminating q ,. . . ,x, and t in the equations above, for f chosen from Table 2, one finally gets the BD's for boundary singularities. Thus the same calculations as above can be carried on. Table 2. Simple critical points of functions a manifold with boundary. In the second column one can read the codimension of the germ in &n, the third and fourth columns are devoted to the germs and versa1 deformations, respectively. 1
Name B i k ,
k 22
c + k , k 2 3
Fi4
k k 4
Germ hXf
h 22'
zixz&xzk &Xf
*
Xz3
Perturbation
+ ... + . . . f &-iXik-' 50 f 5 1 x 1 + 5 2 x 2 + 5 3 x 1 1 2 60 f ( 1 x 1
+
5 ~ x 1 ~
~ 0 f f i X z f 5 2 x 2 ~. .f . + . . . f S k - l x ~ ~ - l
Singularity B2: From the normal form appearing in Table 2, after identify~ x?-xz2 or & = -pl and ing Eo = pl and 61 = p 2 , for the germs ~ : + 2 2and G p2 for the germs -x12 and -x12 2 2 2 we obtain the same BD A ( p ) = pl (4pl - p ; ) = 0. Thinking of A ( p ) as a complete active factor,
+
288
the result is: 4 ~ 1 ~PI 2 8Pi 4Pz With the MIB transformation represented by 41 = 8p1 - p i , 42 = p 2 , we get the the p-matrix solution for ( d l ,d2) = (4,2). Singularity B3: From the normal form appearing in Table 2, after identifying t o = P I , (1 --= P Z and & = p3 for the germs z13 zz2 and z13- ~2~ or = p l , & = -p2 and & = p3 for the germs -q3- zz2 and -z13 + z z 2 we obtain the same BD A ( p ) = 0, where
+
A(P) = P l ( 2 7 8 f 4Pi
-
18Pl PzP3 - Pz Pz
+ 4Pl P:)
.
(17)
Thinking of A ( p ) as a complete active factor, the result is: 4 ~ 1 ~ 2
4
8PlP3
PI +PZP3)
12”) 8P2 4 P3
8 P2
,
.
A )=3: : (
(18)
With the following MIB transformation:
+
41 = 4 ( 2 4 3 ~ 1- 2 7 ~ 2 ~ 4p:) 3
,
42 =
+
- 3 6 ~ 2 8 p 3P2, 2
43=P3 (19)
we get the solution 111.2 (m = 1) for ( d l , d2,ds) = (6,4,2). Singularity B4: From the normal form appearing in Table 2, after identifying t o _= P I , (1 _= p2, 5 2 =- p3 and (3 _= p4 for the germs x14 22’ and z14 - zz2 or t o G - P I , (1 G p2, (2 = -p3 and (3 --= p4 for the germs -q4- 22’ and -zl4 22’ we obtain the same BD A ( p ) = 0, where
+
+
+ ~ -26pip;p: ~ ~ ~ +4144p:p3p: +
A(P)= P I (25613; - 2 7 ~ ;+ ~ ~ ~ P I P-, 1~2P8 ~S ; ~-: 4 ~ 2 ~ 3 3l 6 p l p ; f -192PfPzP4 2
2
+ 1 8 p i ~ 3 -~ 84
2
+Pz P3 P4 - 4Pl P: P; - 4Pg P:
0 ~ 1
+ 18Pi Pz P3 P:
- 27Pf P j )
.
(20)
Thinking of A ( p ) as a complete active factor, we get:
[ zl 4 ~ P1Z
)(’‘
=
8Pl P3
4 (PZP3 +3PlP4) 8 12PiP4 8 (2Pi + P z P ~ ) 4 12P2
AT = ( ~ P z 1, 6 ~ 336P4,64) , .
8 P3
4 P4 (21)
289
With the following MIB transformation:
+
91 = 54 ( 8 1 9 2 ~ 1- 512pzP4 6 4 ~ 3 ~ 92 ~ 2 ), 93 = 3 (32P3 - 9 ~ 2 1) q2 = 108 ( 6 4 ~ 2- 1 6 ~ 3 ~ 34 d ) , 94 = P4
+
(22) we get the solution E3 (s = 1) for (dlId2,d3,d4)= (8,6,4,2).
Singularities c k for 2 _< Ic _< 4: From direct calculation, it is easy to realize that the BD’s associated to the singularities of the series ck are isomorphic to the correspondent ones of the series Bk. Singularity F4:From Table 2, identifying t o _= PI, (1 ~ 3 (2, ~ 2 (3, P4 for germ xf x ~ or~( 0 ,= -pl, t1 = p3, (2 = p2, 63 E -p4 for germ -xf x ~or ~( 0 ,= pl, 61 = p3, 62 = -p2, 63 = --p4 for germ z: - z~~or to= -pl, El = p3, = -p2, = p4 for germ -xf - xz3, we obtain the same BD A(p) = 0, where
+
+
e3
c2
+
+
A(p) = ( 2 7 ~ : 4p:) ( 4 3 2 ~ : 64p;
-
216plpi
+ 27p:+
-96 P i P3 P4 + 72Pi P2 P i 30P2 P i P i - 36Pl P3 Pz + P ~ P ~ - P i P ~ + P Z P 3 P ~ - P6l P 4 )
(23)
Thinking of A(p) as a complete active factor, we get the non-trivial elements of the p-matrix and the X components: 4 4 Pii(P) = -2P2 (2p2P3 f 3 P i p 4 ) , p12(P) = 5 (9PiP3 - 2132174) , 4 p22(P) = 4 (2P2P3 3PiP4) = -- ( 4 ~ ;+ 3 p l d ) , 4 83 2 & 3 ( ~ )= - ; ~ 4 (5132 + ~ 3 ~ 4 ) p23((P) = 3 (9231 -P2P4) h
+
R~(P)
7
AT
= ( - 8 ~ 2 ~ 424p3, , -12p$, 96)
. (24)
With the following MIB transformation:
+
41 = 2592 fi ( 3 4 5 6 ~ 1- 4 3 2 ~ : 144p2pg - 4 8 ~ 3 ~ - :pz) , 92 = -324 (128172 -P4 (32P3 +$)) , 94 = P4 ’ 43 = -18 fi (24P3 P z ) ,
+
(25)
we get the solution E4 (s = 1) for (dl,d2,d3,d4) = (12,8,6,2). 5. Conclusions and outlooks
We have shown that every bifurcation diagram of the list of the simple function singularities (whose unfolding depends on less than 5 parameters) corresponds to a solution of the ME. That statement holds true also for
290 boundary singularities classified in ’. The main motivation is the following: the complete active factors entering the ME, once associated to a linear representation of a CLG, may be viewed as real functions A ( x ) = A ( p ( x ) ) , x being a point in the real, n-dimensional, order parameter space. Any arbitrarily small neighborhood of x = 0 encodes all the information about the orbit space structure. In fact, since the isotropy subgroups at points lying in a line through the origin coincide, any typical point xo corresponding to some symmetry stratum can be moved arbitrarily close to the origin through a convenient re-scaling xo 4 pxo, for /-I E R. It is then apparent that there is a relationship between singularities of smooth functions and orbit space classification. We have shown that simple singularities do not even exhaust orbit spaces of finite reflection groups: it is necessary to resort to boundary singularities. As a consequence, the equivalence relation induced by the ME on the space of germ functions at x = 0 is finer than the standard R-equivalence. As a double check, the condition for determinacy (and finite codimension) for simple singularities we have mentioned at the end of $2 is not obeyed by boundary singularities. The phenomenon is essentially that a solution which is un-stable for some equivalence relation may become stable under a finer one, in this case an equivalence including the notion of symmetry (see for instance lo). Since the classification of orbit spaces is a framework naturally including symmetry, we propose that the restriction of R-equivalence should stem directly from the ME structure. The consequences of the conjecture above are under investigation ll.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
V. I. Arnol’d, Funct. Anal. Appl. 6,254 (1972). G. Sartori, Mod. Phys. Lett. A 4, 91 (1989). V.I. Arnol’d, RUSS.Math. Suw. 33, 99 (1978). G. Sartori and V. Talamini, Commun. Math. Phys. 139(3), 559 (1991) and J . Group Theory in Physics 2, 13 (1994). G. Sartori and G. Valente, J . Phys. A : Math. Gen. 29(1), 193 (1996). G. Sartori and V. Talamini, J. Math. Phys. 39(4), 2367 (1998). C. G. Gibson, Singular points of smooth mappings, Pitman Pub. Lmt., London (1979). G. Sartori, Riuista del Nuovo Cimento 14, 1 (1991). V. I. Amol’d, Commun. Pure Appl. Math. 29, 557 (1976). M. Golubitsky, and D. Schaeffer, Commun. Math. Phys. 67,205 (1979) and Commun. on Pure and Applied Math. 32, 21 (1979). G. Sartori and G. Valente, in preparation.
SYMMETRIC SOLUTIONS IN MOLECULAR POTENTIALS*
LUCA SBANOt Mathematics Institute University of Warwick, Coventry CV4 7AL, IJK
[email protected] 1. Introduction
The study of the dynamics of N particles in the plane R2 interacting through a potential of the form 1 i>j
xz - x j p
with
Q
E (0,foo)
is called the planar N - body problem. The aim is to find the possible motions taking place in the configuration space M = R2N.To prove existence of periodic orbits in a non-perturbed regime the calculus of variation and variational methods have a major role, for a general introduction to the modern approach one can refer to Very recently, in the gravitational case, new orbits have been found by combining extensively variational methods and symmetries of the system, symmetries have a crucial role to find those orbits in which each body follows the same curve according an appropriate phase shift, the so called choreographic solutions see More generally we can define symmetric orbits as follows: let G be a group and x + g-x be its action on the configuration manifold M ; this action naturally extends to paths in M as follows: 633.
x ( t ) -+ (9 . x ) ( t ) = g.z(t) for every t. *PROCEEDINGS SPT-2004 t Work partially supported by grant m.a.s.i.e.EU-TMRnetwork
291
292 We assume that G is symmetry group, namely the G-action preserves the dynamics and then we call z(.) a symmetric orbit if z(.) is a solution and there are T ( 9 . z ) ( t )= z ( t T).
+
>0
and g E G such that V t
Choreographic orbits can be recovered when the permutation group (bodies re-labelling) is isomorphic to a subgroup of the symmetry group G. In the study of the gravitational N-body problem only very recently a general strategy has been found in 5,8 to tackle systematically the problem of finding symmetric orbits. Among the interesting open problems we might ask ourself whether and which symmetric solutions may exist in other physically interesting N-body systems. In this paper we shall consider symmetric orbits for a system of 3 particles of equal masses in a plane interacting through a Lennard-Jones type potential:
We present some general observations on the symmetric orbits, and we consider the circular relative equilibria, which are trivialchoreographies and their variational structure. We show also that is possible to find easily some interesting symmetric orbits with rotational symmetry. Then we consider homotopy classes of choreographies and show the existence of choreographic solutions as mounting-pass critical points. Such analysis, inspired by '>', used the description of the space of choreographies in terms of Fourier series. 2. Lennard-Jones potential and symmetries
A system of equal particles in the plane interacting through a potential of type VLJ has the same symmetries of the usual gravitational N-body problem. In the considered case let z = ( X I ,z 2 , ~ E) M = IR2 x R2 x R2 be a configuration, the dynamics can be prescribed by the Lagrangian:
Note that L is not defined in the coincidence set K , = {z = (zl,z2,z3) E
R6 : 3i # j , zi = z j } . The Lagrangian is invariant under a group containing a subgroup isomorphic to the permutations. Indeed the symmetry actions are:
293 0 0 0
+
Translations: x + x a with a E R6. Rotations: x + Rx with R E SO(2) x SO(2) x SO(2). Re-labelling: z + ( ~ ( x where ) (T is an element of the group of permutations of 3 elements Z3, in particular: ( z l , x 2 , z 3 ) + O ( ( z 1 , x2723)) = (x27 z 3 , z l ) .
2.1. Variational methods o n s y m m e t ~ %paths The equations of motion are given by the Least action principle applied to the functional
We choose the domain of A[.]to be a suitable set paths/loops whose qualitative property has been prescribed by symmetries. The standard choice is to consider paths in H' Sobolev class, so we define:
Simple choreography space: A = { x ( t ) = ( x 1 ( t ) , x z ( t ) , z 3 ( t )E) H1([O,2'1, R6): x ( t ) $ K,, z ( t T / 3 ) = o ( z ( t ) ) } . For the given (T,it is known that for each z(.) E A we can find a loop z ( . ) valued in R2 such that x ( t ) = ( z ( t ) ,z ( t - T / 3 ) ,z ( t - 2 T / 3 ) ) . Rotational symmetric loops: C" = { z ( t ) = ( q ( t ) , z z ( t ) , z S ( t )E) H1([O,TI,R6): z ( t ) K,, z(t 2') = ei" z ( t ) ) } for some a E R (see '). We shall term A0 and Cg the paths/loops in which the centre of mass is the origin.
+
+
We want to find z*(.) in A0 or C: such that DA[x*]= 0. This is a criticality condition for x* (.) equivalent to solving the equations of motion. VLJ has interesting properties:
0
A is unbounded below on H 1loops, A satisfies a form of the so called strong force condition, in fact the action diverges on any loop intersecting K,, this was introduced in This property allows us to prove that any critical loop does not cross the coincidence set K,. '3'.
In fact one can prove: Proposition 2.1. The potential VLJ satisfies following conditions: (1) lim,,Kc
V L J (= ~ )+m.
294
(2) The strong force condition: there exist U E C'(R6\Kc,R), a neighborhood A of the origin and c1 0 such that:
>
lim U ( z ) = +03,
x+K.
VLJ(X)2 IlVU(x)llZ- c1
Vx E A\{O}
(3) There exists m > 0 such that VLJ(X)2 -m for all x E M . (4) For every x E M < V V L J ( X ) ,> ~ c2 for Isome c2 > 0.
Proposition 2.1 permits us to show that the action d satisfies Palais-Smale condition at any positive level:
Proposition 2.2. If (x,(.)) such that d[x,] + c # 0 and Dd[x,] then (x,(.)) converges up to a subsequence in A0 (or C,"). Proof. The proof can be reduced to a similar proposition in strong force condition and preservation of the centre of mass.
'1'
+0
using the 0
3. Relative equilibria (trivial choreographies)
The system has equations of motion given by:
We look for circular solutions (critical points of d ) which are the simplest possible choreographic solutions in the form: xk( t) = p exp(i w (t 2x(k 1)/3))with w = 2 x / T , x ( t ) E Ao. These are relative equilibria and p and T satisfy:
+
For TO= 2~/@&)
-&
there are two coincident circular solutions with ra-
(w)
dius pa = solutions with radii
p1
. For T > TOthere are two distinct circular and p2 with p1 < PO < pz.
3.1. Variational characterisation
For each relative equilibria x*, we evaluate the Hessian D2d[x*](u,u), where u(.) is a tangent vector to x*(.). Writing u(.)in terms of its Fourier
295 expansion we have: 00
k,6
+G(p)
/ T O
3T
k
d t x ( < zi(t) - xj(t),ui(t)- u j ( t ) >)2 i>j
where G(p) is: G(p) = negative for p < po and positive for p
m. >
G(p) is zero for p = PO,
po. Then we c a i deduce:
(1) The circular orbit with radius p1 is a saddle point. Its negative directions are spanned by {exp(iwt),exp(-iwt)}. In fact G(p1) < 0 and consider Fourier components k = fl. (2) The circular orbit with radius p2 is a local minimum, and moreover
Abll > Ab21. (3) Each critical point is degenerate along u(.) tangent to itself. This reflects the SO(2) symmetry. (4) At T = TO, the circular orbit po is a local minimum of the action, it is degenerate along SO(2) orbit and also along {exp(i w t ),exp (-i w t )} . The above conditions do not permit us to find any new critical points by studying the homotopies deforming the circular orbits one into the other. 4. Simple non-trivial rotationally symmetric orbits
We consider x(.) E Cz'3 wherezi(t) = p ( t ) exp(i4(t)+227r(z-1)/3), 1 = 1,2,3. Then the dynamics in ( p ( t ) ,4(t))is described by the Lagrangian:
L = ; ( 4 ( t )+ P 2 ( t ) 4 2 ( Q )
- 3VLJ(P(t))
1 where V ' J ( ~ = ) - -. The dynamics is integrable and we can ( A P Y (6)construct the integrals for p ( t ) , $(t) and also 4 ( p ) . In particular:
A$ =
lr
du
fi513 u2 J E / 3 -
- VLJ(U)
PO,^,] is the interval of allowed values of p for fixed energy E and the total angular momentum J. For a rotationally symmetric periodic orbit we have: A 4 = 27r Jq, for some q E M.
296
X
Figure 1. 7-fold symmetric orbit
We have z ( t )
=
( z ( t ) z, ( t )ein/3,z ( t )ei2a/3), with z ( t )
p ( t ) exp(ic$(t)). Along such solution the shape is preserved: llzi(t) zm(t)ll = & p ( t ) and there is q E N such that: p(t T / q ) = p ( t ) and Ac$ = $(t T / q ) - $(t) = 2x/q,and zi(t T / q ) - q ( t ) = z ( t ) [ei"/q 11 exp(7r i(Z - 1)/3) with Z = 1 , 2 , 3 , For an example see figure 1.
+
+
+
5. Choreographies in Fourier space Planar choreographies are solutions determined by just one plane curve. For three bodies in the plane we write z ( t )= ( z ( t ) , z ( t - T / 3 ) , z ( t - 2 T / 3 ) ) , where z(.) is a closed curve in R2. This suggests to think of a choreography as a Fourier series: z ( t ) = CnEZ z, exp(i writ) where w, = and SO as an element of the Hilbert Space:
9,
{
X1 = z = (z,) : z, E @, with norm
11~11:
I
= c ( 1+w;))z,~~ nEZ
here we consider z ( t ) E @ for every t. Note that fixing the canter of mass in the space for the choreographies is equivalent to choose zo for any z E X1.
297 Without loss of generality we can consider zo = 0. From now on we assume 20 = 0. We would like to point out that the idea of using the basic loop to investigate choreographies has been also used recently in the gravitational N-body problem in '. In X1 the set of collision curves has a description in terms of bundle of hyperplanes. In fact for example the condition q.(t)- q ( t ) = 0 turns out:
zk(t) - zl(t) = ~ k , l ( z , t = ) Cz,~,(k,~)exp(iw,t)= o nEZ
here K,(k,Z) = [exp(2~i(k - l)/N)), - (exp(axi(1 - 1)/3)),]. The action functional on X1 can then be written as:
Fourier description allows us to analyse the geometry of space of choreographies and to carry on a variational study of Lennard-Jones potential similar to 1 , 7 , showing that it is possible to fmd new solutions as critical points of A[.]with mountain pass geometry. and consider the following subsets of X1: Let 211(z) = CnEZ~zIzn12
Br+,b = {Z E x1 : B [ z ]= u'((S(z))' where 6(z) = mink,[ inft
- 2 b211(z)- r 2 = 0)
JRk,l(z,t)l.
Proposition 5.1. There existsr such thatinftEBr,o,b d [ z ]= C O ( T , U , b )
> 0.
Proof. From the definition of Rk,l(z,t) we can show that 6 ( z ) 5 2c for some c > 0. Taking z E Br,a,b implies 6 ( z ) 2
d
m
Now for large r we have that v ( z ) therefore d [ z ]2 f C, W ~ We define co = infgv,n,b A.
J Z ~The ) ~ action .
