Differential Manifolds and Theoretical Physics
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Differential Manifolds and Theoretical Physics
This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks
Editors: SAMUEL EILENBERG A N D HYMAN BASS A list of recent titles in this series appears at the end of this volume.
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Differential Manifolds and Theoretical Physics W. D. CURTIS F. R. MILLER Department of Mathematics Kansas State University Manhattan, Kansas
1985
ACADEMIC PRESS, INC. (Harcourt Brace Jovanovich. Publishers)
Orlando San Diego N e w York London Toronto Montreal Sydney Tokyo
COPYRIGHTO 1985, BY A C A D E M I C P R E S S , INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE A N D RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC. Orlando,Florida 32887
United Kingdom Edition published by ACADEMIC PRESS I N C . (LONDON) LTD 24/28 Oval Road, London NWI 7DX
Library of Congress Cataloging in Publication Data Curtis W. D. Differential manifolds and theoretical physics. Bibliography: p. Includes index. 1. Geometry, Differential. 2. Mechanics. 3. Field theor (Physics) 4. Differentiable man if ol ds. I. Mirler, F. R. 11. Title. QC20.7.052C87 1985 530.1'5636 84-16861 ISBN 0-12-200230-X (alk. paper)
PRINTED IN THE UNITED S T A T E S OF A M t R l C A
85868788
9 8 7 6 5 4 3 2 I
To our wives Beverly and Helen
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Contents
xv
Preface
Chapter 1. Introduction Mathematical Models for Physical Systems
1
Chapter 2. Classical Mechanics Mechanics of Many-Particle Systems Lagrangian and Hamiltonian Formulation Mechanical Systems with Constraints Exercises
5 7 11 14
Chapter 3. Introduction to Differential Manifolds Differential Calculus in Several Variables The Concept of a Differential Manifold Submanifolds Tangent Vectors Smooth Maps of Manifolds Differentials of Functions Exercises
Chapter 4.
16 23 26 28 33 35 36
Differential Equations on Manifolds Vector Fields and Integral Curves Local Existence and Uniqueness Theory The Global Flow of a Vector Field Complete Vector Fields Exercises
vi i
39 40 53 55 57
...
CONTENTS
Vlll
Chapter 5. The Tangent and Cotangent Bundles The Topology and Manifold Structure of the Tangent Bundle The Cotangent Space and the Cotangent Bundle The Canonical I-Form on T*X Exercises
61 66 68 70
Chapter 6. Covariant 2-Tensors and Metric Structures Covariant Tensors of Degree 2 The Index of a Metric Riemannian and Lorentzian Metrics Behavior Under Mappings Induced Metrics on Submanifolds Raising and Lowering Indices The Gradient of a Function Partitions of Unity Existence of Metrics on a Differential Manifold Topology and Critical Points of a Function Exercises
Chapter 7.
72 74 74 77 79 83 84 84 87
90 92
Lagrangian and Hamiltonian Mechanics for Holonomic Systems Introduction The Total Force Mapping Forces of Constraint Lagrange’s Equations Conservative Forces The Legendre Transformation Conservation of Energy Hamilton’s Equations ~-FOIITIS Exterior Derivative Canonical 2-Form on T*X The Mappings # and b Hamiltonian and Lagrangian Vector Fields Time-Dependent Systems Exercises
94 95 96 99 99 103 105 106
110 112 114
114 115 121 124
Chapter 8. Tensors Tensors on a Vector Space Tensor Fields on Manifolds
127 129
CONTENTS
IX
The Lie Derivative The Bracket of Vector Fields Vector Fields as Differential Operators Exercises
Chapter 9.
Differential Forms Exterior Forms on a Vector Space Orientation of Vector Spaces Volume Element of a Metric Differential Forms on a Manifold Orientation of Manifolds Orientation of Hypersurfaces Interior Product Exterior Derivative Poincare Lemma De Rham Cohomology Groups Manifolds with Boundary Induced Orientation Hodge *-Duality Divergence and Laplacian Operators Calculations in Three-Dimensional Euclidean Space Calculations in Minkowski Spacetime Geometrical Aspects of Differential Forms Smooth Vector Bundles Vector Subbundles Kernel of a Differential Form Integrable Subbundles and the Frobenius Theorem Integral Manifolds Maximal Integral Manifolds Inaccessibility Theorem Nonintegrable Subbundles Vector-Valued Differential Forms Exercises
Chapter 10.
132 135 137 138
141 146 149 150 151 154 156 156 161 161 162 163 165 168 168 170 171 172 172 173 176 184 185 187 188 189 191
Integration of Differential Forms The Integral of a Differential Form Stokes’s Theorem Transformation Properties of Integrals w-Divergence of a Vector Field Other Versions of Stokes’s Theorem Integration of Functions The Classical Integral Theorems Exercises
196 199 20 1 203 204 207 208 210
CONTENTS
X
Chapter 11. The Special Theory of Relativity Basic Concepts and Relativity Groups Relativistic Law of Velocity Addition Relativity of Simultaneity Relativistic Length Contraction Relativistic Time Dilation The Invariant Spacetime Interval The Proper Lorentz Group and the Poincare Group The Spacetime Manifold of Special Relativity Relativistic Time Units Accelerated Motion-A Space Odyssey Energy and Momentum Relativistic Correction to Newtonian Mechanics Conservation of Energy and Momentum Mass and Energy Changes in Rest Mass Summary Exercises
213 220 222 222 223 223 224 225 227 229 233 234 235 236 236 237 237
Chapter 12. Electromagnetic Theory The Lorentz Force Law and the Faraday Tensor The 4-Current Doppler Effect Maxwell’s Equations The Electromagnetic Plane Wave The 4-Potential Existence of Scalar and Vector Potentials in R3 Exercises
239 243 245 246 248 250 25 1 253
Chapter 13. The Mechanics of Rigid Body Motion Hamiltonian Systems and Equivalent Models The Rigid Body O(3) and SO(3) Space and Body Representations The Geometry of Rigid Body Motion Left-Invariant 1-Form Symmetry Group Adjoint Representation Momentum Mapping Coadjoint Representation Space Motions with Specified Momentum Coadjoint Orbits and Body Motions
255 256 256 259 26 I 263 264 264 265 266 266 267
xi
CONTENTS
Special Properties of SO(3) Stationary Rotations Classical Interpretation-Inertial Tensor, Principal Axes Stability of Stationary Rotations Poinsot Construction Euler Equations Phase Plane Analysis of Stability Exercises
Chapter 14.
27 1 274 274 277 280 282 283 284
Lie Groups Lie Groups and Their Lie Algebras Exponential Mapping Canonical Coordinates Subgroups and Homomorphisms Adjoint Representation Invariant Forms Coset Spaces and Actions Exercises
286 289 289 290 29 I 292 293 296
Chapter 15. Geometrical Models Geometrical Mechanical Systems Liouville’s Theorem Variational Principles Forces Fixed Energy Systems Configuration Projections Lorentz Force Law Pseudomechanical Systems Restriction Mappings Rigid Body and Torque Gauge Group Actions Moving Frames and Goedesic Motion Basic Theorem Local (Lemma 15.36) Basic Theorem Global (Theorem 15.39) Principal Bundle Model Using a Special Frame The Souriau Equations Structure of the Lie Algebra of the Lorentz Group Construction of a Gauge Invariant 2-Form Curvature Form The Souriau GMS Appendix: Conservation Laws Exercises
297 298 300 30 1 304 305 306 306 307 308 310 31 1 314 316 319 32 1 322 322 327 328 329 332
xii
CONTENTS
Chapter 16. Principal Bundles and Connections; Gauge Fields and Classical Particles Principal Bundles Reduction of the Structural Group Connections on Principal Bundles Horizontal Lifts of Vectors Curvature Form and Integrability Theorem Horizontal Lifts of Curves Associated Bundles Parallel Transport Gauge Fields and Classical Particles Natural 2-Form on Coadjoint Orbits Pseudomechanical System for Particles in a Gauge Field Sternberg’s Theorem Geometrical-Mechanical System for Particles in a Gauge Field Affine Group Model Exercises
335 337 337 338 339 34 I 34 1 342 342 343 345 346 347 349 35 1
Chapter 17. Quantum Effects, Line Bundles, and Holonomy Groups Quantum Effects Probability Amplitudes Probability Amplitude Phase Factors DeBroglie and Feynman Phase Factors and 1-Forms COW and Bohm-Aharanov Experiments Complex Line Bundles and Holonomy Groups Integral Condition for Curvature Form Bundle Description of Phase Factor Calculation Remarks-Geometric Quantization Holonomy and Curvature for General Lie Groups Exercises
354 355 355 356 356 358 360 362 365 366 367 368
Chapter 18. Physical Laws for the Gauge Fields Gauss’s Law in Electromagnetic Theory Charge Conservation Curvature and Bundle-Valued Differential Forms Covariant Exterior Derivative Covariant Derivative of Sections and Parallel Transport The Group of Gauge Transformations
37 1 372 373 375 376 377
...
CONTENTS
Xlll
The Killing Form The Source Equation and Currents for Gauge Fields Exercises
379 379 382
Bibliography
387
Index
389
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Preface
There has always been a fundamental relationship between the concepts of geometry and the study of the physical world. The tremendous achievements of this century in both mathematics and physics, together with active current research efforts, indicate that this affair will continue. We will not attempt a comprehensive description of all of the recent developments-this would require many volumes-but will present the concepts of modern differential geometry in the study of classical mechanics, field theory, and simple quantum effects. In doing so, we start with Newton, Lagrange, and Hamilton and continue through Riemann, Maxwell, Einstein, Cartan, Chern, Yang, Souriau, and Sternberg. The idea of invariance is an essential ingredient and we introduce gauge invariance, bundles, and connections. This book has been successfully used for introductory one-year courses in the theory of differential manifolds and their relationship to various areas of physics. We begin by assuming modest mathematical preparation (advanced calculus, linear algebra, and some point-set topology) and maintain a high level of rigor throughout. The text carefully relates basic mathematical and physical concepts in much detail, but also includes material of a more advanced nature, bringing the reader close to the frontiers of current research. Some of the exercises present important enhancements and extensions of the theory. The following paragraphs give a detailed outline of the book. In the first two chapters, we introduce some basic ideas of classical mechanics of systems of particles and some ideas about the description of physical systems. We discuss the evolution of a physical system as a curve in a space ofstates. The evolution of the system is governed by a vector field on state space which specifies how the state changes in time. In the very simple cases, the state space can often be identified as an open set in some Euclidean space. When more complex cases are considered, we find this is no longer true. The presence of constraints, for example, xv
xvi
PREFACE
results in state spaces that are not open sets. In all cases, however, we find that the spaces which arise look locally like open sets in some Euclidean space, that is, they admit local coordinate systems. These examples motivate the introduction of differential manifolds as spaces on which the concepts of tangent vector to a curve and of vector field are meaningful. In particle mechanics the state space is naturally seen to correspond to the space of tangent vectors on conjiguration space, thus motivating introduction of the tangent bundle of a differential manifold. In Chapter 3, after some preliminaries on multivariable calculus, we introduce the basic ideas about differential manifolds, charts, atlases, submanifolds, tangent vectors, and smooth maps of manifolds. Then in Chapter 4 we introduce the notion of vector field and prove the fundamental result that a smooth vector field has a smooth flow. This result is one of the cornerstones on which the theory of manifolds rests. In Chapter 5 we describe the construction of the tangent and cotangent bundles of a differential manifold. These will serve as the state space and phase space for certain mechanical systems to be considered later. In Chapter 6 we describe general metric structures on manifolds. We discuss questions of existence for Riemannian and Lorentzian metrics. Raising and lowering indices is discussed here. Chapter 7 is a critical chapter. We use the mathematical formalism introduced in Chapters 3-6 to give an extremely elegant description of the mechanics of particle systems (begun in Chapter 2). We obtain the beautiful geometrical result that for a holonomic system subject only to forces of constraint the trajectory of the system is always a geodesic of configuration space for the kinetic energy metric. We introduce 2-forms, in particular the fundamental canonical 2-form on the cotangent bundle of a differential manifold. We establish the basic connection between the Lagrangian and Hamiltonian formulations of mechanics by means of the Legendre transformation. On state space and phase space we construct vector fields whose integral curves satisfy Lagrange’s and Hamilton’s equations, respectively. This relates back to the ideas of Chapter 1. At this point we have developed numerous important ideas about differential manifolds and have seen how naturally they relate to basic concepts in classical mechanics. Our purpose has not been to develop techniques for actually solving particular problems in mechanics. Rather we hope to have shown how the geometrical ideas in manifold theory can elucidate the structure of classical mechanics. Also, we believe that the physcial ideas have served as concrete motivation for introduction of much of the manifold theory. In subsequent chapters we shall try to maintain this interplay between the geometrical ideas of manifold theory and the physical ideas which arise in various branches of theoretical physics. The
PREFACE
xvii
geometrical notions play a basic role in conceptual understanding of physical ideas. And further, the geometry can play an important role from the computational point of view as well, by focusing attention on those quantities that really are basic. That is, the geometry points the way to the correct formulation of a physical problem and suggests which computations should be made. Conversely, the study of various physical notions continually motivates the study of particular mathematical ideas. In Chapters 8 and 9 we introduce further aspects of manifold theory. In Chapter 8 we introduce tensors of arbitrary degree and explore the fundamental Lie derivative in considerable detail. Chapter 9 is one of the most important chapters in the book. Most of the later work depends on the ideas contained here. We give a detailed development of Cartan’s calculus of differential forms. Orientation of manifolds is discussed in detail. We discuss the Hodge *-operator for a metric of arbitrary index and use it to define the classical differential operators, e.g., divergence, curl, Laplacian. We then discuss the fundamental Frobenius integrability theorem. This is the second major theorem of the book and is basic for subsequent work. We prove the inaccessibility theorem for integrable subbundles. In Chapter 10 we investigate integration on manifolds. We define integrals of n-forms over oriented n-manifolds. We relate the behavior of the integral of a form under motions generated by a vector field with the Lie derivative of the form with respect to the vector field. We prove the general Stokes formula relating the integral of an ( n - 1)-form over the boundary of an n-manifold to the integral of the exterior derivative of the form over the manifold, and we also give a proof for cylinders. Then we show how the classical integral theorems of Green, Gauss, and Stokes follow from the general theorem. In Chapters 11 and 12 we study the basic features of Einstein’s special theory of relativity and Maxwell’s equations in relativistic form. The structure of the spacetime of special relativity provides a beautiful example of the manifold theory we have developed. Spacetime is the set of all “events.” On spacetime there is a privileged class of charts which are the global coordinate systems known as Lorentz frames. These correspond to the physical notion of a Cartesian coordinate system rigidly attached to an inertial reference body. We show that the coordinate transformations relating these charts are the usual Lorentz transformations. These charts then form an atlas and hence determine a differential structure on spacetime. We then see that the 4-vectors of special relativity are just the tangent vectors for the differential structure determined by the Lorentz frames. We proceed to develop many of the standard topics of special relativity.
xviii
PREFACE
In Chapter 12 we develop Maxwell’s equations in terms of differential forms. We establish existence of the electromagnetic field tensor using the physical properties of the Lorentz force law. We then give some properties of this tensor and some examples. In Chapter 13 we study the mechanics of a rigid body moving in 3dimensional space with one point fixed and no external forces. We show that the state space for this system (it is a system with holonomic constraints) can be identified with T S 0 ( 3 ) , the tangent bundle of the group S 0 ( 3 ) , and that the kinetic energy metric is invariant under left translation. We use this chapter as a first example of concepts that will be developed later such as Lie groups and Lie algebras, adjoint and coadjoint representations, and symmetry groups and conservation laws. We relate these constructions to the usual angular momentum, inertial tensor, and principal axes, and to the Euler equations. We give a proof of the stability theorem (in both space and body coordinates) for rotations around the major and minor principal axes. In Chapter 14 we develop the properties of Lie groups, some of which were encountered in Chapter 13 for SO(3). These include the associated Lie algebra, subgroups, coset spaces, invariant differential forms, and invariant measures. We also study orbit spaces and vector fields obtained when Lie groups act as transformation groups. In Chapter 15 we further geometrize the study of mechanical systems by considering the trajectories (in phase or state space) as families of curves tangent to the kernel of a 2-form. We first show how this is related to Hamiltonian mechanics and obtain a proof of Liouville’s theorem on the invariance of phase space measure. We show the relationship to variational principles-locally if the 2-form is closed, globally if the 2-form is exact. Second, we see that the 2-form contains the classical forces and can describe nonconservative forces (for example, the rigid body of Chapter 13 subject to an external torque). Finally, we see that this geometrical approach in which the explicit time parameter is lost is well suited to the description of relativistic mechanics and interactions involving gauge fields. We define pseudo-mechanical systems which can incorporate gauge invariance with phase space, and we construct an invariant 2-form. We then obtain the particle phase (or state)-motions from equivariant restriction mappings. We use moving frame methods in these constructions leading to the concept of principal bundle. We show how the 1-form defining the metric connection is a factor of the 2-form defining the pseudo-mechanical system which gives geodesic motion. We also see that the restricted systems can be described by specifying a 2-form on a principal bundle. We give a development of the equations of Souriau which govern the evolution of the states of a
PREFACE
xix
charged particle with an internal spin in the presence of an electromagnetic and gravitational field. We end the chapter with an appendix on conservation laws. In Chapter 16 we study the structures of principal bundles, associated bundles, connections, and parallel transport. We consider homogeneous spaces and general affine connections in detail to present the work of Sternberg and co-workers in describing the interaction of classical particles with gauge fields. We describe how this relates to Chapter 15. In Chapter 17 we describe the concept of holonomy groups and show how complex line bundles are determined by closed 2-forms subject to a quantization condition. We explain the use of these concepts in describing quantum interference effects. The Bohm-Aharanov effect and the Colella, Overhauser, and Werner (COW) experiment are examples. In Chapter 18 the physical laws for the gauge fields are developed. Using the principle that a gauge transformation generates a divergence free current and a generalization of Gauss’s law, the Yang-Mills equations for the gauge fields are explained. The frame dependent nature of the charge densities is discussed. We use 0 to denote the empty set and Ito indicate the end of a proof.
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I Introduction
MATHEMATICAL MODELS FOR PHYSICAL SYSTEMS When we wish to describe a physical system in a “mathematical” way we try to construct some sort of mathematical structure which, in some sense, “represents” those aspects of the system which are of interest to us. This structure is then a “mathematical model” of the physical system. In this chapter we discuss some basic ideas concerning models for particle motion in classical mechanics. We will see that we are led to consider mathematical “spaces” on which the notion of “tangent vector” is meaningful. We will see how “physical laws” can be incorporated into the model as additional structures on these spaces. EXAMPLE 1.I Consider a single particle, of mass m, free to move along a straight line. We are not interested, at present, in the particular nature of the particle but, rather, we want to follow its motion. Thus, at each instant of time, we are interested only in the position and velocity of the particle. If x is the instantaneous position of the particle and u is its instantaneous velocity, we say the ordered pair (x,u ) is the instantaneous state of the system. The set of all possible states, called the state space, can thus be represented mathematically as the xu plane RZ. Suppose the particle is acted upon by a force field
F(x) =
~
k > 0.
kx,
(1.1)
Let the state of the particle at time t = to be (x,,, u,,). What will be the state at other times? To answer this we need a physical law, that is, a law of motion. That law is, of course, Newton’s second law, which says mx”
=
F
(1.2)
or, in the present case, x“ = -(k/m)x. 1
(1.3)
2
1 . INTRODUCTION
Note that the law (1.3)does not tell us directly what the state will be at other times. Rather, it tells how the state changes at each point. Thus (1.3) is equivalent to XI
( 1.4)
u‘ = - k x / m .
= u,
If we define a v e c t o r j e l d on R2 by (a, b) E R2
@(u, b) = (b, - ka/m),
(1.5)
then we see that, if the state of the system at time t is c(t) = (x(t),
VW),
then c‘(t) = @(c(t)).
A curve c satisfying (1.6) is said to be an integral curve of the vector field @. Thus, the physical law (1.2)determines a structure on the state space, namely, the vector field @. Given the state ( x , , v,) at t = t o , the vector field CD determines uniquely the state at all other times. That is, there is a unique curve, called the trajectory of the system with the given initial state, c: R
--f
R2,
such that c(to) = (x,, u,)
and
c’(t) = @(c(t))
for all t E R .
The curve c is to be found by “integrating the vector field,” which means by solving the system of differential equations (1.4) subject to the condition x(to) = x,, u(t,) = u,. This can be explicitly done for the simple example at hand. In fact the reader can easily check that we have c(t) = ( A cos(wt
+ 6),
-
A o sin(ot + 6)),
where w = (k/m)”2
A
= (xi
+(u~/o~))”~,
and
6={
-arctan(u,/ox,) -arctan(vo/wxo)
ot, - ot, -
+n
if x, > 0, if x, < 0.
The system considered in Example 1.1 is a time-independent mechanical system. If the force field (1.1) were replaced by a force depending on position, velocity, and time, F
= F(t, X , V )
(1.7)
MATHEMATICAL MODELS FOR PHYSICAL SYSTEMS
3
then @ would be replaced by @(t,X, U) = (u, 112- F ( f , X, u)).
(1.8)
Because of the explicit time dependence, we say this system is a timedependent mechanical system. Here, @ is not a vector field on state space R 2 but rather, a time-dependent vector field. The mathematical theory of vector fields and time-dependent vector fields is developed in Chapter 4. EXAMPLE 1.2 A mass m is fixed on the end of a rigid rod of negligible mass having length 1. One end of the rod is fixed at a point P in space so that the mass can move about P subject to the condition that it always be a distance 1 from P . Figure 1.1 depicts the situation. Figure l . l a shows the physical system, while Fig. l . l b shows the sphere M , of radius 1, centered at P , which is the space of all possible positions for m. This sphere is called the configuration space of the system. Suppose we are only interested in the motion of the particle. Then we take, as the state of the particle, the pair of x2, x3), u = (u', u2, u3), where x is three-dimensional vectors (x,u), x = (XI, the position vector of m and v is the velocity vector of m (with respect to some Cartesian coordinate system as shown). Since the mass must stay on the sphere M , we see u must be tangent to M . Thus our state space S does not consist of all pairs of 3-vectors but, instead, we have
S
=
{(x, u)lx E M and u is tangent to M at x).
configuration space M
mass m FIGURE 1.1
4
1 . INTRODUCTION
Although S is not a Euclidean space, nor an open set in one, we shall see that S is a space on which notions such as tangent vector, vector field, and time-dependent vector field have meaning. If we have a force field then the force field will determine a vector field on the state space S, as it did in Example 1.1. However, the construction of this vector field is not so simple as in Example 1.1. First, because S is not just an open set in Euclidean space; secondly, because, if we specify an external force field (e.g., a gravitational field or an electromagnetic field, if rn has a charge), we also must consider the force of constraint exerted by the rod to keep the mass on the sphere. This latter force is not known in advance. These difficulties will be overcome when we develop Lagrangian and Hamiltonian mechanics on manifolds in Chapter 7. The preceding examples exhibit the following general features: There is a state space S that represents those aspects of the system which are to be studied. Given a state so at time t o , the state of the system at other times t is determined by the physical laws which apply to the system. More precisely, the physical laws determine, for each state so and time t o , containing to, and (a) an open interval I(so.ro), s such that 4 ( s o , r o ) ( t O ) = so. (b) a map 4 S 0 . t " i I ( S " . t O ) -+
We interpret &,to,(t) as being the state of the system at time t if the state at time to is so. Frequently, we have = (- 00, co) but this is not always so. The curve in (b) is called the trajectory of the system determined by the condition that, at t = t o , the state is so. When we say that the physical laws determine the trajectories of the system, we do not mean that the trajectories are necessarily explicitly calculable. Rather, we mean we can prove that, given so and t o , the laws determine an and a mapping satisfying (a) and (b). As already disinterval I(so,ro) cussed, the physical law makes its appearance as a mathematical structure, e.g., a vector field or time-dependent vector field, which says what the tangent vector to the trajectory must be at each (s, t). Thus we need, as a state space, a space on which the notion of tangent vector to a curve makes sense. Such a space is called a differential manifold and the study of differential manifolds is a central theme of this book. We shall see that the manifolds which arise in the construction of models for physical systems have additional structures on them. These structures, which include various types of metrics, tensor fields, and differential forms, will be discussed in detail in the coming chapters. The significance of the dialogue between C. N. Yang and S. S. Chern will become clear to the reader.
2 C I assica I Mec ha n ics
As discussed in Chapter 1, we want to develop Lagrangian mechanics and generalized coordinates (i.e., manifold theory). The purpose of this chapter is to introduce the reader to some of the basic ingredients of Lagrangian and Hamiltonian mechanics for the special case when configuration space is an open set M in Euclidean space R" and state space is M x R". We will generalize these constructions to the manifold setting later. For now, we also restrict attention to time-independent systems. We begin with Newton's law of motion and show how to rewrite it so as to get Lagrange's equations. We then introduce phase space, defined to be M x (R")*,and obtain Hamilton's equations. We obtain vector fields on state and phase space whose integral curves are the trajectories of the system. We see that, in the Hamiltonian form, the vector field is obtained in a very simple way from the energy function. The trajectories of the Lagrangian and Hamiltonian systems are related by the Legendre transformation. These constructions easily generalize to the manifold setting of Chapter 7.
MECHANICS OF MANY-PARTICLE SYSTEMS Consider a system of n particles moving in space. Conjiguration space is a subset M c R3". A state is a pair (4, u) E M x R3", q = (q', . . . ,q3")rep. . . , u3") representing veresenting position of the n particles and u = (d, locities. If the masses of the particles are m , , . . . ,m,, define M i , i = 1, . . . , 3n, by M 3 i - 2 = M3i-1= M 3 i = mi.
This is just convenient notation.
5
6
2. CLASSICAL MECHANICS
EXAMPLE 2.1 (Two-body problem) Consider a system of two particles of masses m , and m, moving freely in space. If q , = (ql,q2, q3), q, = (q4, q 5 , q6) are positions, then we define configuration space to be
M
=
{(ql, q 2 )E R~ x
R ~ I ~ z, q,).
The requirement q1 # q2 means that our model does not describe collisions. Given a position q E M the velocities are arbitrary so our state space is S
=
M x R6.
Note that M is open in R6. For a given (4, u) E S the trajectory &,") may not be defined on all of R since, if the initial state is such that the particles will collide at some later time, the trajectory will be undefined after the collision time. EXAMPLE 2.2 (Rigid body motion) We will look at this important example in more detail later. We haven particles n 2 4.Let q i = (q3'-', q3').The particles are constrained by the requirement that there be constants cij such that
lqi - qjl = cij
for i , j
=
1,. . . , n.
The configuration space M is a subset of R3".Here M is not open. We shall see later that M is a diflerential submanifold of dimension 6 in R3". The state space will not be all of M x R3" but rather a subset called the tangent bundle of M and denoted T M . Now suppose M is open in R3" and S = M x R3". Let F = ( F ' , . . . , F3") be the total force. Generally we assume forces depend on position and velocity so
F': M x R3" + R ,
1 = 1,. . . , 3 n .
Note we are assuming our forces are time independent. If (q'(t))is a position as a function of t then Newton's second law says Mi
d'q'
= F'(q(t),~ ( t ) ) ,
u(t) = dq/dt.
(2.1)
Suppose we assume our forces are conservative. Then there is a function, called the potential energy function, V:M+R
such that F'(q, u) = -(dV/aq')(q), i = 1, . . . , 3n. Thus, in a conservative system, F depends on q but not on u. If q(t) is a path followed by the system in
7
LAGRANGIAN AND HAMILTONIAN FORMULATION
configuration space and u(t) = (dq/dr)(t),then (q(t),u(r)) is a trajectory in state space. The trajectories are solutions of dqi/dt = vi dd/dt = - l/Mi
I = 1, . . . , 3n
If we define a vector field on state space, (D: M x R3" -+ R3" x R3n,
by
then trajectories of the system, i.e., solutions of (2.2) are exactly integral curves of 0. This 0 is called the infinitesimal generator for the system.
LAGRANGIAN AND HAM ILTO N IAN FORM ULATION DEFINITION 2.3
Let T :M x R3"
--f
R be given by
3n
iMi(~i)2.
T(4, U) = i= I
T is called the kinetic energy function. Let
K:
M x R3" --i M be projection.
DEFINITION 2.4
The Lagrangian of the system is the map
L: M x R3" + R
defined by L
=
T - ( V o x).
Now dL/8ui = Mid, dLl84' = - i-JV/dqi.If (q(t),u(t))is a trajectory, then
(_ 8L) dt doi
-
=
("
do' (q(t),~ ( t ) ) )= Mi -. dt doi dt
So Newton's law reads
Equations (2.4), which in the present context simply represent rewriting Newton's law, are called Lagran ye's equations.
8
2. CLASSICAL MECHANICS
Later, when we discuss mechanics in the presence of constraints, we will see that similar equations still hold but that they are no longer simply a rewriting of Newton's law. Now, still assuming M is open, we turn to an alternative formulation of mechanics, the Hamiltonian formulation. We have
M = configuration space, S = M x R3" = state space. We define phase space to be
S* = M x (R3")*.
(2.5)
Let ( e l , . . . , e3,J be the standard basis for R3",( e l , . . . , e3")the corresponding dual basis in (R3")*. Define a map
9: M x R3" -+ M x (R3")* 9( 4 ,
dei)
= (4,
7
Misiei).
If we write only the coordinates, 9 has the form 9(4',. . . , 4 -I n , u 1 , . . . , u3") = (4', . . . ,q3", M,u', . . . , M3,,u3"). It is traditional to write points in S* as
( d , . . , q3n>P I , . . .
7
P3").
Note that the p i are just momentum components. We see that 2' has an inverse given by
9l(4, P ) = ( 4 , P l l M I , . . . > P3fl/M3J
(2.7)
Now the transformation 9, which is called the Legendre transformation, seems rather trivial since it is just multiplication of various coordinates by ' can also be viewed in the following way which certain scalars. However, 2 will be generalized later. We have an inner product 9: R3" x R3" -+ R defined by g(u, w ) =
3n
C i=
. .
M~u'w'.
1
This defines an isomorphism
6: R3" + (R3")*
by
[ ~ ( u ) ] ( w=)g(u, w).
9
LAG RANG IAN AND HAM I LTON IAN FORM U LATlON
We then have Y'(q, u ) = (q , G ( U ) ) .
The proof of (2.9) is left as an exercise. Note also that ?L/du' = M i d so that we may rewrite V as
The point here is that these last two descriptions of V will generalize to mechanics on manifolds, to be covered later, whereas the simple idea of scalar multiplication on certain coordinates will not determine a well-defined (independent of coordinates) mapping in the context of manifolds. Now, as before, a trajcctory of our system is a curve in state space, c: I -+S, satisfying certain differential equations. We shall refer to Y c as a trajectory in phase space. The image of Y' c will be called an orbit in phase space. DEFINITION 2.5 An integral on the phase space of a mechanical system is a function 4: M x (R3")*+ R that is constant on each orbit. An integral is often called a first integral in the literature. PROPOSITION 2.6 4: M x (R3")*+ R is an integral if and only if for each trajectory in phase space, c ( r ) = (q(t),p ( t ) ) , we have
where V is the potential energy function. PROOF: Left as an exercise.
1
Suppose U is an open set in a finite-dimensional vector space E . A irector jield on U is just a map f : U + E . An itzteyral curve off is a map c: 1 + U such that c'(t) = ,f(c(t)) for all t E 1. (To differentiate a curve in a vector space pick a basis and then differentiate the components of the curve.) Thus, a vector field on M x (R3")* is a mapping 0, I c J open intervals, we say that a C'-map 4: I -, U is an &-approximatesolution if, for all t in I, we have ((4'(t)- f ( t >4(Nl(5 8. LEMMA 4.13 (Gronwall's inequality) Let j , g: [a, b) -+ R be continuous and nonnegative. Suppose A 2 0 and, for t E [a, b), we have f ( t ) I A f ( s ) g ( s )ds. Then f ( t ) I A exp[Ji y(s) ds] for t E [a, b). In particular, if A = 0, then f ( t ) = 0 for t E [a, b).
+
1;
PROOF: First do the case A > 0. Let h(t) = A + j: f(s)g(s)ds. Then h(t) > 0 and h'(t) = f ( t ) g ( t )I h(t)g(t). Thus h'/h 5 g so In h(t) - In A I j: g(s) ds. Therefore, h(t) I A exp[ji g(s) ds] so, since f ( t ) I h(t), we have the lemma for A > 0. In case A = 0 we have
f ( t )I
+ j'f ( s ) g ( s )ds
so f ( t ) I ~exp[f, g(s) ds] for all
E
for
E
> 0,
> 0 and hence, f ( t ) I 0 as desired. I
PROPOSITION 4.14 Let f : J x U -+ E be Lipschitz on U uniformly on J with Lipschitz constant K . For i = 1 , 2 let ci > 0 and let 4i:l-+ U be an E,-approximate solution. If t, to E I and E = E~ + e 2 , then
Il4l(t)- 42(t)lJ 5
Il4Ato) - 42(to)ll
eKlf-*ol+ ( E / W eK'f-zo'.
PROOF: First consider the case t 2 t o . We have
IldXt) - f ( t , 4i(t))llI
Ei,
t
E 1,
i = 1,2.
44
4. DIFFERENTIAL EQUATIONS ON MANIFOLDS
By Gronwall's inequality,
K This is the desired result in case t 2 t o . For the case t I t o , define the functions -
f ( - J ) x U-tE,
&, 32:- I
f(s,x)=
-
f(-s,x),
u,
3i(s) = cPi( -s). Then, f is uniformly Lipschitz with constant K and 6, is an &,-approximate solution. Now - t o I - t and each lies in - I , so J J i l ( - t ) - 82(-t)ll
+
I (I181(-to) - 4 4 - t o > ( l
+ (E/W
eK(lo-')
or
(I41(t)-
+
I (I141(to) - 42(t0)(1 ( E / K eKlt-tol. )
I
PROPOSITION 4.1 5 Let f : J x U + E be continuous, with Ilf(t, x)ll s L on J x U and let f satisfy a Lipschitz condition, uniformly on J , with con-
45
LOCAL EXISTENCE AND UNIQUENESS THEORY
stant K . Let x, E U . Then there is an open interval J , about 0, J , c I , and an open set U,, x, E U , c U , such that (a) f has a unique local flow 4: J , x U , -+ U , (b) q5 is continuous and satisfies a Lipschitz condition. PROOF: By Proposition 4.1 1, J , , U , exist so that (a) holds. Also, if t, s E J,, x, y E U , we claim, for J , small enough, we can prove
( ( 4 kx) - 44%YIII
5 eKllx - Yll
+ Llt - SI.
(4.2)
It is all right to shrink J , , for by Proposition (4.11), (a) will not be affected. So assume J , c ( - 1, 1). Now
Il4(4 x) - 46, Y)ll I Il4k x) - 4 4 Y)ll + Il4k Y ) - 4% Y)(l and II$(t, x) - 4(t,y)ll I IIx - yl(eK by Proposition 4.14. But since llfll I L we have Il$(t, y) - 4(s, y)ll I Llt - .TI, so (4.2) is proved, which proves the proposition. 1 PROPOSITION 4.1 6 Let f : J x U + E be Lipschitz uniformly on J . Let I c J be an open interval, t o E I , I $ ~ 42: , I + U integral curves of f with 41(to) = 4Jto). Then 41 = 4 2 . PROOF: Take E = 0 in Proposition 4.14. I PROPOSITION 4.17 Let f : J x U -+ E be Lipschitz uniformly on J = (a, b). Let 0 E I = (u,, b,) c J and let a: I -+ U be an integral curve such that a has no extension to an integral curve on any larger interval. Assume exis-
tence of (a) E > 0 such that a(b, - c, b,) c U , (b) B > 0 such that I f ( t , a(t))l I B for t E (b, - E, bo). Then bo = b (a similar result holds for left end points). PROOF: Let a(0) = x,. Then for t a(t) = x,
+
E
I we have
1;
f ( s , a(s))ds.
So, for t , , t2 in (b, - E , b,) we get
Hence, lim,,,, a(t) = u exists and, by (a), u E U . Suppose b, < b. We claim there is an open interval I‘ about b, and an integral curve 8:I‘ -+ U s.t. p(b,) = u. For let f ( J - b,) x U + E be defined by y(t,x) = f ( t b,, x). Let I” c J - b, be an open interval about 0, I” -+ U an integral curve of
+
46
4. DIFFERENTIAL EQUATIONS ON MANIFOLDS
?with p(0)= u. Then we define (a) I' = I" + b,, (b) PO) = k t - bo). Now, P(b,) = &O) = u and p(t)= F(t - b,) =f(t - b,, F(t - b,)) = f ( t , P(t)). Consider a, P on I' n (b, - E , b,). Each is an integral curve off. P is defined on some interval (b, - E ~ b,, + E J where (b, - E ' , b,) c I' n (b, - E, b,). On this interval define
: :{
bo - ~1 < t < bo, b, I t < b, E ' ;
YO) =
+
limt+bo-a(t) = u = P(b,) so y is continuous. If we show Y(t)= u
+
Jb', f(s, Yb))
ds,
(4.4)
then y is an integral curve off, so we have extended a to a larger interval. On b, - E' < t < b,, (4.4) reads a(t) =
+
oJ:
f(s, ~ ( s ) )ds.
Since a'(s) =f(s, a(s)) on t < s < b, and lims+bo-a(s) = u, (4.4) follows, for t < b,, from the fundamental theorem of calculus. For bo < t < bo e l , (4.4) says P(t) = 2) + yoP'(s) ds, which is true for the same reason. I
+
EXAMPLE 4.18 Define f:(- 1, 1) x R + R by f(t, x) = x2/(t - 1). Consider the initial value problem
dx/dt
= f(t,
x),
x(0) = -2. By separating variables, we find the solution to be x = - l/(lnJt- 11
+ +). + 1) for, as t + 1
This solution is valid only on (- co, -e-'/' - e-1/2from the left, lnlt - 11 + -$ and x + - 03. How does this compare with Proposition 4.17? If a(t, x) satisfies D,a(t, x) = f ( t , a(t, x)), a(0, x) = x,f is C' and a is C', then D2a is a solution of the linear differential equation
444 x) = D2fk a ( t , x ) ) satisfying
qo, x) = 1,.
w,x)
47
LOCAL EXISTENCE A N D UNIQUENESS THEORY
We will prove that a is C’ with respect to x by showing that the solution A(t, x) is D2a. We first prove a theorem about linear differential equations (see also Exercise 4.12). PROPOSITION 4.19 Let J be an open interval about 0, E and F vector spaces, V an open set in F . Let
g : J x V + L(E, E )
be continuous. Then there is a unique A: J x V -+ L(E, E ) such that (a) D,E.(t, x) = g(t, x) 0 A(t, x) for ( t , x) E J x I/ (b) i(0, x) = 1, for x E I/. Furthermore, this A is continuous. PROOF:
Fix x E I/. We show there is a unique A,: J
(a) AX(0) = I,? (b) 23)= g,(t)
0
-+
L(E, E ) such that
A&), where gAt) = g(t, 4.
Define f :J x L(E, E ) -+ L(E. E ) by f ( t , w) = g,(t) w. Let J’ be an open interval contained in J with J’ compact and J’ c J . Then, on J’ x L(E, E), 1 is Lipschitz uniformly on J’, because 0
Ilf(4 4 - f k
411
Ilsx(t)lI
IIU - 011
and g, is bounded on J’ (since J” is compact). If we have A: J’ -+ L ( E , E ) such that A(0)= l,, X ( t ) = f ( t , A(t)), then A is unique by Proposition 4.16. We claim there is a maximal interval J , about 0, J , c J , such that there is a A : J , -+ L(E, E ) which is an integral curve off with A(0) = 1,. For take J , = { t E 513 a J’ about 0 as above and a i defined on J’}.J , is the union of all open intervals J’ about 0 such that J’ is compact, J’ c J and 3:J’ -+ L(E, E ) with A(0)= l,, A‘ = f ( t , A). We claim J , = J . Let J = (a, b),J , = (a,, b,) and suppose b , < b. We show this leads to a contradiction, so b , = b (and a similar argument shows a, = a). Choose finite numbers a,, b, such that a, < a, < 0, b, < b2 < b. Consider f on ( a 2 ,b2) x L(E, E). Proposition 4.17 applies to ] ” : ( a 2b,) , -+ L(E, E). By maximality of J , , A cannot be extended to an integral curve on any larger subinterval of ( a 2 ,b,). Let E > 0 be such that b , - E > 0. If we show (a) W , - E, b,) = W , EL (b) 3B > 0 such that [If@, A(t))ll I B, t
E (b,
- 8, b,),
then, by Proposition 4.17, b , = b 2 , a contradiction. Now (a) is clearly true. Consider (b). Now (If(t, A(t))II I Ilg,(t)ll IIA(t)ll. But g,(t) is continuous on
48
4. DIFFERENTIAL EQUATIONS ON MANIFOLDS
[0, b,], so let C > 0 be such that IIgx(t)ll I C for 0 I t I b,. Then note that
i(r)
=
I
+ ji y,(s)
3'
A(s)ds,
t
E
[0, b,),
which gives
IlW)ll 5 1 + c J;
IlA(s)ll ds
So, by Gronwall's inequality, for
Il;l(t)ll I ect I eCb2
t
E
[0, bl).
So, for t E (b, - E , b,), Ilf(t, E,(t))llI CeCb2= B. Thus we have proved J , = J . We have shown that for each x E V 3 a unique A x : J + L(E, E ) with A,(O) = 1, and &(t) = f ( t , A,(t)) for t E J . This defines A: J x V L(E, E ) satisfying (a) and (b) and it remains only to prove continuity of A. Fix ( t o ,x,) E J x V . Let I be a compact subinterval of J having 0 and t o in its interior. Choose C > 0 so that Ill(t, xo)ll I C for t E I . Choose an open neighborhood V , of x, I/, c V , such that there is a K > 0 with
-
for all ( t , x) E I x V,.
Ilg(t, x)ll I K
For
(r, x) E I
11%
x V, we have
x) -
W,, .xo)ll s I l i ( r , x)
-
i ( r , xo)ll + 1(W,xo) - W,, xo)ll.
The term IlL(t, xo) - i(ro, xo)ll can be made small by making t near t o , so we need to see that IlA(t, x) - i ( t , xo)ll can be made small by making t near to and x near x,. We have
JIW(t,xo) - d t , 4
O
A(4 xo)l(
IpAG xo) - y(t, xo) " 4 4 X0)ll + Ils(4xo) 5 Cllg(4 x)
-
-
Ilnk X0)ll
g(4 4 11
s(t,x0)II.
Given E > 0 there is, by a simple compactness argument, an open neighborhood V, of xo, V, c V, such that for ( t , x) E I x V,,
Ils(L x) -
s(t3
X0)ll
5 E/C.
For such (t,x) we see that
-
1p14t,x,)
-
y(t, x)
('
3 4 , xo)(lI E.
So, t A(t, x,) is an &-approximate solution of the differential equation of which the map t + A(t, x) is a solution. Therefore, by Proposition 4.14, there is a constant K , such that
Il44 x) -
w,x0)II I
EK1.
49
LOCAL EXISTENCE AND UNIQUENESS THEORY
We can take eK[lenplh of I ]
K, =
K
We have used the fact that 1(0,x) = 1, COROLLARY 4.20
Let f :E
+E
= A(0, x,).
I
be linear. There is a unique a: R x E
+E
such that (a) a is continuous, (b) Dla(t, x) = f ( a ( t ,x)) for t E R and x E E , (c) a(0, x) = x for x E E, (d) if a,(x) = a(t, x), then ( a J f E Ris a 1-parameter group of linear transformations of E, that is, No =
1,.
a,, t 1 2 = atl "at,.
PROOF: Define g: R x E + L ( E , E ) by g(t, x) = j . Let 1:R x E L(E, E ) be as in Proposition 4.19. Define a(t, x) = A(t, x)x; a is continuous, (b) holds and a(0, x) = A(0, x)x = l,(x) = x. By Proposition 4.16, a is clearly unique if (b) and (c) hold. Only (d) remains. If two integral curves R E agree at one point they are equal everywhere, by Proposition 4.16. We claim d(t, x) does not depend on x. To see this, show A(t, x,)y = d(t, x,)y for all t , xl,x,, y. Let q5i(t)= L(t, xJy, for xi and y fixed. Then, for i = 1,2,4;(0) = i(0, xi)y = y. .X;))Y = (s(t,xi) * 4 4 xJ)y = f ( W ,XJY) = Also, 4Xt) = D , ( W , XJY) = (Dl .f(q5i(t)).Here, as in the verification of (b), one uses the chain rule as follows: Let F : R + L ( E , E ) be defined by F(t)= A(t, x i ) and let G: L(E, E ) + E be defined by G ( A ) = Ay. Then 4 J t ) = G F ( t ) so &(t) = DG(F(t))F'(t)= F'(t)y = D,A(t, xi)y.We conclude 4 , = ( p 2 , so 2(t, x) does not depend on x and hence, a(t, x) = lb(f, 0)x. Then clearly, a, = l ( t , 0) is linear. Finally, we verify the group property. We want a ( t , t,, x) = a(t,, a(t,, x)). Fix t , and show, for all t , q5(t) = a(t t,, x) and $ ( t ) = a(t, a(t,, x)) are equal. Now, (p(0)= a(t,, x) = $(O), so if we show both 4 and $ are integral curves, we are done. Clearly $ is an integral curve and -+
-+
w>
+
+
= D,a(t
so
+ r 2 , x) = f'(a(r+ r2,x)) = f ( W ) ) ,
4 is an integral curve. I
THEOREM 4.21 Let J be an open interval about 0, U open in E , f : J x U -+ E a time dependent CP-vector field, p 2 1. Let x, E U . There is an open subinterval J , containing 0 and an open set U,, x, E U , c U such that
(a) f has a unique local flow a : J , x U , + U , (b) a is Cp,
50
4. DIFFERENTIAL EQUATIONS ON MANIFOLDS
(c) For ( t , x) E J o x U , we have D1D244 X) = D J ( t , a(t, x)) D 2 4 t , X) and D2a(0, X) = 1,. PROOF: By Proposition 4.15 we can choose J , , U o s.t. there is a unique flow a: J , x U o + U which is continuous and satisfies a Lipschitz condition. Define 0
9: J 1 x Uo --t L(E, E )
g(t, X) = D2f(t, a(t, x)).
by
By shrinking J , , U , , if necessary, we may assume g is continuous and bounded on J , x U , . Let J , be an open interval about 0 with J, c J1. Corresponding to g:J, x U,
+ L(E, E ) ,
there is a unique 1:J, x U , -+ L(E, E ) for which (a) D , W , x) = g(t, x) (b) l(0, X) = 1,.
0
W,x)
We now claim D2a:J o x U o + L(E, E ) exists and, in fact, D,a(t, x) = l ( t , x) for (t, x) E J , x U , . Since, by Proposition 4.19,lis continuous, we will have shown D2a is continuous. Now D,a(t, x) = f ( t , a(t, x)) on J , x U , so D,a is continuous on J o x U , . Once we show D2cx = A, we conclude (a) c1 is C' on J o x U o and (b) statement (c) of the present theorem holds. We verify the claim as follows:
Fix x E U,. For h small and t E J , we have O(t, h)
defined by
+ h) - a(t, x) and D,e(t, h) = ~ , a ( tx, + h) D,a(t, X) = f ( t , a(t, x + h ) ) O(t, h) = a(t, x
-
-
f ( t , a(t, XI).
Thus we have p1m
h) - g ( 4 x ) W , h)ll
4,x + 4 )- f ( t , a@,XI) - D 2 f ( t , 44 x))% 411 5 IlW?h,ll supllD2f(t9Y) - D2f(4 a(t, X))ll? = Ilf(4
+
where the sup is over all y on the line segment joining a(t, x) and ~ ( tx, h). By the Lipschitz condition satisfied by a, there is a K > 0 s.t. IlO(t, h)ll 5 Kllhll, so
p,O ( 4 h)
-
g(h M t , h)ll 5 KIJhllS U P IID2f(t2Y) - D Z f k Y
46 x))((.
51
LOCAL EXISTENCE AND UNIQUENESS THEORY
As stated above, by Proposition 4.15 we can assume that Ila(t,, x l ) a(t,, xz)ll I Alt, - t,l Bllx, - xzll on J , x U,, where A > 1 and B > 1. For each t E 7, c J , , the continuity of D 2 f at (t, a(t, x)) gives an open interval I , c J , and 6, > 0 so that s E I , and llz - a(t, x)ll < 6, implies that IID,f(x, z ) - Dzf(t, a(t, x))ll < 4 3 k . Let V, = I , n {sl(s - tl < 6J2A). Use and take 6 = min hti/2B. the compactness of J,, to get J, c Now suppose llhll < 6. If t E J,, then s E for some i. Thus
+
uy= V i
Vi
and so that which implies
Hence, and which gives
so that
This shows that, if
then lim $(h) = 0. h+O
Now IlD,O(t, h) - g(t, x)O(t, h)l(Illhl($(h)for t E J,, so O is a (Ihll$(h)-approximate solution of the differential equation satisfied by t -,A(t, x)h, that is, D,(A(t, x)h) = g(t, x ) 0 A(t, x)h. Note that x is fixed here, so the differential equation we are talking about at the moment is y:J, x E
+
E,
y(t, U) = g(t, X ) U .
52
4. DIFFERENTIAL EQUATIONS ON MANIFOLDS
Since O(0, h) = h = L(0, x)h, we conclude, by Proposition 4.14, there is a constant C,,such that p(t,
h) -
w,x,q
for
5 C,llhll$(h)
t E Jo.
Note in Proposition 4.14 the constant does not depend on which Eapproximate solutions we are considering, so C, is independent of h. Since limb+, $(h) = 0, we have proved D2a(t,x) = L(t, x), as asserted. We have also shown
4 ) D,a(t,
DID244 x) = D2fk
x)
O
and D2a(0,X)
=
1.,
Now that we have established a is C' and that Eqs. (c) hold, we proceed to establish by induction, that a is actually C p i f f is Cp. Suppose, inductively, that the theorem is true for p - 1 and that f is Cp.Then, given x, we get a J , and a U , and a unique flow a : J , x U , + U such that (c) holds and with a being CP-I.To show a is C p we show D,a and D,cc are C p - ' . By the chain rule, we conclude that D,cr(t, x) = f ( t , a(t, x)) is C p - ' , by the inductive hypothesis. To show D2a is C p - ' on J' x U',J' a subinterval of J , containing 0, U' open, with x, E U' c U , we show D 2 a is "part" of the flow of a C p - ' vector field. Define F: J , x U , x L(E, E ) + E x L(E, E ) by F(t, x, w ) = (0,D 2 f ( t ,
XI)
O
w).
Clearly F is C p -'. Define p: J , x U , x L(E, E ) + E x L(E, E ) by
At, x, w ) = (x, D,a(t,x)
w).
Now we have P(0, x, w ) = (x, D 2 4 0 , X ) w ) = (x, w)
and DIAL, x
W ) = (0, D1D2a(t, X)
= (0, D Z f ( 4 =
0
W)
@, $1
F(t, X, D2a(t, X)
O
0
D,a(t, 4 w ) F(t, p ( t , X, w)). O
W)=
So p is a flow for F , so p is Cp-' on some J' x U' x W', J', U' as above, W' a neighborhood of 1, in L(E, E ) . So, on J' x U', the map (t, x) + 7r2 p(t, x, lE)= D2a(t,x) is C P - ' . I 0
We now translate our results to manifolds.
53
THE GLOBAL FLOW OF A VECTOR FIELD
THE GLOBAL FLOW OF A VECTOR FIELD PROPOSITION 4.22 Let ( be a vector field on X , x o E X . Then there is an interval I about 0 and an integral curve of (, c: I + X , with c(0) = x o . Further, if c^: f + X also satisfies the above conditions then c = t on I n f. PROOF: Choose a chart ( U , 4) about xo and consider the equation dxldt = (&x). The existence results already proved show that this differential equation has a solution c": I -+ &U), E(0)= 4 ( x o ) .Then 4 - l c": I -+ X is an integral curve of ( with 4- c F(0)= xo.The uniqueness results for differential equations imply that, if c, t are integral curves of ( with c(0) = c^(O), then c = c^ on some neighborhood of 0. To prove c = 2 on I n f, let 0
A
=
{r E I
n r^lc(t) = t ( t ) } .
Clearly A is closed in I n f (Here we use the assumption that our manifold X is a Hausdorff space.) If t o E A, let d(t) = c(t + to), J(t) = t(t to). Then d, d^ are integral curves of ( defined near 0 and d(0) = J(O), so d = d^ on some neighborhood of 0. Hence c = t on some neighborhood of t o . Thus A is open so A = I A f. 1
+
Given xo E X , let
9 = { I 11 is an open interval about 0 and there is an integral curve c: I + X with c(0) = xo}. Let I,,
=
UIE9I ; I,,
is an open interval about 0.
PROPOSITION 4.23 There is an integral curve of 4, with initial value xo, defined on I,, and I,, is the largest interval on which such an integral curve exists. PROOF: Define c,,: I,,
X as follows. Given t E I,, choose I E 9 with -+ X with c(0) = xo, so define c,,(t) = c(t). In other words, cAolI= c. By Proposition 4.22, cx0 is well defined and is clearly an integral curve at xo. Also it is clear that I,, is maximal as stated. 1 +
t E 1. Then there is an integral curve c: I
DEFINITION 4.24 We call c,,, the maximal integral curtie at xo and call 1," the maximal interval of existence at xo. REMARKS: (a) If 5 is a vector field on X , . x 0 e X , we sometimes speak of "the integral curve through xO" meaning the maximal integral curve ex": l x , +
x.
(b) In Definition 4.1, the notion of a C"-vector field on a manifold was introduced. It is clear that, by replacing ''00" by a nonnegative integer p ,
54
4. DIFFERENTIAL EQUATIONS ON MANIFOLDS
we can define the notion of a CP-vector field. It is equally clear that a vector field is C" if and only if it is C p for all p 2 0.
0 such that, for r E (b - E , b), u,,(t) 4 A . PROOF: Suppose not. Then there is a sequence ( t , ) in I,, converging to b, such that axo(tn)E A for all n. Since A is compact, some subsequence converges. By a change of notation we may suppose uxo(t,)+ y , E A . Now 9(5) is open, so there is a 6 > 0 for which there is a neighborhood U of y o with (-6,6) x U c 9(() Choose . IZ so large that u,,(t,) E U and b - t , < 6. Then f l : ( t , - 6, t, + 6) + X , given by B(t) = a(t - t,, uxo(t,)),is an integral curve with P(t,) = u,,(t,). So P extends ax,,to t, 6 > b, contradicting maximality o f b , b). I
+
COROLLARY 4.34
If
5 has compact
support, then ( is complete.
is a compact set K c X such that 5(x) = 0 if x 4 K . Now choose x E X and show I , = R . If x 4 K , then the maximal integral curve at x is c,(t) = x for t E R, so I , = R in that case. If x E K , then cx(Ix)c K , since integral curves outside K are all constant. But then Proposition 4.33 implies I,= R . PROOF: There
COROLLARY 4.35
A smooth vector field
i on a compact manifold is
complete. PROOF: 5 automatically has compact support. EXAMPLE 4.36
diffeomorphism
I
Let x o , y 0 E R", Ilroll, llyoII both 4(U)
THE CANONICAL 1 - F O R M ON T*X Let ( U , 4 = ( q l , . . . ,q")) be a chart on X . If w E T Z X , where x is in U then we can write w = pi&'. Thus we have a map z*-'(U)
+ &U)
x R"
defined by
and this map is precisely (%)*.
w + (q',. . . ,q", p i , .
. . , p,) (5.7)
69
THE CANONICAL 1 -FORM ON T ' X
Now T*X is a manifold, so we may speak of differential 1-forms on T * X . Such a form then, is a map a: T*X + T*(T*X)such that ~ ( wE) T:(T*X)
for each w E T* X .
Let x E X , w E T,*X, and u E T,,(T*X). We have we have
T*:
T*X
-+
X and hence
T,T*: To,(T * X )+ T,X. Define O(o):T,( T * X ) -+ R by
8(0)(4 = 47-J*(41;
(5.8)
0 is a 1-form on T * X , called the cunonicul l-forrn. Note that there is no analogous construction on T X ; 8 represents a very special feature which T*X has. To check 0 is C", use a natural chart. PROPOSITION 5.1 6 P1>.
8 = p i dq', in the coordinate system
( 4 l , . . . , 4",
. > PJ '
PROOF: We will prove this by showing that if o has coordinates (q', p i ) ,
then 0(o)(d/84')= pi 0 ( o ) ( d / 8 p , )= 0
l 0,
since no term is negative while, at least one must be positive (namely, choose i for which x E a,:'((O, 13)). This shows g is a Riemannian metric on X . I DEFINITION 6.38 A C"-Jield of lines on X is a choice, for each x E X , of a one-dimensional subspace L ( x ) c T,X. The following smoothness condition is required: For each xo E X there is an open neighborhood U of xo and a C"-vector field 5 defined on U , such that for x E U , the line L(x)is the span of ((x), that is, L(x) = (((x)) = {lPI.
(7.17)
Then (7.15) becomes gpvqv
+
[BY, ,1]4%j"
(7.18)
= 0.
Multiply by 9'" and sum on p to get the set of equations q"
+ rFv@q" = 0,
CI =
1,
. . . ,d .
(7.19)
Equations (7.19) have as solution curves those curves which are geodesics on M for the Riemannian metric T = gpv dq" dq' on M . Geodesics are important objects in Riemunniun geometry and will be studied in more detail later. are called the ChristofSel symbols of the j r s t and The symbols [pv, p], r$" second kinds, respectively. We summarize the above results by saying: For a system of particles moving under the influence of holonomic constraints, but with no applied forces, each trajectory is a geodesic of configuration space with respect to the kinetic energy metric.
TH E LEG ENDR E TRANSFO R MATlON Let M be a manifold, g a metric on M . The Legendre transformation 9: 7'M sponding to g is given by DEFINITION 7.10
L?(v)(w)= gx(u, w )
for v, w E T,M.
+
T * M corre-
104 b;p
7. HOLONOMIC SYSTEMS
is a bundle isomorphism, i.e., the following diagram commutes,
TM
5T*M
and 9 gives a linear isomorphism T,M PROPOSITION 7.1 1
+ T,*M for
each x E M.
Y is a C"-diffeomorphism.
PROOF: Let ( V ,d, = ( q ' , . . . , q " ) ) be a chart on M . We get charts (T-'(U), Td,),(z*-'(U), (@),) on TM, T*M, respectively. We must show
(G),F c-
0
(Td,)-'
is C" with a Cm-inverse. We have
and
so 9'o (Td,)-'(q, q) = qigiXq)dqj(d,-'(q)).Then note that (5)*(digij(q) dq'(4- l ( 4 ) ) )= (q, gij(q)qi),
so (%)*o 9 o (Td,)-'(q,4) = (4, gij(q)qi).This is clearly C" with C" inverse given by (qi, Pi)
+
(qi, d'(q)Pj).
I
Recall if g is the kinetic energy metric of a particle system, we have the kinetic energy function
T :T M and the Lagrangian L = T L(q, 4 ) = $g,,@'q"
T(q,4) = &py(q)4p4v
R, -
V z. So we have
-
0
V(q)
and
dL/dqp = gpvqv.
Thus, the local representative of 9 is 2 ( 4 , 4) = (4,(dL/%?) ( 4 , 4 ) ) . See Exercise 7.12 for an invariant description of equation (7.20).
(7.20)
105
CONSERVATION OF ENERGY
CONSERVATION
OF ENERGY
Consider a system with configuration manifold M , kinetic energy function T : TM + R , potential I/: M + R, and Lagrangian L = T - I/. 5. DEFINITION 7.1 2
v
L
The energy function is E: TM
-+
R , given by E
=
T
+
5.
THEOREM 7.13 (Conservation of energy) If c is a trajectory of the system in T M , then E c is constant. 0
PROOF: We work locally to show (dldt) (E(c(t)))= 0. We first claim
E(q, 4.)
= (aL/aq')q' - L(q,
4).
(7.21)
This is seen as follows:
aL/di'
= dT/d~j',
774,4) = +gij4'q'
so that aT/aLj' = gij$; thus (aL/dq')q'= gij4jq'= 2T(q, q), so
(aLla4')ci' - U q , 4) = 2T(q, 4) - (T(q,4) - V q ) ) = T(q,4) + V q ) = E(q, 4). so
The first and third terms add up to zero, since Lagrange's equations hold, and the second and fourth terms cancel, so the proof is complete. The quantity (dL/&j') (4, 4)$, which appeared in the preceding proof, is the coordinate representation of a function on TM as follows: Define a function A: TM
+R
by A(v) = U ( u ) u . We see that
We also have E=A-L.
which is clear from the local equation E(q, 4) = (dL/dLj')4' - L ,
derived earlier.
106
7. HOLONOMIC SYSTEMS
HAMILTON'S EQUATIONS Consider a system with configuration space M . W e have shown that the equations of motion are the Lagrange equations, that is, if c is a trajectory of the system in state space T M , then, in local coordinates, c satisfies Lagrange's T M -+ T * M , we can conequations. Given the Legendre transformation 9: sider phase space trajectories, i.e., curves 9 c where c is a trajectory in state space. We want to know what differential equations are satisfied by these phase space trajectories. 0
DEFINITION 7.14 The Hamiltmian a ( 9 - l(a))- L ( 9 - '(a)). LEMMA 7.15 H = E PROOF: E = A
-
H:T*M
-+
R is defined by H ( a ) =
9-'.
0
L so we must show
A(.rl(a))
= +rl(a)).
But A(u) = ~ ( u ) SO u
A ( Y - y a ) )= 9(9-l(a))(9-'(Lx))= a(9-ya)).
I
Now consider H : T * M -+ R . Given coordinates (q', . . . , qd) on M , we get coordinates ( q l , . . . ,qd, q', . . . , qd) on T M and ( q l , . . . , qd , p l , . . . , p d ) on T*M. The Legendre transformation is described by pi = (dL/aq')(q,4). The Hamiltonian is given in local coordinates by H ( q , P ) = pi$
-
L(q, 4).
+
Now dH = (aH/dqi)dq' ( d H / d p , ) d p , in general, but we also have d H = pi dq' + 4, d p , - dL. Now really q i = Lji 9-', L = L 0 9-', so we are interested in 0
d(q' o
9 - l )
= (2-')* dq'
and
d(L 9-') = (2-')* dL. 0
Then
=
($
o
9-l)
dqi + p , ( Y - l ) * dq'.
107
HAMILTON'S EQUATIONS
Therefore dH
= pi(9-')*
dqi + (4'
0
9 - l )
dp,
-
d(L
0
9 - l )
Thus we have
Let c(t) = (ql(t),. . . , qd(t),ql(t), . . . ,d d ( t ) )be a state space trajectory. Then Lagrange's equations hold (we assume a conservative system), L dt( g adi ( q , d ) ) - : ( q ,84 q)=O,
i = 1 ,..., d.
Now 9 0 c is a trajectory in phase space. Let this be written (ql(l), . . * qd(t), 7
...
9
Pd(t)).
Then (8H/dpi)(2 c(t)) = q'(t), by Eqs. (7.22), and 0
We summarize as follows: THEOREM 7.16 Let (ql(t), . . . , qd(t),pl(t), . . . , p d ( t ) ) be a phase space trajectory in a natural cotangent bundle chart. Then (Hamilton's equations)
(7.23) EXAMPLE 7.1 7 (Central force motion-inverse square law) Consider a mass m moving in space under the influence of a force field generated by the potential
V:R3 - ( 0 ) + R,
V ( X )=
-
k/lxI,
k > 0.
There are no constraints, so configuration space is M = R 3 - {0}, and (ql, q2, q 3 )are the usual Cartesian coordinates. We identify T M with M x R3 with coordinates (ql, q2, q3, q l , q2, Q3)
= (494).
108
7. HOLONOMIC SYSTEMS
Also, we identify T * M with M x R 3 with coordinates
GI1>qZ. q3,Ply Pz > P 3 ) = (4, PI, We have the Lagrangian
m,4)
=
fm14I2 - WI).
TM The Legendre transformation 2: Y(q,4) = (4, m4)
and
-+
T * M is given by
2- l(4, P) = (4, P / 4 .
The Hamiltonian is H(q, P) = (lPI2/W + U q ) .
Simply writing down Lagrange's or Hamilton's equations is not very illuminating. Consider GI, G,, G,: T * M -,R defined by Gl(q7
P) = q 2 p 3 - q 3 p 2 ,
G2(q,P) = q3pl - q1p3,
(7.24)
G3(q, P) = q1PZ - q2Pl.
Recall that an integral, or constant of the motion, of a system is a map 4: T * M + R which is constant on each phase space trajectory (cf. Definition 2.5). We claim the functions in (7.24)are integrals. In Chapter 15 we will see how these quantities result from symmetries of the Hamiltonian system. We check the claim for G,, letting (q'(t),p j t ) ) be a trajectory. We have d dt
- Gl(q, P) = 4'6, - q3d2
+ qZp3- ri3pz = q2p3 - q3P2
where, in the second equality, we used p i = mq'. But aH/aq'= aV/aqi = kqi/)q13,so we get
The quantities G I , Gz, G, are familiar as the components of the angular momentum vector of the system and we have proved conservation of angular momentum (for this system). Consider now motion beginning with initial condition (qh, q i , 0, 46, &, 0).
109
HAMILTON‘S EQUATIONS
Then we will always have (see Exercise 7.2).
q3(t)= 0 = d 3 ( t ) = p 3 ( t ) Consider cylindrical coordinates, q’
=
r cos 8,
q2 = r sin 8, 43
= z.
4’ 4’
= cos
We have
=
43 =
sin ~i + r cos 86,
i.
+
Then L(r, 8, z, i, 6, i) = &(i’ r’@ that the Lagrange equations are
d -
dt
d -
dt
~i- r sin 04,
+ 2’) + (
(mi) = mrg2 -
k / J m ) . One easily sees
kr (r’
+z~)~/’’
(mr2@= 0,
d - kr ( m i )= dt ( r 2 + z2)3/2
-
We know z
=0 =
i, so the equations reduce to
mi: = mro2 - ( k / r 2 ) ,
(d/dt)(mr26)= 0.
(7.25)
We will not solve these equations here, but refer the reader to classical mechanics texts, e.g., Goldstein. The Legendre transformation is given by
Y ( r , 8, z , i, 0, i) = (r, 0, z , mi, mr’8, mi),
y -‘(r,6, z , p r , p e , P A
= ( r , 0, z ,
p,./m, peImr2,p,/m).
The Hamiltonian is
=
5
k i (p; + + Pi) 2m Jm-
110
7. HOLONOMIC SYSTEMS
and Hamilton’s equations are
If, due to initial conditions, z and i are 0, we get mi
= p,,
mr2e = pe, 0 = p,,
p.r = - -Pe-2
k
mr3
r2’
fie = 0, p, = 0.
We conclude d dt
- (rnr’e) = 0,
k d (mi) = Pe2 - dt mr3 r2‘
-
~
(7.26)
Note Eqs. (7.25) and (7.26) are the same. We now want to consider the equations of mechanics from the point of view of flows of vector fields. We shall show that there exist vector fields on T M and on T * M whose integral curves are precisely those curves which, when written in natural coordinates, satisfy Lagrange’s equations and Hamilton’s equations, respectively. We will see that the construction of these vector fields is accomplished in a very natural way by means of a canonical 2form on T * M . This canonical 2-form represents what is called a symplectic structure on T * M . First some general facts about skew-symmetric tensors will be discussed.
2-FORMS DEFINITION 7.18 A skew symmetric 2-tensor on a vector space V, is a bilinear map o:V x V + R such that o ( u l , u2) = - o ( u 2 , vl) for all u l , v2 E V . We often call a skew symmetric 2-tensor on V simply a 2-form on V . The set of all 2-forms on V is a vector space, denoted A2(V).
111
2-FORMS
DEFINITION 7.1 9
The antisymmetry operator .d:T 2 ( V )-+ A2(V) is def-
ined by .do(v,,v 2 )
= t(W(U1, " 2 ) - w(v2, UJ).
We say .dois the skew-symmetric part of if o is skew-symmetric. DEFINITION 7.20 CI A
The wedge product
(to.
A
p = 2,d(C!0p).
Note that .do= o if and only
: V* x V* -, A2(V) is defined by
Let e l , . . . , e n be a basis of V with dual basis e l , . . . , e n . PROPOSITION 7.21
{ e i A ejI
1I i < j 5 n ) is a basis for A2(V), so
PROOF: Write Q E A J V ) as (11 = wijei 0ej. Then
o = .do= wii.d(ei8 e') = (w$)e'
Now
(toij = w ( e , , ej) = - w ( e j , e i ) = --toii
A
ej
so that
This proves our set spans For indcpendence, note that if then, for k < e we have
0=
alje' A
XI
j and wii = 0, then w wijei eJ.
o
EXAMPLE 7.22
tl13e1A e3
=
In R3, any 2-form is uniquely expressible as tl12e1A e2 +
+ t123e2A e3.
A differential 2,form on a manifold X is a tensor field F i ( X ) such that o(x) is skew-symmetric for every x E X. The set of differential 2-forms on X is denoted by A2(X). We sometimes refer simply to a 2-form on X . Let ( U , 4 = (q', . . . , 4")) be a chart on X. Then there exist uniquely defined functions w i j such that DEFINITION 7.23
w
E
w = wij dq'
0dqj
on U
and we also have o = olijl dq'
A
dqJ
on U .
We often consider a function of class C", fi X 4R to be a differential O-form. We then use the notation Ao(X) as an alternate notation for Cm(X,R).
EXTE R I0R D E R IVATIVE Also we may write Al(X) rather than Fy(X). Then we have the map d: Ao(X)-+ A,(X).
We now define an operation d: A,(X) + A2(X)
called exterior derivative. Eventually we shall define differential forms of arbitrary k , denoted A,(X). Then we will have an exterior derivative operator d: A,(X) A,, l(X). At present we have only the case k = 0, so we now define d for k = 1. -+
PROPOSITION 7.24 Let ( U , 4 = ( q ' , . . . , q " ) ) be a chart on X . There is a unique map d u : A l ( U )+ A2(U) such that
(a) d,(a + B) = d u ( 4 + d d B ) for a, B E Al(U), (b) d a ( f j l ) = d f A a fd,u for f~ Ao(U), tl E Al(U), (c) d,(df) = 0 for all f E Ao(U ) .
+
113
EXTER 10R D ER IVATIVE
PROOF: Let a E Al(U). Write a = aidq', where the ai are uniquely determined smooth functions. Define d,(a) = daj A dqi. We must verify (a)-(c). We leave (a) to the reader. For (b), let a = xi 4';then fa = fai dq', so
&(fa)
+ a, df)A dq' = fdC$ A dq' + 4 f A a; dq' = f d a + d j ' A a.
=
d(.fai) A dq' = (.f'da,
For (c),
since d2flaqJ aqi is symmetric in i , j while d 4 ' ~dq' is skew-symmetric. This proves existence. But if any map d, satisfies (a)-(c), then d,(ai d 4 ' ) = dcl; A d4'
so uniqueness is proved. THEOREM 7.25
+ Mjd,(d4') = dai
A
d4',
I
There is a unique map d: A,(X) -+ A,(X)
such that if a E A,(X) and ( U , 4) is a chart on X , then dal,
= dU(aIu).
PROOF: To prove uniqueness, let x E X . Then, if ( U , 4) is a chart at x, (da)(x)= d,(al ,)(x). For existence we must show that if ( U , $), ( V , +) are charts at x, then
I
d,(a W x ) = d " ( E I V ) ( . 4 .
Let
4 = (x', . . . , xn),
= ( J ) ' , . . . , y").
Then if a
= ai dx' = Jj dy',
du(alu)(X) = da,(.x)A d X ' ( X )
and d,(al,)(u)
= dBj(x) A
~Y'(x).
Now on U n V we have two coordinate systems (xi),( y j ) .Now d,, " a ( x ) = da,(x) A dx'(x), but also d,, "a(.x) = dp,(x) A dyj(x). These two expressions for the uniquely determined form d,, " a must be equal, so dai(x)A d.xi(x) = dp,(x) A dyj(x), as desired. PROPOSITION 7.26
d: A,(X) + A2(X) satisfies
(a) d(a + p) = da + dp for 2, /j E Al(X), (b) d(fa)= d f ~a + f da for .f E A,,(X), a E Al(X), (4 Wf) = 0, f E AO(X).
114
7. HOLONOMIC SYSTEMS
CANONICAL 2-FORM ON T * X In Eq. (5.8) we defined the canonical 1-form 0 E A1(T*X).We now have DEFINITION 7.27
The canonical 2-forrn R on T*X is defined by
=
-do. PROPOSITION 7.28
then R
If (ql, . . . , q”, p l , . . . ,p,) is a natural chart on T * X ,
= dq’ A d p , .
The proof is left as an exercise.
THE MAPPINGS b AND # Let o:V x V - t R be a bilinear mapping. If u E V , let ub E V* be defined by Ub(U’) =
w(u, u’).
If the bilinear mapping w is nondegenerate, then the map u -+ u b is an isomorphism and the inverse V* .+ V is denoted c( + a*. We note that if g: V x V-+ R is a metric, then the corresponding b and # maps are precisely the operations of lowering and raising indices with the metric. LEMMA 7.29 Let e , , . ... , en,u l , . . . , u, be a basis for the 2n-dimensional e’ A u’. space V. Let el, . . . , en, ul, . . . ,un be the dual basis. Let w = Then
(a) w is a nondegenerate 2-form, (b) e j b= uJ, ujb = -&, j = 1, . . . n. 9
PROOF: The matrix of w for the given basis is
This is true because o ( e , ,e j ) = 0 = o ( u i , u j )
for all i, j,
115
HAMILTONIAN AND LAGRANGIAN VECTOR FIELDS
This matrix is nonsingular so (a) holds. To prove (b), write ejbas ejb= rkek S k u k . Then r k = ejb(ek) =
+
w(ej,ek) = 0,
so ejb= uj. Similarly, one sees ujb
= -ej.
We leave the details to the reader.
I Now let T * X be the cotangent bundle of a manifold X . We have the canonical 2-form R on T * X . For z E T * X , we have the map b: T , ( T * X ) -, T T ( T * X ) . The form R is nondegenerate by Lemma 7.29 since, if (qi, p j ) are natural coordinates, we have
i2 = dq' A dp,.
(7.27)
Thus there is an inverse map for b, #: TT(T * X ) + T,( T*X).
Given a vector field
5 on T*X, we have a (tb)(Z)
(7.28)
1-form on T * X defined by
= (((4)b.
(7.29)
Given a 1-form w on T * X , we get a vector field o ' on T*X by defining d ( Z )
(7.30)
= (w(z))'.
HAMILTONIAN AND LAGRANGIAN VECTOR FIELDS DEFINITION 7.30 Let 5 be a vector field on T * X . We say 5 is a Hamiltonian uectorJield if there is a C"'-map H: T * X + R such that
( = (dH)'.
We say H is a Hamiltonian for 5. Note that, if (qi, pi)are natural coordinates on T * X , then, by Lemma 7.29, we have (a/aqi)b = dpi >
(dq')'
= - (ajdp,),
(a/api)b = - d d ,
(dp,)'
=
(7.31)
(a/&').
THEOREM 7.31 Let 5 be a Hamiltonian vector field on T * X with Hamiltonian H . Let (qi,pi) be natural coordinates on T * X . Suppose c(t) =
116
7. HOLONOMIC SYSTEMS
(qi(t),p i ( t ) ) represents an integral curve of
5 in these coordinates. Then
PROOF: The theorem asserts that the local representative of 5 in the given coordinates is
In other words, we are asserting that aH a ( d H ) =-----. api a41
a a q api
aH
THEOREM 7.32 Let M be the configuration space of a conservative, holonomic system. Let H: T * M -+ R be the Hamiltonian of the system. Then the phase space trajectories of the system are the integral curves of the Hamiltonian vector field (dH)? PROOF:
Immediate from Theorems 7.16 and 7.31.
I
REMARK 7.33 The only place in the proof of Theorem 7.31 where we used the assumption that our coordinates (qi, p i ) were natural coordinates was where we wrote = dq'
A
dp,.
If (Qi, Pi) is an arbitrary chart on T * X , then R is a linear combination of dQ' A dQj, dQi A dP,, and dPi A dP,. If we happen to have a coordinate system (Qi, P i ) for which R = dQi A d P i , then just as in the proof of Theorem 7.31, we conclude the integral curves (Qi(t),P j ( t ) ) satisfy Qi = aH/aPi,
Pi = -aH/aQi.
That is, Hamilton's equations hold in the (Qi, Pj) coordinates. This motivates the following definition.
117
HAMILTONIAN AND LAGRANGIAN VECTOR FIELDS
DEFINITION 7.34 A chart (Qi, P j ) on T * X is a canonical chart and Q’, P j are canonical coordinates if
R
=
dQ’ A dP,.
The differential equations
..
Q‘
Pi= -aH/aQ‘,
= 8H/ilPi,
(7.32)
which hold in any canonical coordinate system, are often knownas Hamilton’s canonical equations. There are techniques for constructing canonical coordinates in a given problem so as to make the problem simple. For more information on these methods see [ 2 ] . EXAMPLE 7.35 Consider the system of Example 7.7. We have the coordinate 8 on M and corresponding charts (0, (0, ps) on T M , T * M . O u r Lagrangian, Legendre transformation, and Hamiltonian are given by
e),
L = + m t 2 d 2+ my[ cos 8, H
= ($/2mt2) - mg/
~ ( 0 , s=) (0, mt’d),
cos d.
The canonical 2-form on T * M is given by
R
=
dOAdp,
and we have dH
= mg/
sin 8 d0
+ (pe/m/’)dp,.
The #-operation is defined by [see Eq. (2.31)] dP
= -?/?pH,
dpi
=
8/80,
so
a +Ps a ( d H ) $= -my/ sin 8 ?p, me2 80’ Suppose (e(t),ps(t)) is a phase space trajectory, i.e., an integral curve of (dH)’. Then
4 = pe/mP2,
po = - m g t sin 6.
Of course these are precisely Hamilton’s equations, as required. Now (dH)*E Y h ( T * M ) can be pulled back to T M , by 9, to give 9 * ( d H ) # )E F A ( T M ) . We compute
118
7. HOLONOMIC SYSTEMS
noting that p e = rnt28, so
a - i a ap, me2 ad'
- - ~-
a
a
a8
a8
- ---
so that Z*((dH)')
9 .
= -- sin
e
8
a + 8 -.a a0 a0
If (8(t), e ( t ) ) is a state space trajectory, then d8/dt
=
8,
d8ldt = -(g/d) sin 8.
These are, of course, Lagrange's equations, as obtained in Example 7.7, since the above system of two equations is equivalent to the single second-order equation
8 + ( g / t ) sin 8 = 0. Now we return to our formulation of mechanics on state space. We construct a 2-form on T X and show that the state space trajectories are exactly the integral curves of (dE)', where E: T X -+ R is the total energy function. Suppose L: T X -+ R is the Lagrangian of a conservative holonomic T X + T * X be the Legendre transformation of the system. system. Let 9: DEFINITION 7.36
The Lagrangian 2-form corresponding to L is
QL
=
9*R. We leave the proof that QL is non degenerate as Exercise 7.7. We thus have b and # operations, b: FA( T i ) + F :( T X ) DEFINITION 7.37
with inverse #: F(: T X ) .+ FA(T X ) .
The vector field (dE)# is called the Lagrangian vector
jield for L.
We shall use subscript coordinates to indicate partial derivatives, e.g., L,i = a q a g i , L~~~~ = a2L/aqi a g j . LEMMA 7.38
In a natural chart (q', 4') we have 52,
= L,,,,
dq'
A
dq'
+ L4i4jdq' A dq'.
The proof is left as an exercise. Now our energy function E has differential dE
=
E,i dq'
+ E,,
dq'.
We want to compute (dE)'. If we write (dE)'
= c('(d/dqi)
+ fl(d/dq'),
then
119
HAMILTONIAN A N D LAGRANGIAN VECTOR FIELDS
there is a (2n x 2n)-matrix such that
We want to determine this matrix. Let A = (L4z.ij);A is invertible and, in fact, A = (gij), since L = i g i j q i $ - V(q). Let B = (LqIqj). LEMMA 7.39 The matrix of b is
This says (a) (d/dqJ)b= (L,pqt - L q t q J ) dq' dq'. (b) (d/dq')b = -LqLq,
+
dg',
Lqt4J
The proof is left as an exercise. The matrix inverse of
1 -:1
------+----
is
[
0
i I
A-' A - ' ( B ' - B)A-'
------+-------------
-A-'
I.
We leave it as Exercise 7.10 to show that the above two matrices are indeed inverse to one another. (Compare this with the formula for inverting a (2 x 2)-matrix,
[:,-[='-I:
1
d
.I
-b
.)
So the components of (dE)# for the basis dldq', d/dq' are given as the column
120
7. HOLONOMIC SYSTEMS
This says =
[ T I ]
A
Eqn
which proves the lemma. Now we compute fl. These are the entries in the column
+ A-'(B'-B)A-' Thus, by Lemma 7.40, we see Lqkgjp'
But recall that L,k$
= gkj.
=
-E y k
+ (L@ik
-
L$kdj)qJ.
Then we get
pi = -gikEqk + gik(LqJqk- L,kqj)4'.
(7.33)
Therefore we have (dE)'
4' 7+ ( -gikEqk l3
=
a9
+
. a
yik(L,jqk -
L4kqj)q')@.
Let t = (2,d') be an integral curve of (dE)'. Then d ( t ) = t'(t)and
ci = -gik(?))E,k(4 + gik(4(L,,k
-
L4kYj)?'.
This gives L4r4' . c'
=
+
-(2Lqr4, - Lqr) (Lqj,. - L,,,,)t'
or L 4'4' . ?i
-
L,,
+ LgrqJi.J = 0.
(7.34)
121
TIME- DEPEN DENT SYSTEMS
This can be rewritten as
(7.35) Thus, we may summarize as follows: In a natural coordinate system, the differential equations describing the integral curves of (dE)’ are the first-order equations resulting from Lagrange’s equations by the standard trick of “doubling the number of variables.” We also see: The Lagrangian vector field (dE)‘ is the iizfinitesiinul generutor of the dynamical group of the conservative holonomic system having E : T X + R as total energy function.
TIME-DEPENDENT SYSTEMS Mechanical systems involving time-dependent forces or dissipative forces (producing time-dependent energies) can sometimes be described using a Lagrangian function which depends explicitly on time. Thus we consider
L:R x T X + R . The Lagrange equations corresponding to this Lagrangian are
(7.36) where (qk,q k )is a natural chart on TX and (q’)k= dqk/dt, 1 5 k I n = dim X . The evolution of such a system can also be described by the flow of a vector field on T * X obtained using the symplectic structure of T*X, just as in the time-independent case. However, now the Hamiltonian will depend on time and hence, the vector field will be a time-dependent vector field. For each t we define (a) (b) (c) (d)
.Yr:T X + T * X , A,: T X + R , E,: T X + R , H,: T*X -+ R ,
by holding t fixed in L and proceeding as before. Thus, in local coordinates,
.IP,(q,4 ) ( 4 = (;L/(%k)(f,4, 4)uk, At(q, 4) = y’r(q> Ci)(Y,
4 1 9
E,(q, 4 ) = A , ( % 4) - L(f, q>4, ffr(y, P ) == Et(9’; ‘ ( q ,P ) ) .
122
7. HOLONOMIC SYSTEMS
We assume, as before, that 2, is a diffeomorphism for each t. The phase flow, i.e., the motion of the system in phase space, is constructed as before, using the canonical 2-form Q on T * X and the time-dependent Hamiltonian H ( t , q, p ) = H,(q, p ) . DEFINITION 7.41
The Hamiltonian vector field X , on T * X is defined by
THEOREM 7.42 If q = q(t) is a curve in X which satisfies the Lagrange equations (7.36) and (q(t),p ( t ) ) = gt(q(t),q'(t)),then (q(t),p ( t ) ) is an integral curve of X , on T * X . PROOF: The proof is obtained just as in Theorem 7.32, by replacing H by H , and 2' by Yt.We arrive at
and
When we describe the evolution by a vector field on T X , the situation changes more drastically due to: (a) time-dependent 2-forms QL,, = YTSZ, (b) the appearance of the dissipative vector field Z,, given by Zr(q, 4) = T y ; '((d/dt)(yt(q,
4))).
Because we have a time-dependent 2-form Q,,t, we have a timedependent sharp operation which we denote #*. Then we have DEFINITION 7.43 The time-dependent Lagrangian vector field X , on T X is defined by X,(t, 5 ) = (dE,)3f(t). Then, for each fixed t, we have
T 9 t ( X , ( t , q 7 4)) = X,(t, 2 ' t ( 4 , 4 ) ) . DEFINITION 7.44
Define a time-dependent vector field X , on T X by 4, 4) = x E ( t , q? 4.1
-
zt(q,
4).
THEOREM 7.45 If q = q(t) satisfies the Lagrange equations (7.36), then (q(t),q ( t ) )is an integral curve of the time-dependent vector field X , = X , - Z i on T X .
123
TIME- D EPENDENT SYSTEMS
PROOF: Define
4:R
x T X +. R x T * X by @(t,q, 4 ) = ( t , Ip,(q, q ) ) and let
dt) = 4k q ( 0 7 4(t))Then Now
L'
is an integral curve of X ,
W t , 4>4)(1,4 6 )
= (1,
+ (i3/&)
on R x T * X by Theorem 7.42.
il + d YLq,4 ) )
T Y t ( q , 4)(4",
and
EXAMPLE 7.46 Suppose that a conservative mechanical system is described by L(q, q). We will see in Chapter 15 that the form R, is preserved under the motion. Suppose additional work is done on the system resulting in FTR, = f(t)R, where F,: T X + T X gives the trajectories on T X , f ( 0 ) = 1 and f ( t ) > 0. We then take e(r,q, q ) = f(t)L(q,4), which gives E"(t,q, 4) = f ( t ) E ( q ,4). We also get Ri, = f(t)R, and, since E = f ( t ) E , the Lagrangian vector field XE(r, q, 4) = X,(q, q ) is the original time-independent field. The dissipative vector field is given by Z't(4>4) = ( f " / f P ) ) T Y - ' ( $ ( 4 , 4 ) ) ,
where Y is the Legendre transformation determined by L(q, 4 ) and
{a% 4) + fY(4,4))
a q , 4) =
in
T,,,.,,(T*X).
t=n
Thus f l * ( t , q, q) = X J q , 4) ( f ' ( t ) / f ( r ) ) T Y - ' ( $ ( q4)). , z(t,q, q) = ea'($q2 - f q 2 ) leads to -
For
example,
m,
494) = X d q , 4) = 4(VW - q ( a / W ,
Z,(q, 4) which gives d q / d t
=
=
kj(?/?q),
q, d q / d t
=
-q
8 , = q(a/aq) - ( q + kj)(a/?4), j.4, so ( d 2 q / d t 2 )+ L(dq/dt) + q = 0.
-
EXAMPLE 7.47 Suppose that a conservative mechanical system is described by L(q, q), so the equations are (d/dt)(dL/&jk)= dL/dqk, and external
1 24
7. HOLONOMIC SYSTEMS
+
forces are imposed to give us (d/dt)(dL/dqk)= (dL/dqk) Fk(t) as dynamical equations. Then take L(t,q, q) = L ( t , q, 4) Fk(t)qk. In this case 9 , ( q , 4) = 3 ( q , 4.) which implies that 2, = 0 and RL, = R,. Then E(t, q, q) = E(q, q) F,(t)q", so that X E is time dependent and its flow determines the dynamics because 2, = 0. For example, L(t, q, q ) = f ( q 2 - q 2 )+ f ( t ) q gives 3 , ( q , q ) = 4, SZet = d q A d 4 , 2, = 0,
+
j77L.2 2q
+ t q 2 - f(t)q,
x, = 4(d/%)
- (4 - f ( t ) ) ( d / W
So the dynamics is given by dq/dt = q
and
dq/dt = -q
+f(t)
or simply, (d2q/dt2)+ 4 = .f(t). COMMENT: In Chapter 15 we will describe time-dependent systems in terms of structures on the manifold R x X . Our analysis will, in fact, also include systems which cannot be described using Lagrangian functions.
EXE R CI S ES 7.1 See the section on conservative forces in the text. Let i: M inclusion. Then
i*F"(u,) Show that if F"(u,)
=
E
T,*M
so i*F": T M
+ R3" be
+ T*M.
-dV(x), then i*F"(v,) = - d ( VIM)(x).
7.2 This exercise refers to Example 7.17. Show that if n is the plane in R 3 determined by the initial position and velocity, then the particle always moves in n. (Use vector notation, r = (q', q2, q3), v = ($, q2, q3), and show n is the plane determined by r x v.) 7.3 Show that if a, p are 1-forms on the vector space V , then a A p = a 0 B - B O a. 7.4 Let ei, 1 I i 1 4 , be the standard basis for R4, ei,1 I i I 4, the dual basis. If w : R4 x R4 -+ R is given by w((a', a', a 3 , a4), (b', b2, b3, b4)) = u3b2- u2b3 a2b4 - a4b2, express w as a linear combination of the basis forms e' A 8,i < j .
+
7.5 Prove Proposition 7.28.
125
EXERCISES
7.6 If X is connected and H , , H , are Hamiltonians for the vector field on T * X , then HI and H , differ by a constant. 7.7 Prove that R,, as defined in Definition 7.36, is nondegenerate. 7.8 Prove Lemma 7.38. 7.9 Prove Lemma 7.39. 7.10 Show that if A , B, and C are n x n matrices with A and C invertible, then
7.11 Let SZ be the canonical 2-form on T * X (see Definition 7.27). For functions f:T * X + R and g: T * X + R define the Poisson bracket { f,g } : T * X -+ R by { f , g } = SZ(dfr,dyff). If H is a Hamiltonian function for a vector field ( on T * X show that f is constant along the integral curves of ( if and only if { H , f } = 0 (see also Exercise 2.5). 7.12 Let M be a manifold and L : T M + R be a smooth mapping. For u E T m M define LF(u): T m M + R by P ( u ) ( w )= (d/dt)l,=,L(u + tw), using the fact that T,M is a vector space so that u + tw is well defined. We thus obtain 2':T M + T * M , which is called the Legendre transjormation determined by L . When L = T - V z as considered in the text, LF is a diffeomorphism. However, show that if ( M , g ) is a then ,9 is not a Riemannian manifold and we take L(u) = ,/= diffeomorphism. 0
7.13 Let M be the configuration space of a holonomic system, H : T * M + R the Hamiltonian function, and X , the Hamiltonian vector field. If S: M x R + R is a smooth function, then we have
where t is the coordinate on R , and we have d,S: M x R
+ T*M
defined by holding t fixed. Define a time dependent vector field X , on M by X , = Tz* X , c- d,S. Show that if 0
(7.37)
126
7. HOLONOMIC SYSTEMS
on M x R, then every integral curve (m(t),t ) of X , produces a Hamiltonian trajectory d,S(m(t), t). Equation (7.37) is called the HamiltonJacobi equation. If F = 51 + dH A dt on T * M x R where R is the canonical 2-form on T * M and g: M x R --t T * M x R by g(m, t ) = (d,S(m, t), t), show that the Hamilton-Jacobi equation implies that g*F = 0. Show that the preceding constructions work for time-dependent Hamiltonian functions H: T * M x R -+ R.
8 Tensors
TENSORS O N A VECTOR SPACE Let E be an n-dimensional real vector space. DEFINITION 8.1
A tensor of type (:) on E is a multilinear map t:E x
... x E
Gi2
x E* x . . . x E * + R ;
z z
r is the cooariant degree oft, s is the contravuriunt degree oft, r s is the total degree oft.
+
The set of all tensors of type (:) is a vector space denoted T;(E).We agree that T:(E) = R. EXAMPLES 8.2
(a) There is a canonical isomorphism e: E -+ E**, defined by [e(u)](a) = c( E E*. By definition, we have T k ( E ) = E**. Whenever it is convenient, we can make use of the canonical isomorphism and identify TA(E) with E. (b) A 1-form on E is a member of E* which, in the present notation, is
a(u) for u E E,
TW).
(c) Second-degree covariant tensors, discussed previously, are elements of T;(E).
127
128
8.TENSORS
REMARK: If a E T;;(E),p E T"(E), y E T"(E), then (a 0B) 0y = a 0(P 07).
Because of this, we simply write a 0 j0y. THEOREM 8.4 Let e l , . . . , e, be a basis for E with dual basis e l , . . . , e".
Then the set
{ e i l Q . . . O e i ~ ~ e j , O . . . O e<ji ~k I< ln , 1 5 j l < n } is a basis for T;(E).Thus dim T"E) NOTE: In general a 0p # a 0p = p
=
nr+S.
p 0a. However, if
a E T;(E), j E T",E), then
0a E T"E).
If a E T"E) and e l , . . . ,en is a basis of E, then the components of a for the given basis are the numbers . + . , .r = a(eil,. . . , ei,,ejl, . . . ,e'"). a ! Il 'I. '.. PROPOSITION 8.5 a = a{:::.'teilQ . . . @ eir Q e j l O .. . @ ej,, where the summation convention is in force.
We leave this proof as an exercise. EXAMPLE 8.6 (Kronecker delta) There is a canonical isomorphism Ti@) E L(E, E ) given by a + 4,Oi(u) being that vector in E such that, if w E E*, then w[Oi(u)] = a(u, w). Let { e , } be a basis for E. We have components
aj
= a(ej,ei).
On the other hand, Oi E L(E, E ) has a matrix claim = a;. To see this we must show that
Oifi
Oij such that 2(ej) = 2jei. We
ei(B(ej))= a(ej,ei).
But this is true by definition of 8. So when we have a = ajej Q e i , we have also defined a linear map E -+E whose matrix is the matrix of components (a)). We define the Kronecker delta 6 E T : ( E ) to be the tensor corresponding to the identity map in L(E, E). Given 1 I a I r, 1 I b I s there is a linear map C:: T;(E).+ T;: : ( E ) called contraction on the "a"-covariant index and "6''-contravariant index. Rather than give a general definition of contraction, which involves notational difficulties, we consider some special cases. Consider C:: Ti(E)+ T:(E). Let t E Ti(E). Fix u , E E, a , E E*. There is a bilinear map z : E x E* + R , (u, a) -+ t(u, u l , a,, a). This corresponds to a linear map z*: E + E .
129
TENSOR FIELDS ON MANIFOLDS
We define (Cft)(u,,a l ) = trace(?); Cft is well defined. We claim (C:t)(u,,a , ) is a bilinear function of (u,, al). We verify directly that C: t(u, + u',, al)= Ctt(ul, a ; ) + C:t(u;, al). Let
+ u;, a ; , a),
T;(U, a ) = t(l', u1 t2(u,
a ) = r(u, u,,
T3(U,
a ) = r(u, u;, a,, a).
+
@I,
a),
+ +
+
Then z1 = t 2 t,, so ?, = t 2 ?,. Therefore, trace(?,) = trace(?,) trace(?,). This shows Cff(c, (11, a , ) = Cft(u,, a,) Cit(u;, a l ) . Let e l , . . . , e n be a basis for E, e', . . . , e n the dual basis. We have C:t(u,, a , ) = t ( e i ,u,, a l , ei), where the summation convention is used. For the bilinear map t:(u, a ) -+ r(u, I ] , , a , , a) corresponds to a linear map ? whose trace is the sum of its diagonal elements. Then trace(?) = ei(?(ei))= t(ei, ei), so ~ f t ,(a l~) = z ( e i ,e') = t ( e i ,u a ,, ei).
+
,
COMPONENT DESCRIPTION:C f t E T : ( E ) , so Cft has components (C21 t)ji
-
-
Cft(ej, e') = t(e,, ej, ei, ek) = ti: ( k is summed). Similarly, if t E T:(E), then C:t(al, a23 ~
3 =)
t(ei, a,,
a29
ei,
that is, 3 i j k - ijrk
(C,r) - t r .
TENSOR FIELDS ON MANIFOLDS DEFINITION 8.7 A tensorjeld of type (i)on a manifold X is an assignment to each x E X of a tensor T(x)E Ts(T,X). We require the following smoothness condition: Let ( U , 4 = (XI,. . . ,x")) be a chart. Define T j ; ;:;, U+Rby
We require these component functions be C"' for every choice of coordinates. The set of all (;)-tensor fields on X is denoted F ; ( X ) . Consider a tensor field of type (i).Let T
=
0
Ttj dx' 0 dx' 0__ ?.uk
in (XI,. . . ,x")-coordinates,
130
8. TENSORS
T
=
a
Tfj d i i 0df' 0a.wk
in (TI, . . . , ?')-coordinates.
How are the T!j related to the Tfj? We have
For a tensor field of type
(s)
we have (see Exercise 8.6)
REMARK 8.8 In classical differential geometry, an (:)-tensor field was an assignment, to each coordinate system, of component functions Ti::::{; such that the transformation law (8.2) holds. Multilinear mappings were never mentioned. But the whole reason for requiring the transformation law (8.2) is as follows: Let (ufl,), . . . , (of,,), (ail'),. . . , (a?)) be r contravariant and s covariant vectors. We want the expression
T{; ;;{ ,'
,. . .
yir
a(,') . . .
(r)
~1
@.is
(S)
to be a scalar, i.e., independent of coordinates. That is exactly what the transformation law (8.2) guarantees. Suppose T is an (:)-tensor field. If ui E Ft(X),d E F y ( X ) for 1 I i < r, 1I j I s, we have T(u,,. . . , t i r , a', . . . , 2): X -+R, given by x
.+
T(x)(ul(x), . . . , ur(x),a'(x), . . . , as(x)).
-
Thus a tensor field T E F ; ( X ) defines a map, also denoted T , T:Y h ( X ) x . . . x Y h ( X ) x Y y ( X ) x . . . x Y y ( X ) -+ F ( X ) v r factors
s factors
where P ( X ) = C"(X, R). This map is multilinear over F ( X ) , i.e., it is additive in each variable separately and satisfies T ( u , ,. . . , f ~ , ., . . , u,, a ' , . . . , CC')= f T ( u , , . . . , u,, a', . . . , a')
for f
E
P(X).
131
TENSOR FIELDS ON MANIFOLDS
-
THEOREM 8.9 If r-factors
s-factor5
b
T:Y A ( X ) x . . .
x $Y(X) x
x
. . . x Y ? ( X )4 F ( X )
is multilinear over F(X), then T arises from a unique (;)-tensor field as above. PROOF To simplify the notation, we give the proof in the case r So we have T : Y A ( X )x Y A ( X ) x Y Y ( X ) + F(X).
=
2, s
=
1.
Let xo E X , u l , u 2 E T,,X, a E T$X. There exist vector fields t l , t2on X and a 1-form q on X with tl(x0) = u l , t2(x)= t i 2 , q(x0)= a. Define T(XO)(U,? L i z ,
co = T(51>52.v ) ( x o ) .
We need to show T(x,)(u,, u2, a ) does not depend on tl, t2,v. For example, if we replace 4 , by another vector field 5, with f,(xo) = u l , we must show
TtP,, 5 2 , v)(xo) = T(51, 5 2 , v ) ( x o )
or, T(5l - F l ? 5 2 , v)(xo) = 0.
So we show if 5(xo) = 0, then T(5,t 2 ,q)(xo)= 0. Let ( U , (ql, . . . , 4")) be a chart about xo. Let y:X + R be C" with g = 0 on X - C, C c U compact, and 61 = 1 on some neighborhood of xo, say I/. Let 5 = l1a/dq'. Then y2( = y t ' g (7/?4' = g('O,, where the vector field 8, = g d/dql is defined on all of X . Then s2T(5,5 2 , v ) = T(q'5, 5 2 ,
v ) = YS1T(Q,, 5 2 , v).
FIGURE 8.1
132
8.TENSORS
Now g(xo) = 1, (‘(x0) = 0 for all i so we get
T(t, 5 2 , v M x O )
=
T ( g 2 t >5 2 7 v ) ( x ~= ) (gt’)(Xo)T(ei,5 2 9 v ) ( x O )
= 0-
Similarly, T( 0 such that ( - E , E ) x U c 9(X). Then for It1 < E we have a,: U .+ a,(U), a diffeomorphism. So a : Y ( p ) T,M ~ and t -+ a:Y(p) is a Cm-curve in T,M whose derivative at t = 0 is LxY(p). More generally, DEFINITION 8.1 2 If X E F h ( M ) and T E Fz(M), the Lie derivative of T with respect to X is
LxT(P) = ( 4 4It :=.0
VP).
So L,T(p) is obtained as the derivative of a curve in T;(T,M). Here (g*T)(p)(v,,. . . , v,, wl,. . . , U S ) =
T(a,(p))(Ta,(u,),. . . , Ta,(v,), w1 0 Ta-,, . . . , os0 Ta-,).
We now give some useful formulas for computing Lie derivatives. Further results are in the exercises (see Exercise 8.9). CONVENTION 8.13 It is convenient to use the “comma notation” for partial derivatives. According to this, we write aj’/axJas l jand a2Ti/axkaxe
133
THE LIE DERIVATIVE
as T:,lk.Similarly, if f is a function of certain variables including t , we might write f,, for df/dt. We now have the basic formulas for Lie derivative of a function, a vector field, and a 1-form.
PROOF: Let M : 9 ( X ) + M be the flow of X . Given p E U , choose E > 0 and an open set W c U about p such that ( - E , E ) x W c 9 ( X ) and a((-&, E ) x W )c U . We write a on W x ( - E , F.) as a(t, q) = (a'(t, q), . . . , ~ " ( 4)). t , We write a,(q) = a(t, 4). We also note that ai,,(O,q) = X'(q). Now consider (a).
-bf(P)= (~/dt)Jt=o(MjEf)(P) = (Wt)l,=,f(a(t, PI) = .fi(p)~i.t(O~ P) = (.fiXi)(P).
so c ( ' , ~ ~ (pO) ,= 0
and
ai.AO, p ) = bij.
Thus,
(L,Y(p))' =
- X',,(p)Y'(p)
+ b"Yj,J(p)X~(p)
134
8.TENSORS
or
PROPOSITION 8.15
Let X
E
Fh(M). Then:
L,[Y + Z ] = L,Y + L,Z, for Y , Z E FA(M), Lx(ol w 2 ) = L,w, L,w,, for wl, w 2 E Fy(M), L,( f Y ) = fL,Y + (L, f ) Y , for f E 9(M), Y EF : ( M ) , L,( fw) = fLxw + (L,f ) w , for f~ B ( M ) , w E F:(M), L , [ w ( Y ) ] = L,w(Y) + w(L,Y), for Y E F;(M), w E Fy(M). We leave this proof as an exercise. (a) (b) (c) (d) (e)
+
+
Moving Frame Description Let e l , . . . ,e, be a basis for T,M. For It1 < E we have a diffeomorphism a,. Let ei(t)= Tpat(ei). Since Tpat:T p M + Tar(p,M is an isomorphism,
(el(&. . . , e,(t)) is a basis for Tat(p)M. So if c: (- E, E ) + M is the integral curve of X given by c(t) = at@), then el(t),, , . , e,(t) is a basis for T,,,,M. The functions el(t), . . . , e,(t) form what is called a moving frame along c.
Suppose Y is a vector field and Y(c(t))= Yi(t)ei(t). Then, a : ~ ( p= ) ( ~ , a , ) - ' ~ ( c (=t )( )~ ~ a , ) - ~ ( Y ' ( t )= e , Yi(t)ei, (t))
so (d/dt)l,=,cr:Y(p)= Yi(0)ei.We have proved
135
THE BRACKET OF VECTOR FIELDS
PROPOSITION 8.16
LxY(p) = Yi(0)ei.
If c is the integral curve of X through p and v E T p M , then the vector T,cc,(v) E T,,,,M is said to be obtained from v by Lie transport along c.
THE BRACKET OF VECTOR FIELDS The special case of Lie derivative, L x Y , is worth looking at in more detail. DEFINITION 8.17 If X , Y E $,!,(M), the Lie bracket is [ X , Y ] E T , ! , ( M ) defined by [ X , Y ] = L,Y. PROPOSITION 8.1 8
(a) (b) (c) (d)
The bracket satisfies
[ X , X I = 0, [ X , Y ] = - [Y , X I , [ X , aY b Z ] = a [ X , Y ] b [ X , Z ] for a, b E R, [ X , [ Y , Z ] ] + [ Z , [ X , Y l l + [Y, [z,X I ] = 0.
+
+
PROOF: The local formula for [ X , Y ] is
[ X , Y ] i = Y’ 0 such that for It1 < E ~ (sI , < z2 we have
E
M . Assume
there exist
dt, B(s, P I ) = B(s, 4 t , PI). Then [ X , Y ] ( p )= 0. PROOF: Differentiate the assumed equation with respect to s at s = 0. We
get TP%(Y(P))= Y ( a 4 ) .
So a:Y(p)
= (T,,cc,)-'Y(a,(p)) =
Y ( p )and hence,
[ X , Y X P ) = L,Y(P)
= (d/dt)l,=,Y(d = 0.
I
LEMMA 8.26 Assume [ X , Y ] = 0 in some open set U . Suppose E ' ,
0, p
E
U are such that both
44 P(s, P I ) are defined and lie in U , for It/ < E 4
t
3 B(s3
PROOF: Let t be fixed with It1
P(s, @(t> P))
and ~ Is1 ,
<E
~ Then .
P I ) = P(s, 44 PI).
< E , . Let
E~
>
137
VECTOR FIELDS AS DIFFERENTIAL OPERATORS
c2 is an integral curve of Y with initial value cc(t,p). We claim c1 is too. That will prove our result. c;(s) = 7',~(s,p)c(,(Y(p(s,p))). We claim this equals Y ( 4 4 B(s, PI)). We prove
Y(B(s,P I )
= (7',,,,,,,%-
Y Y ( 4 4 P(s, P ) ) ) ) .
(*I
(d/dt)(~p,,*,,~,)- Y ( d L P(s, P ) ) ) ) = ( T p ( s , p )'(L*Y(a(t, w P(s, P I ) ) ) = 0
since L,Y = [ X , Y ] = 0. So the right side of (*) is independent oft. The left equals the right at t = 0. I
VECTOR FIELDS AS DIFFERENTIAL OPERATORS Given X E Fh(M) and f' E F(M), we can form a new scalar d f ( X )E F(M). If we fix X , this defines an operator, also denoted by X ,
X : F ( M )+ 3 ( M )
by
X ( j )= df(X).
As is often done with linear operators, we sometimes write simply X f . Since X f ( m ) = (dW01, = o f ( c ( t ) ) , where c is any curve with c'(0) = X(m), we see that X f is the directional derivative off in the direction of X . In case X = i?/dx' is a coordinate vector field in some chart, 4 = (XI,. . . , Y"), we get
again justifying the (d/dx')-notation. If X and Y are vector fields, consider X , Y and [ X , Y] as operators. Then we have THEOREM 8.27
[X, Y ] f
=
X ( Y f )- Y ( X f ) .
PROOF: Choose coordinates and write X = X id/Jx', Y = Y' c?/2xi. Then
Yf
= df( Y ) = f j Y J ,so
that
X ( Y f ) = ( f j Y j )kXk = f j k Y J X k+ .fjYJ,,Xk. Interchanging X and Y subtracting, we get
X ( Y f ) - Y ( X f ) = Y',,XYj - X ' , Y y j . But by Proposition 8.14, we see
[ X , Y ] j '= f j ( Y',kXk- X',,Yk), which completes the proof.
I
138
8.TENSORS
REMARK: Theorem 8.27 shows that, in terms of operators, bracket is the usual commutator.
EXER C IS ES 8.1 Show the validity of the remark following Definition (8.3). 8.2 Prove Proposition (8.5). 8.3 How are the components of a @ /? related to those of a and /?? How about a + /?? If c is constant, how are the components of ca related to those of a?
8.4 Find the components of the Kronecker delta 6 for any basis {ei} of E . Show that if t E T:(E) and the components of t are the same in every basis, then t = a6 for some a E R . 8.5 (a) (b) (c) (d)
If t = tjej@ ei, find Ctt. Find Cis where 6 is the Kronecker delta. If t = tfFei @ ej @ ek @ e, @ em, find an expression for CfCft. If u l , . . . ,u, E E , a l , . . . ,a, E E*, then t = a, 0 . . . 0a, 0u1 0 . . . @ us E T;(E).If 1 I iI r, 1 I j s s, show that
ai(uj)al * * . @ &,@. . . @ a, @ el @ . . . @ Zj 8 . . . @ e, (where over a term means to delete that term). For example,
Cit
=
"^"
C:(a @ u) = a(u).
(s)
has components T{::.':j;,Tt::::!;in two coor8.6 If a tensor of type dinate systems (xi),(Xi), then
8.7 Show that if the assignment of components to each coordinate system is made in such a way that (8.2) holds, then the components are the components of an (;)-tensor field.
8.8 Prove proposition 8.15.
8.9 Let X be a vector field on a manifold M and p be an element of M . Then: (a) If e l , . . . , e, is a basis for T,M and c is the integral curve of X through p , we have the moving frame ei(t), . . . ,e,(t) as in the text. Then { e i l ( t )@ . . . @ eir(t)
ej,(t)
. . . @ ejJt)l 1 I i, I n, I
I j, I n}
139
EXERCISES
is a basis for Ts(T,,,,M). If T E T"M) let T(c(t))= Ti,!;.';!;(t)eil(t) 0 . . * 0ej,(t).
Then show that
(b) If T E FS(M), T' e Fs:(M), then L,(T 0 T') = (L,T) 0 T' + T 0L,T'. A special case is when s = r = 0, i.e., T = f E F ( M ) . Then T 0 T' is just f T ' and L,(,fT') = (L,f)T' + fL,(T'); (c) Show that Lx6 = 0, where 6 E F : ( M ) is the Kronecker delta tensor; (d) Let TEF;(M), X I , . . . ,X , E F;(M). Then T ( X , , . . . , X,)E F(M)and L,[T(Xl,. .
. ?
Xr)] = ( L X T ) ( X l , . ., X r ) r
+ 1 T ( X , , . . . , L x X i , . . . ,X,); i= 1
(e) L , commutes with contractions, i.e., if T E Fs(M), 1 I a I r, 1 I b I s, then L,(c:T) = C:L,T.
8.10 Prove Lemma 8.24. 8.11 Suppose X , Y E Fh(M) with [ X , Y ] = 0 on M . Suppose a, P are the flows and that we have t, s E R, p E M for which P(S7
PI)
and
P(s,
44 P))
exist. Need they be equal? 8.12 With notation as in the previous exercise suppose P(s, a(t, p ) ) exists. Is it necessarily true that a(t, P(s, p ) ) exists?
8.13 Let X be the vector field on R Z defined by X ( a , b) = (-b, a). At (1,O) E R 2 we have the basis el = (1,0), e2 = (0,1) for T(l,o)R2.Let c(t) be the integral curve of X with c(0)= (1,O). Calculate the moving frame (el(t),e 2 ( t ) )along c obtained from e l , e2 by Lie transport. If Y(a,b) = (a + b, - l), compute L,Y(l, 0). If Z(a, b) = (a, b), compute L,Z. 8.14 Suppose X is a vector field on S 2 , given in spherical coordinates by x(0,4) = 4 8/80. Let Y(0,4) = (ape) + C$(d/8C$).Compute L,Y(0, 4).
8.15 Let
(XI,.
. . ,x") be a coordinate system on a manifold [d/dxi,a/axq
=0
for
1 I i, j I n.
M . Show that
140
8. TENSORS
8.16 (a) Suppose that f :M + N is a diffeomorphism and that X and Yare vector fields on M. If (V, 4) is a coordinate chart on N then ( j - ' (V ) , 4 o f ) is a coordinate chart on M . Use this fact to give a simple proof that [f,X,f,Y] = f,([X, Y]). (b) Suppose that f :M + N is a smooth mapping, that X, and X, are vector fields on M , and that Yl and Y, are vector fields on N satisfying
Tmf(Xi(rn))= X ( f ( r n ) ) Show that T,f(CX,, XZl(4) = CYI,
Vm E M *
Yzl(f(rn)).
8.17 (Tensor bundles) Let M be a smooth manifold. Define T;W) =
u
CUrnM)
meM
and define the projection
z;:T"M)
+M
in the obvious way. Let ( U , 4) be a chart on M . Then, for each rn in U we have the linear isomorphism dm4:T,M + R". Define (T4);:(z;)-' ( U )
+
4(U) x T m " )
by (T4)Xt)= (&Xt)), t o ((drn4)-' x . * . x ( d n d ) - ' x ( d m 4 ) * x . . . x (d&)*).
Since T;(R")2 ' Rk ( k = dr+')),we may use the collection of all such mappings as charts in a differential structure. Show that the topology for which all maps (T4); are homeomorphisms is a uniquely determined, second countable Hausdorff topology. Then show that these charts form a smooth atlas and hence determine a differential structure on T"(M). Show that, with this differential structure, r;: T ; ( M )+ M is a smooth vector bundle. Show that a smooth tensor field of type 0 on M is precisely a smooth cross section of this bundle (which is called the bundle of tensors of type on M ) .
(s)
9 Differentia I Forms
EXTERIOR FORMS ON A VECTOR SPACE Let E be an n-dimensional real vector space. An exterior form of degree p on E (or briefly a p-form on E ) is a tensor w E T;(E) which is skew-symmetric, i.e., interchanging any pair of variables changes the sign of the form. DEFINITION 9.1
NOTATION:Denote by A,(E) the vector space of all p-forms on E . Ao(E) is defined to be R . Thus A,(E) is a linear subspace of T:(E). If A: E + F is linear, then A*w = w (Ax . . . xA) is in A J E ) when w is in A,(F). Let s k be the set of all permutations on the set {1,2,. . . , k } : If o E s k define 6 : E k + Ek, where Ek = E x . . . x E (k-factors), by 8 ( u l , . . . , uk) = 0
...
(uu(l)?
7
uu(k)).
LEMMA 9.2
= z"
0
$.
. . , Uk)=Z"(w1,.. . , wk) where W i = U u ( i ) ; z"(W1,. . . , w,)= . . . , w,(k)).so N'qj) = U u ( , ( j ) ) and so fo8(U,,. . . , u k ) = ( T o z ( u l , . . . , u k ) . I
PROOF:
(W,(i),
If o, z E S k , then
Z"O$(u,,.
DEFINITION 9.3
If T E T f ( E ) ,o E s k , define oT E T@) by oT
=
T 0 8.
From Lemma 9.2 it immediately follows that o(rT)= (a " z)T. DEFINITION 9.4
For T E T f ( E )define d T
(9.1) E Ak(E)by
(antisymmetrization operator). Recall that
sgn(o) =
1, - 1,
o is an even permutation, o is an odd permutation. 141
142
9. DIFFERENTIAL FORMS
PROPOSITION 9.5
T E T f ( E )is in &(E) if and only if, for all a E S,, aT = sgn(a)T.
Proof is left as an exercise. PROPOSITION 9.6 (a) If T E T f ( E )then d T E I\,@), (b) If T E Ak(E)then d T = T , (c) If T E T f ( E )then d ( d T ) = d T . PROOF: (a) Let
7
E
sk.
Then
As a runs over Sk so does 7 a, so 0
1 sgn(z
0
O)(T
0
a)T =
arSr
Hence z d T (b) d T
= sgn(z)dT as = (l/k!)
1sgn(p)pT. PEsk
asserted.
c sgn(a)aT
= (l/k!)
c sgn(a) sgn(a)T
a€Sk
UfSk
= (l/k!) k!T = T.
(c) is immediate from (a) and (b) DEFINITION 9.7
Let
(1) E
&(E),
v] E
I AAE). Define 0 A v]
E Ak+!(E)
by
LEMMA 9.8 (a) A : A,(E) x A,(E) + I\k+/(E) is bilinear. (b) If i: E +F is linear and c1 E I\k(F),j? E A#), then A*(a A j?) = A*ct A A*@. If 0 E Ak(E) and v] E &(E), then 0 A v] = (- l)kdv]A w. (C) PROOF: (a) Left as an exercise. (b) We need (k t)!d ( A * c t 0 i*j?) = i*(ca.sk+d sgn(o)a(cc O j?)). But
+
143
EXTERIOR FORMS ON A VECTOR SPACE
while (k + t)!d(A*a0
A*/$
c
=
sgn(o)o(l*a 0 A*@,
U€Sk+/
so (b) holds. (c) Let p E sk+fbe given by
t. We see easily sgn(p) = (- l ) k Land q
So q A w = (- l ) k ' A~ q as asserted.
@ o = p(m @
q). Therefore
I
PROPOSITION 9.9 Let w E T:(E), 0 q ) = d ( o0 d q ) .
y~ E
T;(E). Then &'((do) 0 q)=
d(O
PROOF: d o = (l/k!)
For
k
OEESk,
let
curSk sgn(a)(oo)so we have
oESk+,
act as o on ( 1 , . . . , k } while leaving k
+ t fixed. Let H c sk+fconsist of all such o. Then
Then we have
For each a E H ,
+ 1,. . . ,
144
9. DIFFERENTIAL FORMS
So summing over c1 E H gives k ! d ( o Q yl) and hence, multiplying by Ilk! leaves d ( o Q yl), as desired. The proof of the second equation is similar. I PROPOSITION 9.1 0
( k + d + m)! 4 k!&!m!
( C O A ~ ) = A ~ O A ( ~ A = ~ )
0
0 4 0 Q,
where o,yl, 8 are forms of degree k, d , m. PROOF:
+ + 0 0) + ( k + d + m)!( k + d)! 4 d B ( W 0 I?) 0 6 ) ( k + L)!rn! k!d! ( k + L + m)! d ( m 0 q 0 0). ( k d m)! ( k &)!m! d ( ( w A yl)
( o A ~ ) A=~
k!L!m!
Similar calculation gives o A (yl A 0). PROPOSITION 9.1 1 o1
A ' .
I
Let a',. . . , orE Al(E). Then
. AW'
= r!d(o'
0 . . . Q or)
The proof is left as an exercise. THEOREM 9.12 Let e l , . . . , e, be a basis for E with dual basis e l , . . . , e". Then {e'l A . . . A eikJ 1 I d, < i, < . . . < i, I n>is a basis for Ak(E).There= (:). fore, dim PROOF: Let o E Ak(E).We write
o = oil . . ',eil0. . . @ eik. ,
Then
Now in this sum we do not have only terms with i , < * . * < i,. But, if i,, . . . , i, are not all different, then eil A . . . A elk = 0. Thus we may sum only over
EXTERIOR FORMS
145
ON A VECTOR SPACE
i,, . . . , i, different. But then, by changing the order of the factors, ascending order may be achieved, so that our proposed basis does indeed span Ak(E). Suppose we have a l i l ...ir,eilA . . . A eik= 0 (recall that this means sum over i, < . . . < ik). Choose a particular set ,jl, . . . , j , with 1 Ij , < . . . < j , < n. Then we have 0 = a l i l ..ik,eil . A . . . A eik(cjl,. . . ,ejk).Since j , < . . . < j k there is only one choice of i,, . . . , i, which gives a nonzero value, namely i, = jl, . . . , ik = j,. But ejl A . . . A ejk(cjl, . . . , ejk)= 1, so the sum above reduces to just a j , .j , . This proves linear independence. [ ,,
PROPOSITION 9.13 Let w ' , . . . , o k ~ A 1 ( E )Then . { w ' , . . . , w k } is a linearly independent set if and only if o1A . . . A wk # 0. PROOF: If { w ' , . . . , w k } is linearly independent, choose a', . . . ,an-, E A , ( E ) such that wl,. . . ,uk,a ' , . . . , is a basis. Then w' A . . . A wk is a member of the basis for Ak(E) given in Theorem 9.12 so w' A . . . A cok # 0. Conversely, if { w ' , . . . , w k } is a dependent set, then one vector in the set is a linear combination of the others. Suppose, without loss of generality, that 0 ' = @2(02
+
+ akak.
' ' '
Then w'
A..
. A wk = (a2w2
+ . . . + akwk)
A
= M2W2 A m 2 A W 3 A
' ' '
A Wk
+
+ (*I2 Lo3 + a k W k A 0 2 A . . . A w =k 0. M4W4 A
A
. . A wk
w2 A . A
A
'
.
'
U3W3 A 0 'A' A Wk
+
'
' A Uk
' ' '
[
NOTE: dim A,(E) = 1 since (i) = 1. A basis for A,(E) is {el A . . . ~ e " } , where e l , . . . , en is any basis for E*. DEFINITION 9.14
ul,.
A k-form w on E is decomposable if there exist w = v' A . . . A uk.
. . , vk E E* such that
REMARK: Exercise 9.5 says if dim E decomposable.
=
n, then every ( n - 1)-form is
PROPOSITION 9.1 5 Let o E A,(E), u l , . . . , u, a basis for E . Let wj = afiui.Then w(w,,. . . , w,) = det(aj)w(u,, . . . , u,). PROOF: Let u l , . . . , v" be the dual basis for u l , , . . , u,. There is a i. E R such that w = la1A . . . A u". Now u1 A . . . A u"(ul, . . . , 0,) = 1 so w(u,, . . . ,
146
u,)
= I.
9. DIFFERENTIAL FORMS
NOW w(W1,.
. . , W,)
= IU'
A.
. . A U n ( W 1 , . . . , W,)
1 (sgn
=A
0),$1)a$2)
. . . ax(,) = I det(aj).
assv
I
The fact that dim A,(E) = 1 is quite important as it lies at the heart of the concept of determinant. DEFINITION 9.1 6 Let I : E unique number a such that
3
E be linear. If w E A,(E) is not 0, there is a I*w
We dejine det(I) to be this
= am.
cl.
NOTE: Ifcw is any other n-form then A*(cw) = cA*w = cclw = acw, so det(I) does not depend on the choice of w. Let el, . . . , en be a basis for E . If I: E + E is linear, then A has matrix, for the basis e l , . . . ,en, (aj), where A(ej) = a:ei.
Then we have PROPOSITION 9.17
det(I) = det(a:).
PROOF: Let 0 # w E A,@). Then
w(Ie,, . . . , Ie,)
= det(I)w(e,,
. . . , en).
But Proposition 9.15 shows w(Ae,, . . . , Ie,) = det(aj)w(e,, . . . ,en). Since w(el, . . . , en) # 0, our result is proved.
1
ORIENTATION OF VECTOR SPACES Let dim E
=
n in what follows.
DEFINITION 9.18 Let ( u l , . . . , u,), ( w l , . . . , w,) be ordered bases for E . We say these bases are similarly oriented if wj = a;uiwith det(a;) > 0. If the two bases are similarly oriented, we write ( u l , . . . , u,) (w,,. . . , w,,).
-
We leave it as an exercise (Exercise 9.7) to show that relation on the set of all ordered bases of E .
- is an equivalence
147
ORIENTATION OF VECTOR SPACES
FIGURE 9.1
DEFINITION 9.19 An orientation of E is an equivalence class of ordered bases. An oriented vector space is a vector space together with a choice of orientation. REMARKS 9.20 If p is an orientation on E, the pair ( E , p ) is an oriented vector space. If ( u l , . . . , u,) is a basis in p , we say ( v l , . . . ,v,) is positively oriented (with respect to p). Note that each vector space has exactly two orientations.
I nt u itive Discussion In Fig. 9.1, we see that ( e l ,e,) ( u ~ u,), , since ( e l ,e,) gives the counterclockwise orientation to the plane while (ul, u,) gives the clockwise. Suppose we want to continuously change ( e l ,e,) into (vl, u,), maintaining a basis at each stage. Intuitively we cannot do this because, in order to change from counterclockwise sense of rotation to clockwise sense, at some stage the first vector must pass through the line given by the second in order to “get to the other side.” This idea is captured by our definition in terms of determinants. For let +
+ afe,, u2 = aie, + ate,,
u1 = a i e ,
a: a:
< 0, a: < 0, < 0, a t > 0.
So det(af) = < 0. Now if our deformation is given by (fl(~),f2(~)), 0 I7 5 1, A(0) = e i , A(1) = ui let
L{T)= a;{T)ei.
148
9. DIFFERENTIAL FORMS
Then
Fo
1 0
(a;@)) =
(a$)) = (txj).
Now det(a;(z)) must be 0 for some z E (0, 1) since it is positive for z = 0, negative for z = 1. So there is a z where fl(z) and f2(z) are dependent, i.e., lie along the same line. On the other hand, if (ul, u2) ( w l , w2), then according to Exercise 9.8 there is a continuous matrix function (a;(z)) with (a;(O))= Identity and a:{l)ui = wj. Thus, if fj(z) = aj(z)ui, then (fl(z),f2(z)) is a basis for R 2 and J(0)= u i , A(1) = w i . Note that what Exercise 9.8 really proves is that GL'(2, R ) is path-connected, where GL'(2, R ) is the group of all invertible (2 x 2) real matrices which have positive determinant. More generally, we can argue that GL+(n,R ) is path-connected, so that the result generalizes to arbitrary dimensions. Thus, to orient a vector space, we must say which bases (ordered bases, of course) are positively oriented. We now give an alternative formulation in terms of n-forms.
-
DEFINITION 9.21
If o,v] are nonzero n-forms on E, write o
Iv] for some I > 0.
-
v]
if o =
This divides the nonzero n-forms into two equivalence classes. Let (ul,. . . , u,), ( w l , . . . , w,) be bases with duals ( u l , . . . , u"), (w', . . . , w"). LEMMA9.22
(ul , . . . , un)-(wl
, . . . , w,) if and only if
~ A . . . A U " -
w l A . . . A Wn.
Assume wj = a$, with det(aj) > 0. Suppose w1 A . . . A w" = show A > O . We have W ~ A . . . A W ~ ( W ~ , . . . , W ,1,) = Iu' A . . .A un(wl,. . . ,w,) = 1 det(a$d A . . . A U " ( U ~ ,. . .,u,) = I det(a$. Hence we see I det(a$ = 1, so A > 0. Conversely, assuming A > 0, the above calculation shows det(txj) > 0. I PROOF:
I ~ A . * . A u " We .
This shows that specification of an equivalence class of bases determines uniquely an equivalence class of n-forms and conversely. Note that, given an equivalence class of n-forms, the equivalence class of bases is as follows: Pick an n-form o in the given equivalence class. A basis ( u l , . . . u,) is in the corresponding equivalence class if and only if o ( u l , . . . ,u,) > 0. If ( E , p) is an oriented vector space and ( u l , . . . , u,) is a positively oriented basis, an n-form u is called positiuely oriented if u(ul,. . . ,u,) > 0. DEFINITION 9.23 The standard orientation of R" is that determined by (el, . . . , en), the standard basis, or, equivalently, by el A * . . A en.
149
V O L U M E ELEMENT OF A METRIC
VOLUME ELEMENT OF A METRIC Let ( E , p) be an oriented vector space; p determines an equivalence class of nonzero n-forms. Suppose y: E x E + R is a metric on E . We claim g picks out a specific positively oriented n-form. Let (el,. . . , en)be a positively oriented orthonormal basis. Such bases do exist. Define w = el
A'.
.Aen.
w is a positively oriented n-form. Suppose (ul,
. . . , u,) is another positively
oriented orthonormal basis. We claim u1
. A U"
= e A . . .A
en,
Let u j = aiei. The matrices A = (.y(uj, u k ) )and B = (g(ej, ek))are equal, since both bases are orthonormal, (see remark below) d u j , uli) = aj&Mei, e,)
or, if C = (aj), A = C'BC.
Taking determinants we see (det(C))2= 1, so det(C) = f 1. But det(aj) > 0 so det(a;) = 1. Hence we see
u"
u l A .A. .
= det(cc'Je' A .
. . A en = e1 A . . . A en
as desired. REMARK: Actually, the statement A = B is not quite correct. Both matrices are diagonal with f l diagonal entries and both have the same number of 1's. But their placement on the diagonal may be different. However, det A = det B # 0.
+
DEFINITION 9.24 The volume element of the metric with respect to the given orientation is e1 A . . . A en, where (el, . . . , en)is any positively oriented orthonormal basis. Let ( u l , . . . , u,) be any positively oriented basis (not assumed to be orthonormal). Write u, = &;e,. Then y(v,, u k ) = a;a$g(e,, el) or, in matrix form, A = C'BC,
where A = (g(vj, vJ), B = (g(e,, eJ), C = (a:). Then det(A) = det(B)(det(C))' so = det(C), where we have used the facts, det(B) = f 1, det(C) > 0. Let g denote det(yJk)= det(y(u,, ok)). Thus, given a basis, g represents the determinant of the matrix of components of the metric. This notation may
Jm
150
9. DIFFERENTIAL FORMS
seem confusing but it is commonly used and is, in fact, convenient in calculations. The above calculations then give
4 = det(c$). Now by Proposition 9.15, we see that, if u l , . . . , u, is any basis with dual then det(clj)u' A . . . A u" = e' A . . . A en. For basis u', . . . , u", and if uj = let Iu' A . . A U" = e' A . * A en. Then 9
9
1 = iv' since e'
A.
. . A u"(ul, . . . ,u,)
A.
= el A .
. . A e"(ul, . . , , u,)
= det(olj)
. . A e"(e,, . . . , en) = 1. We have shown
PROPOSITION 9.25 Let ( u l , . . . ,u,) be a positively oriented basis for E where E has a metric g. Then the metric volume element is A . . . A u", where g = det(g(u,, uj)). If (u,, . . . , u,) is not positively oriented, then ( u 2 , u l , u 3 , . . . , u,) is, so the metric volume element is
mu'
mu2
A Y' A U 3 A ' . ' A U" =
-mu'
A.
. ' A 0".
Here note that, although the matrix of g on ( u 2 , u l , u 3 , , . . , u,) is different from the matrix on (ul, u z , . . . ,u,), the two have the same determinant. So we may state PROPOSITION 9.26 If E is an oriented vector space with a metric, then the metric volume element is f ~ U ' A . ' ' A U " ,
where u , , . . . , u, is any basis, g = det(g(u,, u j ) ) , and positively oriented, "-"otherwise. NOTES 9.27
"+"
(a) For a Riemannian metric, g > 0, so
is for ( u l , . . . , u,)
m is replaced by
(b) For a Lorentz metric, g < 0, so we write the Lorentz volume form as &U'
A.
. . A U".
DIFFERENTIAL FORMS ON A MANIFOLD On a manifold X , all of the preceeding algebraic constructions can be carried out at each point x E X , taking E = T,X. Thus a differential form of degree p on X , call it o,is an assignment w(x) E Ap(TxX)for each x E X . Suppose that ( V , 4) is a coordinate chart on X with coordinates x', . . . ,x".
151
ORIENTATION OF MANIFOLDS
Thenon v w = wlil ,...,i?l dxii A . . . A dxiP.We will say that w is a smooth form if the component functions w l i , , . i,l are smooth for all coordinate charts (V, 4). The set of all smooth forms on X of degree p will be denoted by Ap(X).Suppose that 5 is a vector field on X and o E A p ( X ) .Since w E F:(X), the Lie derivative L p , as defined in Chapter 8, is an element of F : ( X ) . But it is a simple exercise to show that, since w E A p ( X )c F:(X), L p E A p ( X ) . Now, iff: X + Y is a smooth mapping and w E AP(Y)>then we define the pullback f *w E A p ( X )by ,,,
f*o(x)(u,, . . . up) = (Nf(X))(TXf'(Ul), . . . > T x f ( u p ) ) . 2
Recall that we have studied pullbacks in Chapter 6 for the cases p = 1 , 2 [see (6.19), (6.20), (6.22)-(6.24)]. Of course, Ao(Y) is the ring of smooth real valued functions on Y and f * ( g ) = g .f. 0
ORIENTATION OF MANIFOLDS Let X be an n-manifold. For x E X let px be an orientation for T,X. Thus px is an equivalence class of bases. We want the orientations to "vary continuously with x" in some sense. DEFINITION 9.28 An orientation p of X is a choice of orientation px for T x X ,for each x E X , such that for each xo E X there is a neighborhood U of x,, and continuous vector fields t l , . . . , 5, on U such that for x E U , (tl(x), . . . , 0 at all points in the domain of 4j 0 4,:'. PROOF: If X is orientable, then one easily sees that X has an atlas consisting of positively oriented charts for some orientation p. If 4i = (x', . . . , x"), 4 j = ( y l > .. . , y") are two positively oriented charts for p, we claim det(dy'/dxj) > 0. Now d/dxj = (dyi/dxj)(d/dyi). But both bases (d/dxi(,), (d/dyil,) are in px for x E U i n U j . So the matrix relating them must have positive determinant as desired. Conversely, suppose an atlas 0 exists with the stated properties. If x E X , choose ( U i ,di)E R with x E U i .Let px be determined by (a/dxilx,. . . ,a/dx"l,).
152
9. DIFFERENTIAL FORMS
-
If ( U j ,q5j = ( y ' , . . . , y")) is another such chart, then ( 3 / d y ' l x , . . . , d/dy"l,) (d/dx'(,, . . . , 3/3x"l,), since det(dy'/dx') > 0. The vector fields (d/dx', . . . , d/dx"), for charts in R, satisfy the continuity requirement of Definition 9.28.
I An atlas with the properties of Proposition 9.30 is called an oriented atlas. Such an atlas uniquely determines an orientation of X . Thus, specification of an orientation of X is equivalent to specification of an oriented atlas. Now we give another way of specifying orientation in terms of diferential forms. Let p be an orientation of X . Let R = { ( U i ,4i)}ibe a positively oriented atlas. Fix i and let 4' = (x', . . . , x"). Then mi = dx' A . . . A dx" is a posbe a C" partition of unity subordinate itively oriented n-form on U i . Let to the cover (Ui)i. Then aimi is an n-form on all of X vanishing (by definition) outside of U i . Let o = criwi; a is a C" n-form. If ( u l , . . . ,u,) E p,, then aioi(u,, . . . , u,) 2 0 for all i and strict inequality holds for some i, so ~ ( u , ., . . , u,) > 0. Thus we see o is positively oriented for p . In particular, w(x) is never the zero n-form for any x E X . Conversely, suppose o is a nowhere zero C" n-form on X . We can define an orientation of X as follows: For x E X let p , be the orientation of T,X defined by the n-form m(x). For xo E X let ( U , 4 = (x', . . . ,x")) be a chart about x, with U connected. Consider (3/dx', . . . ,djdx"). For x E U consider o(x)(d/dx'l,, . . . , d/dx'l,). This function is always positive or always negative on U . Thus, either (dldx', d/dx2,. . . , d/3xn) or (d/dx2, d/3x1,. . . , djdx") give the local vector fields defined near xo as required in Definition 9.28. If a l ,o2are two nowhere vanishing n-forms we say they are orientation equivalent if there is a positive Cm function ,f:X + R such that col = fa2.The above arguments then prove
El
THEOREM 9.31 There is a 1-1 correspondence between orientations of X and orientation equivalence classes of nowhere vanishing n-forms on X . In particular, X is orientable if and only if there is a C" n-form on X which vanishes nowhere. EXAMPLE 9.32 Consider the atlas R discussed in Example 3.12(d). We have
=
{(U,,
4+),( K ,4 - ) } on
S" as
and
"[
3x' (x')2
X'
I
+ . . . + (.x")2
1x126; - x'(2xj) =
1x14
.
We must consider the determinant of the matrix having this last expression
153
ORIENTATION OF MANIFOLDS
as (i,j)-entry. Consider x
= (1,0, . . . , 0) so
the matrix is
p:
;].
So the determinant at x = ( I , 0, . . . , 0) is - 1. Since the domain of $ + $1' is R" - 0, which is connected, we conclude the determinant is everywhere negative. Thus, this atlas is not oriented. But if $ - is modified by interchanging two of its coordinate functions the resulting chart, together with ( U + ,$+), gives an oriented atlas. Thus, S" is orientable. The orientation defined by the atlas just described is called the standard orientation of S". The following proposition provides a simple method for showing that a given manifold is not orientable. 0
PROPOSITION 9.33 Suppose X has charts ( U , $), ( V , $) with U , V connected. Assume X is orientable. Then det(D($ $ - ' ) ) cannot change sign on $(U n V ) . 0
PROOF: Let p be an orientation of X . Now p defines an orientation on U and on V. Then, since U is connected, $ is either positively oriented, i.e., $ = (x', . . . , x") and (i?/dxllx,. . . , d/dx"l,) E p, for x E U , or (a/dx'I,, . . . , d/Jxnl,) 6 px for all x E U . Similarly for V . If (d/dx'l,, . . . , d/dxnl,) 6 p.yfor all x E U , call ( U , #I = (x', . . . , x")) negatively oriented. Now if ( V , rc/ = ( y ' , . . . , y")), then d/dxj = (LJy'/~xj)(d/?x'). If both charts are positively oriented or both negatively oriented, then they are similarly oriented so det(ay'/Jx') > 0. If one is positive, one negative, then det(dy'/dxj) < 0 on $(U n V ) .So in any case, det(D(rc/ $ - I ) ) = det(Jy'/?.u.') does not change sign on U n V . 0
-
(The Mobius strip-A nonorientable manifold) Let R 2 I - 1 < y < 1). Define an equivalence relation on Z by (x, y) (x 2, - y ) for all x E R , - 1 < y < 1. Let X = Z / - . Let n : Z + X be the canonical projection. Let U , = {n(x,y ) = [x, y ] I - 1 < x < l}, EXAMPLE 9.34
Z
-
= {(x, y ) E
+
u, = {[x, y]lO
< x < 2).
n:(0, 2) x n:( - 1, 1) x (- 1, 1) + U , is a homeomorphism with inverse ( - 1, 1) + U , is a homeomorphism with inverse &. ( U , , $ 1 ) and ( U , , 4,) are charts covering X . U , n U 2 consists of two disjoint open sets, namely,
n((O,1) x ( - 1, 1)) LJ n ( ( l , 2 ) x (- 1, 1)). We have$,
o$;':((O,
l ) u ( l , 2 ) ) x (-1, 1 ) + R 2 , $ 1 ~$;'(x,y)=(x,y)for
154
9. DIFFERENTIAL FORMS
O < x < l , - k y < l , 4 1 0 ~ ; ' ( ~ 7 y ) = ( ~ - 2 , - y ) f 1o 0. But (a;) has the form
1:
0
O '.
156
9. DIFFERENTIAL FORMS
(pf) will
have a positive determinant. This shows ( u l , . . . , v n - J and . . , w n P 1 )are similarly oriented as desired. Let p E M . Choose coordinates (x', . . . , x"-') on M near p so that, at p , (a/ax', . . . , d/ax"-') is
so
(wl,.
positively oriented. We want to show this basis is positively oriented in a neighborhood of p . But if
where ( y ' , . . . , y") are Cartesian coordinates on R", then dy' A . . . A dy"(f(z), 8/ax1I=,. . . ,3/8x"-'I,) > 0 for z near p , by continuity. This shows that we do have an orientation for M . I Note that we have shown that specifyinga smooth unit normal on M is equivalent to specifying an orientation of M . On an oriented Riemannian manifold we have seen that there is a uniquely defined n-form, the Riemannian volume element. In case M is an oriented hypersurface in R" we have an explicit description of the Riemannian volume element on M .
INTERIOR PRODUCT DEFINITION 9.39 Let o be a k-form on T,M, let u E T,M. The interior product of u and o is ivo,an (n - 1)-form on T,M, given by
ivw(ul,. . . , 0,-
1)
= w(u, u l ,
. . . , 0,- ').
PROPOSITION 9.40 Let N be the positive unit normal on an oriented hypersurface M in R". Then the Riemannian volume element on M is ow= i,(dxl A . . . A dx"). PROOF: Let p E M and let ( u l , . . . , u,- ') be a positively oriented orthonormal basis for T,M. Then ( N p ,u l , . . . , u,- ') is a positively oriented orthonormal basis for T,R". Then
. . A dxn(ul,. . . , u,- 1) = dx' A . . . A dx"(N,, u l , . . . , u,- = 1. This shows i, dx' A . . . A dx" is the Riemannian volume element on M . I i,dx'
A.
EXTER I0R D E R IVATIVE PROPOSITION 9.41 Let U be an open set on a manifold M such that U is the domain of some chart. There is a unique family of maps d,: Ak(U)+ Ak+' ( U ) ,k 2 0, such that
(a) (b) (c) (d)
iff
E
Ao(U) = Cm(U,R ) , then d,f
d,(f + Y) if
CI E
d,(d,cc)
=
d,f+
is the usual differential off;
d"g;
A,(U), fl E A,(U), then d,(cr A p) = (d,a) = 0 for every cr E A,(U), k 2 0.
A
+
/l (- l)%A
(&a);
157
EXT E R I0R DER IVATIVE
PROOF: We first prove uniqueness. If d, satisfies the above properties and a E I\k(U), then
a is a sum of terms,
with f E C " ( U , R ) , i , < . . . < i ,.
fdxi'r\...r\dxik
Then d,( f dxil A . . . A dx'") = f P dx' A dx" A . . . A dxik, so we see d,cc is uniquely determined. For existence, let a = c l l i , . , i k l dx" A . . . A dxB and define d,a = a l i l ,. i.k l , L dxl A dxil A . . . A dxik. Note that we are defining the operation d, in terms of a particular coordinate system but, if we show (a)-(d) hold, then it follows that the operation is independent of coordinates because of uniqueness. Now (a) is clear, as is (b). Note that if i,, . . . , ik are all different but are not necessarily in ascending order, then we still have ,
d,( f dxil A . . . A dx'") = J j d x j ~dxil A . . ' A dxik.
Because supposej,, . . . , j , are i,, . . . , i, rearranged in ascending order. Then fdx" A . . . A dxik= (- 1)"f dxj' A . . . A A dxJk,where rn is the number of interchanges needed to effect the reordering. So
d,( f dXi' A . . . A dXik)= d,( ( - 1)"f dXj' A . . . A dXjk) = (- l ) m f t d x ' ~dx" A . . . A dxjk = J t dxeA dxilA . . . A dxik. Now we prove (c). Write LI = a l i l ...i k l
dxil A . . . A dxil,
=
Plj,.. . j,l
dxj' A . . . A dxL
so that CIA
b = c l l i l . . . i k l Blj,. . .,il
dxilA . . . A dxikA dxjl A . . . A dxj'.
so d,(a
A
a
j)= axp [clli,. . . iklbljl,, .j,l] dxp A dx"
A.
. . A dxikA dxjl A . . . A dxj'
-
A . . . A dxj' dxPA dxil A . . . A dxj/ = ( - l)kali,. . .iklPljl. . _jfl.p dx" A . . . A dxikA ~ X P A dxJ1A . . . A dxj' (dux)A P = ( - l)kCcA dub + dua A p.
- ali,.. . i k & ? l. j. j ,l l ., p dxP A dx''
+ PIj,.. .jlpli,,
,,ikl.p
+
This proves (c). To prove (d), it is enough to show d,(d,a) = 0, where a is of the form f dxilA . . . A dxik.But d,r = , f e dxe A dxi' A . . . A dxik so d,(d,a)
= Jl,
by equality of mixed partials.
dx'
A
I
dx'
A
dxil A . . . A dxik= 0
158
9. DIFFERENTIAL FORMS
Note that if V is open in U , then d&l")
= (dual/".
We now globalize this. PROPOSITION 9.42 On any manifold M there is a unique family of maps d:Ak(M)+ Ak+ ,(M) such that:
(a) Iff E A,(M), df is the usual differential off, (b) d(a P) = da dp, d(CI A b) = d E A fl (- l)kCr A dp if CI E I \ k ( h f ) , (C) (d) dda = 0.
+
+
+
PROOF: Let a E I\k(M). Then if p E M we must define da(p). Let p where ( U ,(x', . . . , x")) is a chart on M . Define
E
U,
4 P ) = 4A@Iu)(P). If ( V , ( y ' , . . . , y")) is another chart about p , then choose an open set W with p E W c U n I/. Then d,(al,)(p) = d&al,)(p) = dv(alv)(p), so that da(p) is well defined. Since (da)l, = d,(al,) we see that da is a smooth ( k 1)-form as desired. So we have a map d: Ak(M)4 A,+ ,(M) as desired. We must verify (a)-(d).
+
(a) For f E A,,(M), (df)u = d u ( f l U )= differential o f f on U . Since M is covered by charts, (a) holds. (b) Clear. (c) On any chart domain U we have ( 4 a A P))Iu = du(alu A
PIU)
+ (- l)kaIuA du(PIu) = (da)lu A Plu + (- l)kaluA (dP)Iu = du(aILJ A P I U = (da A
P + (-
l ) kA ~Lip)[,.
(d) Similar. We now prove uniqueness. Suppose a E Ak(M),p E M . We want to write down a formula for da(p). First we argue: If a = ,8 in a neighborhood of p , then da(p) = dP(p). For suppose a = /lon a neighborhood I/ of p . Let k be a C" function on M which satisfies
Then ha
= kb
k
=
k
=0
1
nearp, outside V .
on M , so we get d ( W ( p ) = dk(p) A
+ k(p) A
=ddp).
159
EXT E R I0R D E R IVATlVE
and d(hP)(P) = dB(P) so da(p) = dP(p). Now, given
and express
CI
CI
and p we can choose coordinates around p
locally as a = a l i , ...i k l dx"
A
. . . A dxik.
We can choose functions A i l . , , i kX, i which are C" on M and agree with a i l . .. i k , x'
near p . Then M =
Ali, . . . ikI dXi1 A . . . A dXik
near p. So da(p) = d ( A l i , . . i k l ) ( pA) dX"(p) A . . . A d X i k ( p ) = d(ali1. . . i k l ) ( p ) ~ d x i l ( p .) ~~d. x . '"(p).
This proves uniqueness.
I
PROPOSITION 9.43 (d commutes with pullbacks) Let f : X smooth map, w a k-form on Y. Then f * d o = d ( f * o ) .
+
Y be a
PROOF: The case k = 0 is Proposition 6.26. Now locally o can be writ-
ten as o = w l i l . . i k l d x i l A . . . A dxik.
Then ,f*o = (olil.. . i k l L f ) f * dx" A . . . ~ , fdxik * - oli, , . . i k l 0 f d(x" f ) A . . . A d(xik 0 f ) 0
SO
d f * o = d ( o l i ,. . irl 0 f ) A d(xil f )A . . . A d(xik f ) . Now dw = do+, . . . i k ldx" A . . . A dxik so f * d o = f * dmlj,. . . j k l A f * dxil A . . . A f * dxik.
Since f * dx'j
0
,
= d ( x ' ~o f )
and f * d o i , .. . i k
= d ( o i l. . . ik
0
f ) , we have the result.
I A subset U of R" is said to be starlike with respect to p if, for x in U , the closed line segment from p to x is in U . PROPOSITION 9.44 Let U be an open set in R", starlike with respect to the origin. Let w be a k-form on U such that d o = 0. Then there is a (k - 1)form M on U with da = w.
160
9. DIFFERENTIAL FORMS
PROOF: We define a linear map H : A,(U) + At- l(U) for all 8 2 1 such that, if o is any [-form, then
+ H(do).
o = d(H(o))
Then if d o = 0 we get o = d ( H ( o ) ) ,so that we may take CI = H(o). Let o = f dxil A . . . A dxif (with no assumption about the ordering of the indices). We define e
Ho(x)=
1 (-1Y-I u= 1
(1:
1
A
t t - ' f ( t x ) dt xi, dxil A . . . A dxiaA . . .A dxi'.
If the factors are permuted by 6,so f is replaced by (sgn a ) f , then the form H o ( x ) is unchanged, so that H is well defined. Then e
e
x
Now d o = H do(x) =
xj dxj
A
(j: teJj(tx)d t ) xi- dx' dxil A .
A
-. d x i l A ' . ' A dx'.
A.
. . A dxie.
. . A dxit, so that we have
(j: t y j ( t x )d t ) x j d x i l e + (- 1)" (j:
A.
. . A dxie
t'lj(tx) dt)
-.
xi= dxj A d x i l A . . . A dxla A . . . A dxi'.
a= 1
Therefore dHo
+ H do = /
(1: + (1:
=
)
t'- ' f ( t x ) dt dxil A .
. . A dxiC
t Y j ( t x )d t ) xj d x i l A .
. . A dxiC
(j: ( / t t - ' f ( t x ) + t f f j ( t x ) x j )dt
= ( t y ( t x ) l h ) dxil A .
1
d x i l A . . . A dxi'
. . A dxie = f ( x ) d x i l A .
. . A dxi' = ~ ( x ) . a
161
DE RHAM COHOMOLOGY GROUPS
REMARK: For more insight into the mapping H see Exercise 10.8. The requirement that U be starlike from 0 can be relaxed considerably using Proposition 9.43. Let U be diffeomorphic to a starlike region and let w be a closed k-form on U . Let f: U + V be a diffeomorphism with V starlike from 0. Then ( f - ' ) * w is a closed k-form on V since d((f-')*w) = ( f - ' ) * d o = 0. So write ( f - ' ) * o = dp. Then f * f l E Ak- l(U) and d(f*b) = f * dp = f * ( f - ' ) * c o = w so w = da where a = f*P. Thus the actual requirement of starlikeness is unnecessary. However, some requirement on U is necessary. For example, if 0-
x2
-y
+
thenon U = R2 - {(0,0 ) }we havedw For if there were, then
j:n w(c(t))c'(t)dt =
X
dx+-
x2
y2
=
+y
dY,
2
0. Thereisno a: U
(a c)(t)dt 0
=
rR
(U
+
0
R withda
= w.
c)' dt
= a(c(270) - a(c(0))= 0,
which a direct calculation shows to be false. Here c(r) = (cos t , sin t),
POINCARE L E M M A PROPOSITION 9.45 (Poincare lemma) Let w be a k-form on M with dw = 0. Then for p E M there exists a neighborhood U of p and a (k - 1)form a on U such that da = w on U . PROOF: Immediate from Proposition 9.44.
I
DE R H A M COHOMOLOGY GROUPS Since d2 = 0 it follows there is a quotient vector space
and H o ( M ) = ker(d:A,(M) + A,(M)).Note that H o ( M )E R' where c is the number of connected components of M . A form w for which dw = 0 is said to be closed, and if w = da, then (0is said to be exact. This quotient is called the qth D e Rham cohomology group of M . The result of Proposition 9.44 is that on a starlike subset of R" we have H q ( M ) = 0 for all q > 0.
162
9. DIFFERENTIAL FORMS
MANIFOLDS WITH BOUNDARY Let H" = { x E R"lx" 2 O } . H" is called the upper half-space in R". We denote by dH"the set dH"= { X E R " ~ x "= O } . (9.3) Let U be open in H", f:U -+ R" a map. We say f is C" on U if, for each p E U , there is an open set V in R" containing p and a C" map F : V R", such that P = f on U n V. Thus f is C" if, near each point, f extends to a C" map on some open set in R". It is thus possible to speak of the derivative Df(p), even for points of U n dH".Ordinary rules of calculus, e.g., the chain rule, hold as usual. We want to discuss manifolds with boundary. We consider, for this purpose, charts which have as images open sets in H". If X is a topological space, let U be open in X and 4: U --t 0 be a homeomorphism of U onto an open set 0 in H". We consider such to be charts on X . An n-dimensional manifold with boundary is a second countable Hausdorff space which is covered by charts in the more general sense just discussed. We say charts ( U , $), ( V , $) are C"-related if $ 4-l and 4 0 $- are C" on their respective domains. Just as before, we may consider a C"-atlas or a diferential structure (see Definition 3.13). --f
0
DEFINITION 9.46 Let X be an n-dimensional manifold with boundary. The boundary of X , denoted dX,is given by dX = { p E XI there is a chart ( U , 4) in the differential structure of X such that 4 ( U )is open in H" and 4 ( p ) E dH"}. PROPOSITION 9.47 Let p open in H" and $ ( p ) E dH".
E
ax,( V , $) a chart with p in V . Then $(V) is
PROOF: Let ( U , 4) be as in Definition 9.46. Then both $ 4-l and 4 $ - I are C". If the proposition were false, then, by restricting 4 if necessary, we may assume $(V) is open in R". Now 4 $-' is defined on an open set in R" and D(4 $-')($(p)) is a linear isomorphism. By the Inverse Function Theorem, there is an open set W in R" such that im(4 I,!-') 3 W 3 &). But this cannot be, since 4 ( U ) c H" and 4 ( p ) E dH". [ 0
0
0
0
0
DEFINITION 9.48 If X is a manifold with boundary, the interior of X , denoted Int X , is given by Int X = X If p E Int X and ( U , 4) is a chart about p , then either 4 ( U ) is open in R" or, if 4 ( U ) is open in H", then
ax.
4(P)4 dH"All of our basic constructions on manifolds go through with no change for manifolds with boundary. We have, at each p E X , an n-dimensional tangent space T p X . This is true even if p E ax.We have the tangent bundle, co-
163
INDUCED 0 R I ENTATION
tangent bundle, Riemannian metrics, differential forms, orientations, and volume elements, just as before. The theory of integral curves of vector fields does not go through so well. At boundary points there may be no integral curve at all through the point.
+
EXAMPLE 9.49 Let X = {(x,y) E R 2 Ix2 y 2 5 1). Consider the vector field d/dx on X. What is the integral curve passing through (0, l)?
At points of Int X there is, of course, no problem. L E M M A 9.50 Let X be an n-manifold with boundary. Then dX has a natural structure of an (n - 1)-dimensional manifold without boundary. PROOF: Let A? be the set of all charts ( U , $) on X such that $(U )is open in H " a n d U n d X # $ . L e t Q , , = {(Unax,$I,,,,)l(U,$)~.~}.Note$I,,,, maps U n dX onto an open set in dH" = R"-'. Since, for any two charts ( U , $), ( V ,9) in d,$ 0 9-l and 9 $ - I are C", it follows that Qax defines a natural differential structure on dX. I 0
Let x E ax.Then we may regard T,(dX) as an (n - 1)-dimensional linear subspace of T,X, namely, all vectors in T,X which can be obtained as c'(O), where c: ( - E , E ) -+ X is a Cm-curvewith c(t) E dX for --E < t < E. Suppose g is a Riemannian metric on X . Then g defines a Riemannian metric on dX.There are exactly two unit vectors in T,X, x E a x , which are perpendicular to T,(dX). We distinguish one of these as follows. Let ( U , 6) be a chart about x with U n dX # 4, let u E T,X and let $ = (x', . . . , x"). Then write u = ui d/axi. Now, if u is a normal to a x , then U" # 0. We say u is inward pointing if u" > 0 and outward pointing if un < 0. This depends only on u not being tangent to a x ; being a normal is not needed.
INDUCED OR I ENTATION Let X be an oriented n-manifold with boundary dX. Then the orientation of X induces a natural orientution on dX as follows:
ax
DEFINITION 9.51 The induced orientation on is defined by requiring ( u l , . . . , u,_ 1) E T,(dX) be positively oriented if and only if (u, u l , . . . , u n _ 1)
is a positively oriented basis for T,X whenever u E TpX is outward pointing (see Fig. 9.3). PROPOSITION 9.52
T,(dX), for p E ax.
This definition gives a well-defined orientation at
PROOF: We first argue that, if u, u' are both outward pointing vectors at We p , then if (u, u l , . . . , u n P l ) is positively oriented so is (u', u l , . . . , u,-
164
9. DIFFERENTIAL FORMS
1
0 Pvectors outward pointing
ax, w i t h induced orientation
FIGURE 9.3
have
m i 011
U VI 9
I
1J
a > 0 since both u and u' are outward pointing. Hence the matrix has positive determinant as asserted. Now we argue that, if u is outward pointing and ( u l , . . . ,u ~ - ~ ) , (v;, . . . , V L - ~ ) are bases for T,(dX), then they are similarly oriented if and only if (u, u l , . . . , u,- I ) and (u, u;, . . . , ub- are similarly oriented bases for T p X .We have
The determinant of the matrix is equal to that of the submatrix (a:) so the result is proved. 1 PROPOSITION 9.53 Let X be an oriented, n-dimensional, Riemannian manifold with Riemannian volume element w x . Let N be the outward unit normal field on ax.Then the Riemannian volume element on dX is
wax = hv4wx).
Proof is left as Exercise 9.35.
165
HODGE "-DUALITY
HODGE *-DUALITY Let E be an n-dimensional real vector space which is oriented. Let g be a metric on E , i.e., a nondegenerate symmetric bilinear form. Let E be the volume element on E corresponding to the orientation and the metric (see Definition 9.24). Let 4: E + E* be the isomorphism defined by the metric
4.
4MW)= The map
(9.4)
4 (which is actually the b-operator for y) induces a map
4*: Tf(E)-+ T k ( E ) defined by 4*a(B', . . . , p)= a((flfl1,. . . , 4-'pk).
(9.5)
This is an isomorphism; in terms of components, 4* raises all the indices. Let a E Ak(E).Then 4 * a 0 E lies in Ti(E).Let C: Ti(E)-+ Tf-k(E)be the k-fold contraction contracting the ith contravariant with the ith covariant n). index for 1 I i I k (we are assuming k I
CI
DEFINITION 9.54 The Hodge *-operator *:Ak(E)-+ An-k(E)is given by * a = (l/k!) C(4,a @ E).
-+
Let (el,. . . , en) be a basis for E . Let a = a l i l . . ikleil A . . . A eik,E = E
__
~ ~ ,inleil . A
. . . A e'n; ail_ _ . i k , E
~ . ,. . j,
are the tensor components of a, E. Then 4 * a has components g i i j i . . . gikjkaj,, . j k so * a has components
=
,
(*a)jl...jn-k = (l/K!)d""' - .(i
So if a
= a l i l ., .ikleil A.
I
. . . ikl
kEil..
.ikjl...
E i l . . . i k j l . . .jn
k .
. . A eik,then
* a = alil,,.ikl E
~ . . i~k l j ,. . . .jn-klejl A.
. . A ejn-l..
(9.6)
EXAMPLE 9.55 (a) Let E = R 3 with the usual metric and orientation. Let (el, e2, e3) be the standard basis. We claim
*e 1 -- e 2 ~ e 3 , e1 = 6:e' so 4*e'
= hiei
*el
=
*e2 = e3 Ael,
*e3 --e1Ae2.
(we have identified TA(E) = E** with E). Thus,
(e')'EiljkleJA ek = c123e2A e3 = e2 A e3.
166
9. DIFFERENTIAL FORMS
Similarly, e2
= Sf.'
e3 = 63ej (b) Let E
= R4
so
*e
A e' = ~ 2 1 3 e 'A - 82~jlkIlek j
so *e3 = dj3EjlkllekA e'
= E312e' A
e3 = e3 A e', e2 = e'
A
e2.
with the usual orientation and the Lorentz metric g =
eo @ eo - e' @ e' - e2 0 e2 - e3 0 e3,where (eo, e l , e 2 , e3)is the standard
basis. The volume element is E = eo A e1 A e2 A e3. Verify the equations (a) (b) (c) (d) (e) (f)
*eo = el A e2 A e3, *el = eo A e2 A e3, *e2 = - eo A e l A e3, *e3 = e0 A e1 A e2, *(el A e3) = -eo A e2, *(eoA el A e3)= - e 2 .
Here is the pattern: Suppose a is a wedge of some e"s. The dual is the wedge of the others. The order of the others is such that the wedge in c1 followed by that in * a gives E. Then put a minus on * a each time one ei appears in a with i = 1, 2, or 3. Thus *(el A e 2 )= (- l)'eO A e3
since e' A e2 A e0 A e3 = E. If a, fl are k-forms then 4 * a
E
Tk,(E)so we have the complete contraction
C(4,a
0 B) E R .
(9.7)
Define the inner product of a and /? by (a, P) = (l/k!)C(4*a 0 PI.
(9.8)
Then, in terms of components, (a, p) = alii ...i k l g 1, 1 " ' l k.'
Clearly (a, p) is symmetric and bilinear in a and PROPOSITION 9.56
p.
For any k-forms a, p we have aA
PROOF:
(9.9)
*fl
= (a,
a)&.
167
HODGE *-DUALITY
Then we have
aA*fl
=
alil,.,iklfi~j~~~~j~l cjI . . .jklsl
. . . S n - klei i
A
. . .A
,SI
A.
. . A esn-k.
In this sum we always have
i, 0, (a) 4 ( ~=){ x E (b) Tm4(Bm) = {4(m)}x (Rk x 10)) for all m E U , (c) there are constants a k + . . . , a,, such that S = $-'({x for all i, x j = ajfor k 1 ~j I r z } ) and II/ = 41s.
+
(S, $) is said to be obtained from ( U , 4).
E
R"I lxil < E
186
9. DIFFERENTIAL FORMS
THEOREM 9.86 Let m E M be fixed. Let O(m) be defined by O(m) = {$-'(V)I there is a B-slice (S, $) such that S c W(m),V is open in Rk and I/ c $(S)}. Let t(m)be the topology for W(m)generated by O(m).Let d ( m ) = { ( S , $)l(S, $) is a B-slice and S c W(m)}.Then, with topology r(m),d ( m )is a C"-atlas for W(rn).
Proof is left as Exercise 9.29. Provide W(m)with the C"-differential structure determined by d ( m ) .We have required, as part of our definition of differential manifold, that the topology be second countable. Thus we need to show that t(m) is a second countable topology on W(m).This is a nontrivial fact and is established in the following theorem. THEOREM 9.87 With the topology t(m)the space W(m)is connected and is second countable. PROOF: Let x E W(m).There is a piecewise smooth curve y: [a, b] + M which is tangent to B, with y(a) = m, and y(b) = x. Clearly y([a, b]) c W(m). We claim y: [a, b] + W(m)is continuous, i.e., continuous with respect to the topology t(m) on W(m).Let c E [a, b] and m, = y(c). Choose a chart ( U , 4) on M such that $(U) = {x E R"I1xil< E } , Tp4(B,) = {&I)} x (Rk x (0)) for all p E U and 4(mo)= 0. Then 4 y: J -+ R" (where J is a connected open subset of [a, b] containing c), 4 y is smooth and 4 y(c) = 0. Since (4 y)' ( t )= q(t+$(y'(t)),which lies in { 4 ( y ( t ) ) } x ( R k x {0}), we see that, if 0
0
S
=~ - ' { x ~< ~x E, i ~ =
1,.
0
0
. . , n, xj = 0,j = k + 1,. . . , n},
then y ( J ) c S. Since ( S , 4 IS) is a chart for W(m)we see y: [a, b] + W(m)is continuous at c so, since c was arbitrary, y is continuous viewed as a map into W(m).Thus W(m)is connected. The proof that W(m)is second countable is given in [5, pp. 96-98]. Our W(m)is a submanifold of M according to the terminology of [5] (not necessarily according to our terminology, however). Lemmas 1-4 in [ 5 ] are detailed and clear and hold for C" as well as analytic manifolds. I Thus, we now have a smooth manifold W(m)and the inclusion i: W(m)-+ M. By construction the pair (W(m),i) is an integral manifold of B containing m and is called the maximal integral manifold of B through m. REMARK 9.88 W(m) is a connected differential manifold. However, it may not be a submanifold of M. Let S' be the unit circle in the complex numbers. Take M = S' x S'. We may represent TM as
TM
=
{(z,ilz, w, iiw)lz, w E S',
A E R).
187
INACCESSIBILITY THEOREM
Take X ( z , w ) = (z, iz, w, ivw), where v is a small irrational number. Let
z , ) E M, A E R } c T M . B = {AX(Z,w ) ~ ( W Then B is an integrable subbundle of T M and each W(m)is dense in M . The proofs of these statements are left as Exercise 9.30.
INACC ESS IB I LlTY TH EO R EM DEFINITION 9.89 If x, y are points in M we say y is B-accessible from x, written xBy, if y E W(x).
x B y defines an equivalence relation on M , (b) If S, and S, are B-slices of M and x,By, for some x1 E S, and y, LEMMA 9.90
(a)
E S,,
then xBy for all x E S,, y E S,. Proof is Exercise 9.31. We see that if S is a B-slice and m E M then either S c W(m) or S n W (m)= 0. In the former case, we can speak of the slice S being accessible from m. THEOREM 9.91 (Inaccessibility theorem) Let m E M . Then,in any open set containing m there exist points which are not B-accessible from m. More precisely, let ( U , 4) be a chart about rn such that
(a) 4 ( ~=) {xllx'l < E, i = 1 , . . . , H I , (b) Tp4(Bp)= { 4 ( P ) } x (Rk x (01) for all P E U , (4 4(m) = 0. Then there exist at most countably many B-slices obtained from ( U , 4) which are B-accessible from m (note we are assuming B integrable). PROOF: Let S be a B-slice obtained from the chart ( U , 4) such that S is a B-accessible from m. Then S c W(m)and, by construction of the topology on W(m),S is open in W(rn).If there were uncountably many such slices, then W(m) would contain a uncountable collection of disjoint open sets, contradicting second countability. 1
If we denote the set of B-accessible equivalence classes by M/B, then M/B
=
{W(m)lmE M )
and we have the canonical projection IT: M .+ MJB
by
.(m)
=
W(m).
188
9. DIFFERENTIAL FORMS
NONl NTEGRABLE SU BBUN DLES Let B be a nonintegrable k-dimensional vector subbundle of T M . The definition of the set W(m) still makes sense, even though B is not integrable. However, without the assumption of integrability we cannot give W(m)the structure of a k-dimensional manifold as we did before. Also the inaccessability theorem no longer holds. We present the following example.
EXAMPLE 9.92 Let (x,y, 8,4) be coordinates on R4 and define a subset B of TR4 by the requirement that each of the following l-forms vanish on B: cl1 =
Thus, B
= ker(cr')
dx - sin 8 d4,
c12 =
dy
+ cos 8 d 4 .
n ker(c12). By Lemma 9.71 we see
B
= ker(w),
where w
= cll A ~1,.
Thus, B is a two-dimensional subbundle of TR4. Now we have
o=dx~dy+sinedyr\d4+cosBdxr\d4. By Theorem 9.79, B is integrable if and only if i,,i,, dw = 0 whenever V,, V, are vectors in ker(w). Now it is easily checked that ker(w) is spanned by the vector fields
By Theorem 9.78(c) we see . . E ~ , dw Z ~=,d 4
+ cos 8 dy
-
sin 6 dx # 0,
so that B is nonintegrable. We claim if m E R4 then W(m)= R4. That is, any two points are accessible from one another. It is enough to show any point (xo, y o , 8,, &) is accessible from (0,0, 0, 0). For this, move along appropriate integral curves of V , , V,. Details are left as Exercise 9.32. REMARK 9.93 The subbundle B of the preceding example actually arises in mechanics. If we refer to [12, p. 151, we see that our subbundle B is the state space for a mechanical system which consists of a disk of radius 1 which rolls without slipping in the xy plane. The fact that all points are accessible from each other can be seen from the physical picture presented on p. 15 of [12]. The disk can be rolled to any desired point in the plane and then we can arrange any desired values of 8, 4. The desired value of 8
189
VECTOR-VALUED DIFFERENTIAL FORMS
is obtained by simply rotating the disk about the vertical axis through its center. Then the desired value of 4 is obtained by rolling the disk about a circle returning to the original x , y , 0 but with 4 changed by an amount depending on the radius of the circle. This is an example of a mechanical system subject to nonholonomic constraints. The configuration space is R4 but the no-slip condition causes state space not to be all of TR4 but rather, the subbundle B. If B were integrable, we could consider the system with configuration space restricted to one of the maximal integral manifolds W(m). The restricted system is holonomic in the sense that the state space is the complete tangent bundle of the configuration manifold. In the example just given this cannot be done, since B is not integrable.
VECTOR-VALU ED DIFFER ENTlAL FORMS In many situations, including some discussed later in this book, there arise differential forms whose values lie in some vector space other than the real numbers. Much of the discussion closely parallels what we have previously done but, as there are some new features, we give a brief discussion here. Let E be a real vector space of finite dimension, E* its dual. DEFINITION 9.94 A smooth E-iiulued k-form on M is an assignment to each m E M of an alternating multilinear map of degree k, a(m), from TmM into E such that the following smoothness condition holds: For every j- E E*, (Acr)(m)(ul,.. . , u k ) = A[a(m)(ul,.. . , u,)] defines a smooth real valued k-form 1-a on M . REMARKS 9.95 (a) The E-valued form a can be expressed in terms of components by choosing a basis for E. Then the smoothness criterion in Definition 9.94 is equivalent to smoothness of all components. (b) We could have constructed a smooth vector bundle flk(M;E ) = fl,(M) @3 E = (&(TmM) @ E). Then an E-valued k-form is a smooth cross-section of this bundle.
UmGM
Choose a basis e l , . . . , e, for E and write R-valued k-form. DEFINITION 9.96
dcr, is the ( k
M = Miei
where each cii is an
The exterior derivatiue of the k-form c!
+ 1)-form defined by
da = (da’)e,.
=
criei,denoted
190
9. DIFFERENTIAL FORMS
It is a simple matter to check that da does not depend on which basis is used in E. The notions of tensor product and wedge product parallel those previously given except that a bilinear “pairing” is used to “multiply” function values. DEFINITION 9.97
Suppose v: E x F
Let a be an E-valued k-form and p an F-valued [-form. is a bilinear mapping. Define
+G
(a 0, p ) ( m ) ( u l , .. . 9
Ok+l)
= v [ a ( m ) ( U l ?. . *
3
Uk)? p(m)(uk+
1,.
..
9
uk+t)].
Also, define
Note that S is the antisymmetry operator and is defined exactly as before. The fact that the values are vectors makes no difference here. REMARK: In applications, v is often some very natural pairing and we often omit writing it as a subscript. THEOREM 9.98 Suppose e l , . . . , e, is a basis for E, fl, . . . ,f,is a basis for F and v: E x F -,G is a pairing. Then we have
(a) If a = aiei, p = Bjfj and gij = v(ei, f j ) , then cc A,P = aiA p j g i j , (b) d(a A, p) = da A, p (- l ) k aA, dp where k = deg(a).
+
PROOF:
(a) a A,P(U1, . . . > uk,
Uk+ 1 ,
.. .> Uk+t)
EXERCISES
191
EXERCISES 9.1 Prove Proposition 9.5. 9.2 Prove Lemma 9.8(a). 9.3 Prove Proposition 9.11. 9.4 If w E A,(E), where dim(E) = n, then there exist u l , . . . , u" in E* for which w = u1 A . . . A 0". 9.5 (a) If dim(E) = n, show that every ( n - 1)-form on E is decomposable. (b) Let (d,u2, u 3 , u") be a basis for (R4)*. Show that u 1 r \ u 2 u3 A u4 = o is not decomposable.
+
9.6 Let dim(E) = II. (a) If w E A,(E) is nonzero and e l , . . . ,en is a basis for E, then o ( e l , . . . , en)# 0. (b) If y, I.: E + E are linear show, using Definition 9.16, that det(y 3.) = det(y) det(2). 0
9.7 Prove that -, as defined in Definition 9.18, is an equivalence relation on the set of all ordered bases of E . 9.8 Let ( u l , u2), ( w l ,w2) be similarly oriented bases for R2. Then there is a matrix of continuous functions (af(z)),0 I z I 1, det(aj(z)) > 0 V t such that (ai(0))= I and ai(l)ui = w j . 9.9 (a) Let p, p' be two orientations of X . Suppose for some xo E X , pxo= p i o . Then p x = pk. for all x in the component of X containing xo. (b) If X is orientable, then X is connected if and only if X has precisely two orientations. 9.10 Suppose X has charts ( U , 9) and ( V , i+b) with X connected. Then X is orientable.
=
U v V and U n V
9.1 1 (a) In Example 9.35 show that each U iis open and each homeomorphically onto R2. (b) Show P2(R)is not orientable.
9imaps U i
9.12 (Higher-dimensional real projective spaces) Define P 3 ( R )= (R" - O)/-, where p Ap for p E R4 - 0,A E R - 0. Let n: R4 - 0 + P 3 ( R )be projection. Define charts, this time four of them, by analogy with Example 9.35. Show that P 3 ( R )is orientable. What about P,(R) for arbitrary n?
-
9.13 Consider the standard spherical coordinates on R3, (Y, 8, 4). r > 0, 0 < 8 < n, 0 < 4 < 2n.
192
9. DIFFERENTIAL FORMS
(a) Show that this is a positively oriented chart. (b) Calculate the standard Riemannian metric on R 3 in the spherical coordinate system. (c) Calculate the Riemannian volume element in spherical coordinates.
9.14 If ( u l , . . . , u " - ~ )is a positively oriented basis for T,M and (Np, 019. ..>
un-1)
is a positively oriented basis for T,R", then if (u;, . . . , V L - ~ ) is any other positively oriented basis for T,M, show that (N,, u ; , . . . , is positively oriented. This shows the definition of positive unit normal is unambiguous.
9.15 (a) Let e, = (d/ax')l,, 1 I i I n. Let n
n.
j,:dx'A...r\dx'/\...r\dx"
w = i= 1
be an ( n - 1)-form where, as usual, the "hat" over dx' indicates omission of dx' from the wedge product. Then show J ( p ) = 4p)(el. .. .
,4, . . . , en).
(b) Let N = n'e,. Show that
(c) In the case M
=
S2, N
i, dx A dy A dz
= (x,y , z )
=
so
x dy A dz - y dx A dz
+ z dx A dy.
Does this agree with the formula given in Exercise 9.13? That is, take the formula given by that exercise for the Riemannian volume element and compute the interior product with N = a/&. Then compare with the above result.
9.16 (Components of p-forms) Given (bjl.. j k ) , 1 I ji I n, we introduce the corresponding skew-symmetrized quantities b,, . .j,, by ,
,
(a) Show that (bj,.. .jk) is skew-symmetric in j , , . . . , J k if and only if bj, . .jk = brjl .j k l . (b) A tensor, given as A = A i l . , i,dx" 0 . . . 0 d x ' ~ is , a p-form if . = A,',.. and only if A i l . .i,, ,
,,
,
193
EXERCl S ES
(c) Let A
=
A i l . ,. i,
dx" A . . . A dxiP. Show that
-
A , i , . . ip]
= ( l/p!)Ail... .ig.
(d) Prove that if A is a p-form, with components chart, then the coordinates of d A in that chart are
in some
(dA)ii...ip+l= (P + l ) ( - l ) P A I i , . . . i p , i p + I ~ .
9.17 For each of the following l-forms, see if it is closed and, if so, find a O-form whose exterior derivative is the given form. (a) w (b) w
+
= xy dx (4x' - y ) dy = ex cos y dx ex sin y
+
dy
9.18 If X is a manifold with boundary, we have defined the notations of Show that the inward and outward pointing vector at a point p E definition does not depend on the choice of chart.
ax.
9.19 If X = {(x,y)lx2 + y 2 5 1 ) and ( r , 0) is the usual polar coordinate system, then, if we use the usual Riemannian metric in the plane, show that a/& is the outward pointing unit normal at each point of d X .
9.20 (a) In Minkowski space we use the usual time coordinate xo = t , but replace the spatial coordinates (x', x2,x3) by (r, t3,$). Find the volume 3-forms d3C,, d'C,,
d 3 C , , d3C,.
(b) Let f be a C"-function on R 4 . Express Af in the coordinates ( 4 r> 0, 4). 9.2 1
In a manner analogous to that used for the tangent and cotangent bundle, put a topology and differential structure on T:(M) and R,(M) so that both become smooth vector bundles over M .
9.22 Let W be open in R",j W + L( R', R' x Rk-' ) smooth. Let f = (f1, f 2 ) , where fi, f 2 take values in L(R', R')), LIR', Rk-'), respectively. Assume fi(x) is an isomorphism for all .x E W . Then for each x E W we have R' x Rk-'
=
im(f(x)) @ ( ( 0 )x Rk-').
Let 0 2 :W -+ L(Re x Rk-', Rk-') be given by projection on the second summand above. Prove that o2 is smooth. 9.23 Prove that if a is a 2-form on the n-dimensional vector space E , then dim(ker(a)) = n - 2 if and only if a is decomposable. 9.24 Suppose p: B + M is a smooth vector bundle and that, for each rn E M , C, is a k-dimensional linear subspace of B , = p-'(rn). Suppose for
194
9. DIFFERENTIAL FORMS
each m E M there is a neighborhood U of m and k Cm-vectorfields on U , X,, . . . , X,, such that X , ( p ) , . . . , X,(p) is a basis for C, for each p E U . Then C is a smooth k-dimensional vector subbundle of B. 9.25 Prove that conditions (a) and (b) of Theorem 9.77 are equivalent. 9.26 By working in a local coordinate system, prove that, iff is a 0-form, V a vector field, then L , df = dL,f: 9.27 Complete the proof of part (8) in Theorem 9.78. 9.28 Prove Theorem 9.81. 9.29 Prove Theorem 9.86. As part of the proof show that O(m)does in fact generate a topology. This means if Y,, Y2 E O(m)and p E Y, n Y2,then there is a Y, E U(m)with p E Y3 c Y, n Y,. 9.30 Verify the comments in Remark 9.88. Proceed as follows: Define p : R 2 + M by p(s, t) = (e2nis,e'""). Given (so, t o ) E R 2 let L(so, to) = {(s, t ) I t - to = v(s - so)}. Show p : L(so,to) + M gives an integral manifold p(so, to).Since p is locally a diffeomorphism it can be used to construct appropriate charts showing integrability. Alternately, p could be used to locally represent B as the kernel of a 1-form and then use Corollary 9.80. For the denseness property, first argue that p(L(so,t o ) ) is dense in A4 for all (so, to). Since each W(m)contains p(L(so,t o ) ) if p(s,, t o ) E W(m),this shows each W(m)is dense. In fact, if p(so, to)E W(m),then W(m)= P(L(S0, t o ) ) . 9.31 Prove Lemma 9.90. 9.32 In Example 9.92, prove that W((0,0, 0,O)) = R4. 9.33 Consider H" as an oriented manifold with the standard orientation. View dH" as R"-' in the obvious way. How does the induced orientation on R"-' compare with the natural orientation? 9.34 Consider the standard orientation of S". How does this compare with the orientation obtained by viewing S" as the boundary of the ( n + 1)-ball B"+l = (X E
R n f l 11x1 I l }
the ball having the standard orientation from R""? 9.35 Prove Theorem 9.53. 9.36 Let E be an oriented vector space, g a metric on E, E the metric volume form in An(E).Show that if a E R is a 0-form, then * a = a&.
195
EXERCISES
9.37 Let E be an oriented real vector space with metric g and metric volume element E. Let LY E A,(E) and let 2' = a # , that is, v results from r by raising the index. Show that * a = i,s. 9.38 Let M be the sphere of radius ro > 0 about the origin. We have spherical coordinates (r, 0 , 4 ) in R 3 . Find the three area 2-forms d2C,, d2C,, d2C,. Find expressions for the restrictions of these forms to M in the coordinate system (0, 4) on M .
+
9.39 Continuing the preceding exercise, let 4 = Se(i3/a0) t4((a/84)be a vector field on M . Calculate Sf as a function of (0, 4). Iff: M + R is a scalar, calculate Af as a function of (0, 4). 9.40 Suppose that M is a manifold and T M = B 0 C, where B and C are smooth integrable subbundles. Show that for each rn, E M there is a chart ( U , 4 = xi, . . . , x") with rn, E U such that
for each m E U . 9.41 Suppose that B c T M is a smooth subbundle and that X is a complete vector field on M with flow ( 4 J r c RSuppose . also that if Y is any B-valued vector field on M , then L,Y is B-valued. Show that, for = B,t(m). every t E R and m E M , Tm4r(Bm) 9.42 In a manner similar to the construction in Exercise 8.17, give the set Q,(x) =
u
A,(T,X)
xex
the structure of a smooth vector bundle over X , and show that a differential p-form on X is just a smooth section of this bundle. Show that this bundle is a smooth vector subbundle of the bundle of covariant tensors of degree p defined in Exercise 8.17.
I0 Integration of Differential Forms
In this chapter we consider integration on smooth manifolds. We shall define the integral of a continuous n-form with compact support (this is defined below) over an oriented n-manifold. We shall see that this allows us to integrate a continuous function with compact support over an oriented Riemannian manifold. This generalizes the notion of surface integral, familiar from calculus. The main result of the chapter is Stokes’s theorem (Theorem 10.6). This theorem is of fundamental importance in manifold theory and has as consequences the three classical integral theorems, usually associated with the names of Green, Gauss, and Stokes.
THE INTEGRAL OF A DIFFERENTIAL FORM We begin by recalling some properties of smooth partitions of unity on a smooth n-manifold M . From Theorems 6.34 and 6.36 we know that, given there is a locally finite atlas ((V,, $ j ) ) j , J with any atlas on M , ( ( U i ,4i))ia,, each 6 contained in some U i (the cover (y)jEJ refines the cover (Ui)isl)and a family of P - m a p s (pj)jaJ,p j : M + [0, 11, such that the support of p j lies p j ( x ) = 1 for all x E M . Recall that, because in 5 for eachj and such that of the local finiteness of (V;.),iEJ, each m E M has an open neighborhood W on which all but finitely many p j vanish. This implies that, if K c M is compact, then there is a finite subset J’ c J such that if j E J - J‘, then p j ( x ) vanishes on K . In this chapter, when we refer to a partition of unity ( p j ) j s J , we shall assume it is related to a locally finite atlas as above. In analogy to the familiar definition for functions, we have
cj
DEFINITION 10.1 If o is a differential form on M , the support of o is the closure in M of the set of x E M for which w(x)# 0. This set is denoted supp(w). If supp(w) is compact, we say o has compact support. 196
197
THE INTEGRAL OF A DIFFERENTIAL FORM
sLI
We let p denote Lebesgue measure in R" so that integrals ,f d p , for U c R", should be interpreted as Lebesgue integrals. However, in most cases to be considered, f will be continuous with compact support so Riemann integrals will suffice. A basic result from analysis that we will need is the change of variable formula, which we state. THEOREM 10.2 Let U be open in R", f:U + V a diffeomorphism onto the open set V in R". If g E L ' ( V ) , then (g -f)Idet(Df)I is in L'(U) and
lv
9 dP = Ju (9 f)ldet(Df)l dP.
A proof can be found in [18].
Henceforth, we assume M is a smooth n-manifold (possibly with boundary) and assume M is oriented; thus we have chosen a particular orientation v for M . Let o be a continuous n-form with compact support on M . We want to define the integral of o over M , denoted jM
w.
We first define the integral in case there is a positively oriented chart ( U , 4) with supp(w) c U . Indeed, in that case, we may write o = h d.u'
A.
. . A dx"
on U ,
where 4 = (x', . . . , x") and observe that supp(h) = supp(w) is a compact set in U . Since h: U -+ R is continuous, we may define (10.1)
To see that this is well-defined, we prove LEMMA 10.3 Let ( U , 4) and ( V , $) be positively oriented charts with supp(o) c U n V , 4 = (x', . . . , x") and $ = (y', . . . ,y"). Let o = h dx' A * . ' A dx" on U and o = k dy' A . . ' A dy" on V . Then
PROOF: Since supp(h) c U n V and supp(k) c U n V , we may replace n V ) and $(V ) by $(U n V ) in the equation to be proved. On U n V we have
b( U ) by 4( U
h dx' Let F
= II/
0
4- ': $(U
A..
. A dun = k dy'
A'.
. Ad?/".
(1 0.2)
n V ) + $(U n V ) . Then, by Proposition 9.15 and
198
10. INTEGRATION OF DIFFERENTIAL FORMS
Eq. (10.2), we get
h
0
4-l
= (k 0
4-l) det(DF).
Thus on 4 ( U n V ) , we have h
0
4-l = ((k I+-') F ) det(DF). 0
0
(10.3)
Then Theorems 10.2 and 10.3 give, using det(DF) > 0 (recall that both charts are positively oriented),
which proves the lemma. [ Suppose (PJi.1 is a partition of unity and that w and ( U , 4) are as above. Then p i o is a continuous n-form having compact support in U for each i. Since supp(w) is compact, we see that p i o is identically zero if i is not in a certain finite subset I' c I . Then
Thus, from (10.1) it easily follows that
Now drop the assumption that supp(o) lies in a single chart. Let (Pi)iEI be any smooth partition of unity (recall that we always assume that our piu partitions of unity are subordinate to some locally finite atlas). Then is already well defined, by (10.l), so we may define
sM
If supp(w) lies in a chart, then we have, in (10.1) and (10.5), two definitions of
IM o.However, by (10.4), the two agree. It remains to show that, if (qj)jeJis another partition of unity, then
But, for each i,
199
STOKES'S THEOREM
so that we have
where only finitely many terms are nonzero. Similarly,
Thus,
j M w is well defined for any (0 having compact
support.
STOKES'S THEOREM We begin with some preliminary results, which will lead to the general form of Stokes's theorem. In this section, assume M is an oriented n-manifold with boundary M. Let w be a smooth ( n - 1)-form on M . Recall that the interior of M is defined by Int(M) = M - dM and that (ul, u 2 , . . . , in T,(dM) is positively oriented if ( N , ul, . . . , u,is positively oriented in T,M, where N in T,M is outward pointing. LEMMA 10.4 Let ( U , 4) be an oriented chart on M with U n dM empty and $ ( U ) a bounded open set in R". Suppose o is an ( n - 1)-form on M having compact support in U . Then
PROOF: Let
4 = ( X I , . . . , x") and
let
(11
be given on U by
Then
so that we get (10.6) Choose a cube C, = {x E R"1 lxil 5 u for i = 1, . . . , n} containing $(U) in its interior. If g is any smooth function with compact support in $(U), then -2d57p = O ,
+(a)c7x'
for j = 1 , . . . , n.
200
10. INTEGRATION OF DIFFERENTIAL FORMS
This is because the integral is equal to
which is seen to be zero by treating it as an iterated integral, integrating with respect to xj first, and using the fact that g vanishes on the boundary of the cube. Since the right side of (10.6) is the sum of such integrals, we have the lemma. LEMMA 10.5 Let ( U , 4) be an oriented chart on M with U n i3M = r! a nonempty set. Assume that 4 ( U ) is a bounded open set in H". Let o be an ( n - 1)-form on M having compact support in U . Then
PROOF: Express o on U as 0
(- l)"Ei
=
dx'
A . . . A z i A . .
. Adx".
i= 1
We see it is enough to prove the result in case 0 = (-
for some j. Let ,f = ,f
1)j-Y dx' A , .
. . A dx"
4- ',so that f has compact support in 4 ( U ) .We have
so that
If j = 1, . . . , n - 1 this integral is zero since we can integrate first in the xjdirection and use the fact that f vanishes outside a compact set in $(U). For j = n we get
Now consider J a M o;we see ( V , 6)is a chart on ciM, where &in) = ( ~ ' ( m .) ., . , x"-' ( m ) ) . The orientation of the chart ( V , 8) is ( - 1)" times the induced orientation which we use on d M . The form o on dM has compact support in V . If j = 1,. . . , n - 1, then clearly w is 0 on V , since dx" = 0 on V . Thus,
201
TRANSFORMATION PROPERTIES OF INTEGRALS
the lemma holds for j < n. Thus, consider = (~
1y -
d.yl . . .
If
because of the orientation of the chart 8
I)',-
= (-
J'M
I(
I(.*-', ...,
-
-J$w which gives the lemma in this case. -
(I/, +),
1.
we see
6-
f'
I)
dp
0) dx' . . . d x " - ' ,
.Y"-',
I
We can now prove our main result. THEOREM 10.6 (Stokes's theorem) Let A4 be an oriented n-manifold with boundary aA4. Let w be a smooth ( n - 1)-form on M having compact support. Then
IM
dw =
j 0 since at t = 0 we have X = Agx. Thus (1 1.4) becomes
(11.5)
By the principle of relativity, the equation expressing ( t , x, y, z ) in terms of
(f,2,j,5)is obtained by interchanging barred and unbarred coordinates and replacing u by -u. Thus, t = A:(
x
=
Y= Z=
-A:(
-
u)t
+ A:(
- u)( -
-
u)X,
u)I + A:(
-
u)X,
(11.6)
A 3 - u)Y, A:(
- u)Z.
21 7
BASIC CONCEPTS A N D RELATIVITY GROUPS
If (11.5) is inverted directly we get
(11.7)
Comparing coefficients in (1 1.6) and (1 1.7) we get (11.8)
(11.9) (11.10) Now A:( - u) must equal A:(u), since the factor by which transverse lengths change cannot depend on whether the motion is to the right or to the left. So Ai(u) = 1 by (1 1.10).Now A:(u) is the factor by which time scales change and hence, A: is an even function of u. By (1 l.9),A: is an odd function of u. Our transformation thus becomes
t = A:t
+ AYx
(A: > O),
z= z . This form depends only on the principle of special relativity [25], homogeneity of space, and isotropy of space. Suppose we assume absolute time, i = t. Then A: = 0, A: = 1 so we get t = t,
x=x j
=
y,
z = z.
-
ut,
(11.12)
21 8
11. THE SPECIAL THEORY OF RELATIVITY
This is the Galilean transformation. It is based on the relativity principle, spatial homogeneity, spatial isotropy, and the explicit assumption of absolute time. These assumptions are in conflict with a fundamental experimental fact: the velocity of light is the same in all inertial frames. Let a flash of light occur at t = 0 at the origin 0. A spherical shell of light spreads out in all directions with velocity c. A point (t, x,y, z ) on this shell satisfies x 2 + y2 + z 2 = c2t2. The corresponding events in 32 have coordinates (f, 2,j , 2). The Galilean transformation, -
t
=
t,
x = x - ut, j = Y,
z= z , implies (X + + j 2 + 2’ = c21z. This is not a sphere about the origin 8, so the light is not spreading out with the same velocity in all directions in the frame 32. The assumption of absolute time is just that, an assumption, based on “common sense.” The approach taken by Einstein was to assume, instead of absolute time, the law of constancy of velocity of light. This leads to a different determination of A: and A:. We now proceed to this. REMARK 11.2
Let us assume (11.11) and the postulate of constancy of the velocity of light and determine A: and A:. Consider an event ( t , 0, ct, 0) which measures the position of a light signal sent along the y-axis, beginning at t = 0 at 0. The coordinates of this event in iZ! are, by ( l l . l l ) ,
t = Att, X = A:(
-
ut),
j = ct,
z = 0. But this light signal travels with velocity c, starting at the origin at l = 0, so its distance from 8 after t seconds is cl. Hence 22
+ y 2 + 52
= c2p,
which gives
(A:)2U2t2+ C2t2 = C2(A:)2t2,
21 9
BASIC CONCEPTS A N D RELATIVITY GROUPS
so that ( Q 2 u 2 + c2 = c ~ ( A : ) ~ . Solving for A: and using A: > 0 gives (11.13)
Now consider sending a signal of light along the positive x-axis beginning at 0 at t = 0. After t seconds the pulse is described in 9 by (t, ct, 0,O) and i n . g by(t,ct,O,O). But b y ( l l . l l ) ,
+
(a) t = A:L Ayct; (b) c t = Ag(ct - ut). Multiply (a) by c and subtract from (b) to get
0
=
AyC2t
+ @t
or, canceling t,
0 = Ayc2
+ A$.
We then conclude
Thus, we obtain the Lorentz transjormation equations -
t= 2=
t
(I
-
(ux/c2)
- (u2/c2))”2
x
-
ut
(1 - (u2/c2))”2’
’ (11.14)
Y’Y,
z= z. Henceforth, we shall use x l , x 2 , x3 rather than x, y , z and xo = ct rather than t. Also, let fi = u/c, y = 1/(1 - fi2)’’2. We then can rewrite (11.14) as
.uo = y(x0 - fix’), 2’ = y(x1 - fiXO), 22
- x 2, -
.f3= x3.
(11.15)
220
11. THE SPECIAL THEORY OF RELATIVITY
In matrix form, these equations become
(11.16)
The transformation (I 1.15) which transforms from an inertial coordinate system (x”) to another inertial coordinate system (2”)moving with velocity u in the x’-direction is called a velocity boost along the x’-axis with velocity u. It is clear that similar equations can be obtained for velocity boosts along the x2- and x3-axes. Formulas for velocity boosts along the coordinate axes are (boost in x’-direction)
(boost in x2-direction)
(boost in x3-direction)
RELATIVISTIC LAW OF VELOCITY ADDITION Let (2”)have velocity u2 in the 2’-direction and let (2”)have velocity u1 in the x’-direction. What is the velocity of (2”) relative to (x”)? We shall derive Einstein’s formula for this which says that the “common sense” answer, u1 u 2 , is incorrect. The origin of (2”) is given in (%”) by
+
(t,u2i,0, 0).
22 1
RELATIVISTIC LAW OF VELOCITY ADDITION
The corresponding events, in (x”)-coordinates, are obtained by inverting (1 1.16), which gives
Y
[
PY 0 0- ci
Brr 0 0
u2T
0 0 1 0
0 0
0 0
0 1
The velocity in (xu)is then (pyci
+ u,yt)c
-
u1
+ u2
yci + Pyu2t 1 + (u1u2/c2) Thus we have the velocity of (2’)measured by (xu).In terms of p we can write
i= ( p + j7/(1 + PF). < 1 we will have Note that as long as IPI < 1 and Define the velocity parameter a by
P = tanh CI. If B = tanh 5, then tanh(tl + a)=
-
1.
(P).
a = tanh-
Then
Ijl
l.
CI =
fi
= 0,
we obtain (a;)’
(a:)2.
-
(a;)’
-
1. Taking determinants in the matrix
=
-
-
a:a;g,,
gives det(ga8)= [det(at)]’ det(g,,). Since det(gab)= - 1, we conclude det(u:) = f 1. It can be shown that 9:is precisely the set of those linear transformations preserving g for which (a) a: 2 1; (b) det(at) =
+ 1.
If we write 2” = atx” and consider points with arbitrary xo and xi = 0 for j = 1, 2, 3, then
xo = a:xo. Thus, a: 2 1 implies that the transformation preserves time sense. Such a transformation is called orthochronous. This is the reason for the arrow in 9 i . The “+” indicates that the transformation preserves spatial orientations. When we refer to Lorentz transformations, we mean transformations in 9;.
THE SPACETIME MANIFOLD OF SPECIAL RELATIVITY As we have discussed, spacetime is the set M of all events. Each inertial reference frame, together with its system of Cartesian coordinates and clocks, gives a chart on M which we shall refer to as a Lorentzjrame. If (xu)is the system of four coordinates associated to a certain Lorentz frame, we shall often refer to “the Lorentz frame (x”).” If (x”) and (2”)are two Lorentz frames, we have seen that the coordinate transformation is of the form
+ b, where b = (b”)is a point in R4 and A E 91. x 4 2 = AX
226
11. THE SPECIAL THEORY OF RELATIVITY
DEFINITION 11.6 Let g be the metric on M whose (x”) representation is ( ~ x O ) ~ ( d ~ ’) ~( d ~-~( d) ~~ ~The ) ~metric . g is a well-defined Lorentz metric on M . We call g the spacetime metric.
Thus, M is a four-dimensional Lorentz manifold which is diffeomorphic to R4.At each point p E M we have a four-dimensional vector space T p M of tangent vectors to M at p . A convenient convention used in relativity is to refer to v E T P M as a 4-vector. This is convenient, since we also want to refer to 3-vectors. Given a Lorentz frame (x”), a 3-vector is a 4-vector of the form (0, a’, u2, u3), that is, a vector which is purely spatial in the given reference frame. But note that this same vector, when viewed in a different Lorentz frame, may have a nonzero time component, so there is no welldefined subset of T p M consisting of 3-vectors. When we speak of a 3-vector we must have a definite Lorentz frame in mind. Consider a particle moving in space. The set of events which make up the history of this particle, defines a curve in spacetime, called the world line of the particle. Given a Lorentz frame (x”), the world line appears as (ct, x’(t), x2(t),x3(t)). We shall use Greek indices p, v, a,. . . as indices on 4vectors and tensors. Latin indices i, j, k, . . . are used as 3-vector indices. So Greek indices take values 0, 1, 2, 3 while Latin indices take values 1, 2, 3. If the world line is (ct, x’(t), x2(t),x3(t)), then (il(t), i2(t), i3(t)) is the usual 3-velocity of the particle, as measured in the given reference frame. We now want to define a four-dimensional velocity vector for this world line. Choose a Lorentz frame (x”) and write xp = x”(t). We cannot define the 4-velocity to be (?(t)) because this involves parametrizing by t and differentiation with respect to t, while t is not independent of the reference frame. What is needed is an invariant parametrization of the curve. We now obtain this parameter, called proper time. Choose a Lorentz frame (x”) and write the world line as (x”(t)). Then xo = ct and (xi@))gives the spatial position in the given reference frame, as a function of t. The 3-vector (ii(t)) satisfies (i’(t))2 + (i2(t))2+ (i3(t))2< c2. Thus the 4-vector having (xp)-components (i”(t)) is a timelike vector, g(dx/dt, dxldt) > 0. Pick to and dejine z(t)
=
c
s‘ /= dt ’ dt to
dt;
(1 1.21)
z(t) is called the proper time elapsed on the world line from to to t. We note that z depends both on to and on the particular world line under
consideration and that dz
1 c
- = - (g(i, i ) ) ’ ’ 2
dt
> 0.
227
RELATIVISTIC TIME UNITS
Let us reparametrize the world line by z. Writing x(z), we see
Now if we choose a different to, z will be affected only by an additive constant. If we use a different Lorentz frame we again get a parametrization of the world line by a parameter f, and
+
It is then easily verified that .7 = z k, for some constant k. Thus we have, for the world line of any particle, a natural parameter z defined up to an additive constant, with the 4-vector dx/dz satisfying dx dx
“(di.di) = c DEFINITION 11.7 The 4-vector dx/dz is called the 4-uelocity of the par-
ticle. So every particle has, at each point on its world line, a 4-velocity vector dx/dz = u satisfying g(u, u) = c2. If we work in a Lorentz frame (xp),and if u is the magnitude of the particle’s 3-velocity, then 1 dt C - -2 2 1/2‘ dz (2- u2)”2 (1 - (0 /c 1 So u = 1/(1 - ( U ~ / C ~ ) ” ~ ) ( Cul,, v2, c 3 ) , where (v’, u 2 , u 3 ) are the 3-velocity components. Thus,
(4 velocity components) u’ =
U‘
(1
-
UZ/C2)’/2
for i
=
1, 2, 3.
RELATIVISTIC TIME UNITS We shall measure spatial distances in meters as is usual. But now, we shall also measure time in meters.
228
11. THE SPECIAL THEORY OF RELATIVITY
DEFINITION 11.8 A meter of time is the time in which a light signal trav-
els 1 meter (m). Since light travels 3.0 x lo8 m/s, we see that 1 m of time =
1
3.0 x lo8
=
3.3 x 10-9 s.
The result of using meters of time is that velocities become dimensionless and c = 1. The speed of any material particle is 0 5 u < 1 in any Lorentz frame. The equations derived up to now remain valid if, wherever c appears we replace it by 1. Note that, since x” = ct, xo is in meters of time. Consider the Lorentz frame (x”).Consider a world line (x”)given by xB = x”(t). Then xo(t) = t and (dx’ldt, dx2/dt,dx3/dt) is the 3-velocity of the particle in the given reference frame. Let v(t) be the ordinary magnitude of this 3-velocity. Consider an interval TI I t I T,. Partition this interval TI = to < t , < . . . < t, = T,. Assume these subintervals are so short that the 3-velocity is essentially constant on each one. Consider a subinterval [ t i t i ] . There is a Lorentz frame (2”)such that, during the interval t i - < t < ti we have the particle at rest in (2”).According to a clock in (2’) the time elapsed as t goes from t i - l to ti is approximately where ti lies in [ t i - l , t i ] . Since the particle is at rest in (X”),this elapsed time is the time which elapses on a clock carried by the particle. Thus, as t goes from TI to T,, the elapsed time, as measured by the particle’s clock, is given approximately by
the approximation becoming better as the lengths of all the subintervals go to 0. Thus, the exact amount of time which passes on the particle’s clock as t goes from T , to T , is dt.
(11.22)
If y is the spacetime metric and dxo dx’ dx2 dx3
then (11.22) is (11.23)
ACCELERATED MOTION-A
SPACE ODYSSEY
229
Thus we have the important fact: The proper time between two events on the world line of a particle is the elapsed time as measured by a clock traveling with the particle.
ACCELERATED MOTION-A
SPACE ODYSSEY
Let z -+x(z) be the world line of a particle, parametrized by proper time. We have defined the 4-velocity of the particle to be u = dx/dt.
We now want to define the 4-acceleration,
for each z. We define a(z) by giving its components in an arbitrary Lorentz frame (x”). If x(z) = (x”(z))in the Lorentz frame (x”) then we define a”(z) = d2x/dt2.If (9) is a Lorentz frame the components (iP(z)) are related to (a”(7))as the components of a vector should be, since the Lorentz transformations are linear. The vector a(z) is thus determined by giving its components in a Lorentz frame. Choose a Lorentz frame (x’). If u, w E TPM we have g(u, w) = U’W’,
the inner product of u and w. If u is the 4-velocity, then 1
(1 1.24)
u’a, = 0.
(1 1.25)
UQU’ =
and hence
Thus, 4-velocity and 4-acceleration are always perpendicular. Now, at any point on the world line (xp(z))we can construct a Lorentz frame in which the particle is instantaneously at rest. We call such a frame an instantaneous rest frame of the particle. In the rest frame at z = zo we have U’(5’) = (1, 0, 0, 0).
Now aw(zo)= (a’, a’, a’, a 3 ) so ( 1 1.25) implies a’ = a’u, = 0.
Thus, the 4-acceleration is (0, a’, a’, a3). The acceleration at any time is a’ = d2xp’/dz2,
230
11. THE SPECIAL THEORY OF RELATIVITY
so we have dx” dx’ dt dz dt dz’ d2xp- dx” d 2 t
dz2
d2x” dt
dt dz2+
dt2 (z)
(11.26)
‘
If we measure these components in a frame in which the particle is instantaneously at rest, then dx’ldt
= 0,
i
=
1,2,3;
dtJdz
=
1.
(11.27)
So we have, from (1 1.26) and (1 1.27): In the instantaneous rest frame of the particle, the spatial components of 4-acceleration are d2x‘/dz2= d2xiJdt2.
(11.28)
In the instantaneous rest frame of the particle, the 4-acceleration is (0, a) where a is the ordinary 3-acceleration in that reference frame. If la1 = a,then u”ap = - a 2 . Thus, the frame independent scalar uPupcan be interpreted as a measurable quantity by noting (- apu,,)’12is the magnitude of the acceleration measured by the observer riding with the particle. A rocket leaves the earth on a mission to explore deep space. It travels in a straight line away from earth. The acceleration of the rocket is used to create an artificial gravity equal to that on earth, i.e., 1 g. Thus, the rocket accelerates so that, in the instantaneous rest frame of the rocket the observer feels an acceleration of magnitude 1 g. Therefore, we have a”ap = - g 2 .
The rocket accelerates with the constant acceleration 1 g. as above. Note this does not necessarily mean that the 4-acceleration is constant. The rocket accelerates away from earth for 10 years. Then the rocket reverses its thrust so as to give a 1-g deceleration for 10 years. At the end of 20 years (these time intervals are measured by clocks on the rocket) the rocket is once again at rest with respect to earth. Then the process is reversed for the return to earth which takes another 20 years of rocket time. The rocket arrives back at earth having traveled for 40 years according to the rocket passengers. We consider the following questions: (a) How much time passed, as measured by clocks on earth, during the rocket’s absence? (b) How far from earth did the rocket go, as measured from the frame of the earth?
ACCELERATED MOTION-A
231
SPACE ODYSSEY
Let ( t ,x, y , z ) be the Lorentz frame of the earth (we ignore the fact that the earth is not quite an inertial frame). The rocket starts at the origin at t = 0 and moves in the positive x-direction. Then all yz-coordinates are always 0. In our calculations, t will be in meters so we will have to convert years to meters to get numerical answers. The following equations hold. Let t be elapsed time as measured by the rocket clock, so that t is proper time along the world line of the rocket. Then the following conditions hold: t=O
dt/dz
=
u0
dx/dt
=
u1
when when
x=O
(u"2
z = 0, z
=0,
u1 = O when z uo = 1 when t
= 0,
- (u')2 =
= 0,
1,
uouo - u'al = 0,
(a0)2 - ( d ) 2
=
-92.
Now (aO)2
= (a')2 - g2 = (uOuO/u~)2- 92,
so 0 2
(a
1 (u112 - ( a 0 2( u0 )2 -- -9
2
1 2
(u)
5
which gives -(a0)2 = -gZ(u')2.
Then we get u0 = gul, since a', u1 2 0 and a' = uouo/u1 = uOgu'/u' = guo.
so duO/dt= qu',
~'(0)= 1,
du'/dr
~ ' ( 0= ) 0.
= gu0,
(1 1.29)
The initial value problem (11.29) has the solution
u0(z) = cash gt ~ ' (= t ) sinh
gt.
Then t
= (l/g) sinh g T ,
x
=
(l/g) cosh gz
-
(l/g).
(1 1.30)
232
11. THE SPECIAL THEORY
OF RELATIVITY
FIGURE 11.1
If we view this motion in the tx-plane, then we see the world line of the rocket is (part of) a hyperbola as shown in Fig. 11.1. Note that the world line is asymptotic to the line x = t - (l/g) as t --* co. Suppose that, at t = 0, a light beam is fired after the rocket, beginning at x = - l/g. Then the conclusion is that, as long as the rocket continues its 1-g acceleration, the light beam will not overtake the rocket. See Exercise 11.2 for further discussion of this space journey problem. For obvious reasons, motion subject to constant acceleration as above is known, in relativity theory, as hyperbolic motion. We now answer question (a) and (b) posed earlier. ANSWERTO (a): Elapsed time is four times the elapsed time during the first 10-year segment of the trip. We see that 10 years = 9.46 x 10l6 m, since we have 9.46 x 1015m/year,
g = 9.8 m/s' = 1.09 x
(l/g) sinh gz
= 9.17 x
m-';
lOI5 sinh 10.3 = 1.36 x lo2' m.
so t = 1.36 x l0''m
=
1.44 x lo4 year.
Total elapsed time for the trip is 57,600 years! ANSWERTO (b): If we compute x when z tance from earth is twice as far;
=
10 years the maximum dis-
x = 9.17 x 1 0 ' 5 ( ~ ~ 10.3 ~ h- 1) =
1.36 x lo2' m
=
14,376 light-years.
So the rocket gets 28,800 light-years from earth!
233
ENERGY AND MOMENTUM
ENERGY AND M O M E N T U M Let (ct, x’(t), x2(t),x3(t))be the world line of a particle of mass m > 0. We introduce proper time z as parameter and write the world line as ( ~ ” ( 7 ) ) . Then the 4-velocity is (u”) = (dxP/dt). DEFINITION 11.9 The 4-momentum is (p”) = (muP).Thus
po
= myc
(where y
=
(1 - u ~ / c ~ ) - ’ / ~ ) .
(11.31)
pi = mu’y
The energy of the particle, as measured in the Lorentz
DEFINITION 11.I 0
frame (x”),is E
= poc.
In relativistic units (c = 1) we have PO = my, pi = mu’?,
E
= po.
(11.32)
Note that the 4-momentum satisfies (conventional units)
p’p,, = m2c2,
(1 1.33a)
(relativistic units)
p”,,
(1 1.33b)
= m2.
Consider the case of a photon. The photon has zero rest mass. Also it travels at the speed of light so that, if we define proper time t along its world line as in (1 1.21), then z = 0 for all r. Thus we can not reparametrize by proper time to define 4-velocity as we did previously. Indeed we shall not define the 4-velocity of a photon at all (nor does anyone else define such a vector). However, we d o define a 4-momentum vector ( p ” ) for a photon. To accomplish this, let ( x p )be a Lorentz frame and let u be a 3-vector (in the given Lorentz frame) which is a unit vector in the direction of motion of the photon. We need to introduce certain physical quantities which are
h h
Planck’s constant, =
h/21~,
o
circular frequency of the photon,
v
frequency of the photon,
i
wavelength:
k
= 2z/i
wave number.
Then the energy of the photon is taken to be E
=
h\j
( = h o = hc/i).
(11.34)
234
11. THE SPECIAL THEORY OF RELATIVITY
The momentum 3-vector (pi)is defined to be
(Pi)= (E/c)u and the ordinary 3-vector length of this vector is the magnitude of the 3momentum of the photon. The wave 3-vector (k') is defined by (k') = ku. We define the wave 4-vector to be
(k')
= (o/c,
k', k 2 , k3),
where (k', k 2 , k 3 ) are the components of the wave 3-vector. Finally, we define the Cmomentum vector of the photon to be (p') = h(k').
(11.35)
Note that in this case, as in the case of particles of positive mass, we have the relationship between energy and the 0-component of 4-momentum, E
= pot,
E
= PO.
or, in relativistic units,
In the case of positive rest-mass particles we have (11.33). The analogous result in the case of zero rest mass would be that
'P'P
= 0,
that is, the 4-momentum of a photon is a null vector. This is true and we refer to Exercise 11.3 for the details.
R E LATlVi STI C CO R R ECTl0 N TO NEWTONIAN MECHANICS The question arises as to the correction to Newton's laws in a given inertial frame W E 9,in the special theory of relativity. The correct result is obtained by simply replacing the usual 3-momentum in W by the relativistic 3-momentum (m dx'ldz). Thus, we get (d/dt)(relativistic 3-momentum) = force Of course, this force is a 3-vector dependent upon W. in 9.
235
CONSERVATION OF ENERGY A N D M O M E N T U M
Similarly, it is the total relativistic 3-momentum that is conserved in an isolated system. Finally, what about the time rate of change of the other component E of dpldz in W?We claim that dEldt = F v, which is the classical idea of energy and work. Note that it is the change in energy which is due to mechanical effects. These claims have been verified by experiment-for example, in particle accelerators.
-
CONSERVATION OF ENERGY AND M O M E N T U M Suppose we have a collection of particles and the sum of their 3-momenta (mdx'ldz) gives a total 3-momentum (Pi). Some of these particles may be photons, neutrinos, some positive rest-mass particles. These particles interact in some way and then we add up the 3-momenta of the particles after interaction. Call this total ( P i ) .The law of conservation of momentum says ( P i ) = ( P i ) .This is supposed to be true in every Lorentz frame. Now suppose E l , E , are the total energies of all the particles before and after the interaction, in a given Lorentz frame. We claim that it follows from the conservation of momentum that El = E,. To see this, note (El, P1),( E , , P,) are 4-vectors and, in every Lorentz frame, PI = P,. The following result then proves E l = E,. LEMMA 11.I 1 Let (b') be a 4-vector with (bl, b2, b 3 ) = 0 in every Lorentz frame. Then bo = 0 so (b') is the zero vector.
PROOF: Let (6') be the components of b in a frame related to (xp) by a boost in the xl-direction. Then
6'
= ybo
-
Pyb',
-
b2 = b2, The second equation says 0 = -bybo
-
b'
=
-Pybo
+ ybl,
b3 = b3.
+ 0, so bo = 0. I
REMARK 11.I 2 Similarly, if energy conservation is assumed, then conservation of 3-momentum follows. Thus we see that actually total 4-momentum is conserved. We state this as the
FUNDAMENTAL LAWOF NATURE: In any physical interaction the total 4momentum of the particles involved is the same before the interaction as after.
236
11. THE SPECIAL THEORY OF RELATIVITY
MASS AND ENERGY The energy of a particle of mass m is E = poc = mc2y, where y = (I - p2)-lI2 and /3 = u/c with u the ordinary spatial speed of the particle (in the given Lorentz frame). If the particle is at rest in the Lorentz frame, then we see E = mc2. Thus, a particle at rest has, as a consequence of its mass, a certain amount of energy given by the above well-known formula. In our discussions, m has always denoted the so-called rest mass of the particle, a scalar quantity having the same value in all Lorentz frames. Sometimes a relativistic muss Z is defined by f i = my. Then, the energy of the particle is always related to Z by E = Zic2. The difference between the energy of a particle measured in a Lorentz frame (XI’)and the rest energy mc2 of that particle is called the relativistic kinetic energy and is given by E , = E - mc2. See Exercise 11.5 for the relations with the Newtonian concept of kinetic energy. The law of conservation of 4-momentum says the total energy of a collection of particles does not change if they interact. However, the amount of rest energy (energy due to rest mass) and the amount of kinetic energy need not to preserved. The following simple example illustrates this.
CHANGES IN REST MASS Energy is conserved in interactions, but there is no law which says the total rest mass of the particles must be the same before and after an interaction. EXAMPLE 11.I 3 Suppose two objects, each of rest mass m, move toward each other and collide, forming a single object as depicted in Fig. 11.2. Specifically, let the 4-momenta of the particles before the collision be (2m, 2m($/2)i), (2m, -2m($/2)i), where i is the unit vector in the x-direc/ ~2 to arrive at the above tion. We are using v = &2 so y = (1 - ~ * ) - l = 4-momenta. After collision the 4-momentum of the resulting particle must be
(4m, 090, O),
237
EXERCISES
FIGURE 11.2
so the rest mass of the particle formed will be 4m, rather than 2m as might have been expected. The kinetic energy of the initial particles is converted into rest mass of the resultant particle.
SUMMARY The spacetime of special relativity is a four-dimensional Lorentzian manifold. The set which forms spacetime is defined as the set of all “events,” where “event” is left as a primitive notion. The topology and differential structure are determined by introducing the notion of Lorentz frames on spacetime, which provide charts. The Lorentz (or, more generally, Poincare) transformations enter as the coordinate transformations relating pairs of Lorentz frames. We see that the 4-vectors of special relativity are precisely the tangent vectors corresponding to this differential structure. We note that threedimensional vectors, such as ordinary velocity, are not well-defined objects on the spacetime manifold; rather they are tied to choices of coordinate system. We define 4-momentum and note how deep consequences result from the fact that energy and 3-momentum form a 4-vector; for example, the (unforeseen) fact that conservation of energy and conservation of (relativistic) 3-momentum actually imply one another.
EXERClS ES 11.1 Refer to the discussion in the text of relativistic time dilation. Show that, if events a, and u2 are the successive ticks of one of the clocks in the (x”)frame, then the time difference between these events, as measured in the frame (?), is shorter by the factor (1 - fi2)-1/2than the time interval as measured in (xu). This shows the two reference frames obtain the same results when measuring each other’s clock rates, as required by the principle of relativity. 11.2 (a) Refer to the discussion of the space odyssey in the text. Suppose a rocket is to travel in the x-direction with constant 1-g acceleration. Let a light beam be fired after the rocket starting, at t = 0,
238
11. THE SPECIAL THEORY OF RELATIVITY
some distance behind the rocket. How many miles behind the rocket must the light beam start, in order that it never catch the rocket? (b) In the space journey in the text, replace each 10-year segment by a 20-year segment. How far from earth does the rocket get? If the rocket leaves in the year 2000 A.D., what year will it be when the rocket returns to earth? 11.3 Show that the 4-momentum of a photon is a null vector by verifying that p”pp = h2(W2
- k2)
=0
11.4 Verify Remark 11.12 by proving: If (bp)is a 4-vector such that bo = 0 in every Lorentz frame, then b’, b2 and b3 must also vanish in every Lorentz frame and hence (b’) = 0. 11.5 Let u be the ordinary spatial speed of a particle of mass m as measured in a Lorentz frame (x’). Show that the energy E can be expressed as a sum of terms equal to (a) the rest energy mc2, (b) the classical kinetic energy +mu2, (c) terms proportional to ( u / c )which, ~ for Newtonian speeds u
BAtN).
(13.4)
The right side of (13.4) is a state of the system in R3N x R 3 N .We would describe this state by saying that the instantaneous velocity is the injinitesimal rotation B applied to the instantaneous configuration ( A t , , . . . , A t N ) . The mapping T@ p is called the space representation of the state space. The matrices A, B which correspond to a given state under this mapping have direct interpretation in terms of the observed motion of the rigid body in physical space. On the other hand, the mapping T @ 1 is called the body representation of state space. This representation is useful in describing mathematical (global) properties of the motion of a rigid body, as we shall now see. 0
0
260
13. MECHANICS OF RIGID BODY MOTION
Recall from the general development of Lagrangian mechanics we have a kinetic energy metric T defined for states u = (xl,.. . , x N ,u l , . . . ,uN), w = ( x l , .. . , x N ,wl,. . . , wN) by T,(u, w) = I mi(ui, wi). Then the kinetic energy function was defined on state space by T(u)= $T,(U,u) for u as above (see Definitions 7.1 and 7.2). If we want to obtain the motion of the rigid body in the absence of external forces, we simply use T as the Lagrangian and proceed as usual in classical mechanics. We first use @ to pull the structures over to G. Define h = @*T to be the metric on G corresponding to the kinetic energy metric on configuration space under the mapping @. Thus
xyz
h(A)(P,4 = T(T,@(P),TA@(4), PROPOSITION 13.6
for all A,
E
B, 6 E TAG.
(13.5)
h is a left-invariant metric on G, that is, h = Lz,h
G.
PROOF: Pick A , , A E G, 8,6 E TAG.Then we want to show
Note we have used the fact that A , preserves the Euclidean inner product < >
>. I
Now the motions of the rigid body that we see in physical space correspond to curves A(t) in G such that the positions of the masses at time t are A ( t ) t , , . . . , A ( t ) t N .We called such curves in G space motions of the rigid body. THEOREM 13.7 The space motions of the rigid body subject to no external force are the geodesics of the left-invariant metric h on G. PROOF: As shown in Example 7.9, the trajectories of a holonomic system without external forces are the geodesics in configuration space for the kinetic energy metric. Under our mapping @ the group G corresponds to configuration space and the metric h corresponds to the kinetic energy metric. Thus the geodesics of our two metrics correspond under @ so the theorem holds. I
261
GEOMETRY OF RIGID BODY MOTION
THE GEOMETRY OF RIGID BODY MOTION We now present an outline of the constructions involved in our analysis. The required details and definitions are then given in the following pages. We encourage the readers to consult this outline while studying the rest of this chapter to avoid losing their way in a mass of details. (1) O n G = SO(3) there is a left invariant metric h such that if A(t) is a geodesic, then y ( t ) = ( A ( t ) t l , . . . , A ( f ) t Nis) a motion of the rigid body. The geodesics are called space motions of the body. (2) Using 1:G x T,G + TG, construct the Hamiltonian system
2
=
(G x T,G,
6, H )
whose Hamiltonian trajectories are those curves (A(t),B(t))= I-'(dA/dt), where A(t) is a space motion. In matrix form this gives B(t) = A(t)-'k(t). (3) G acts as a group of symmetries on 2 which leads to the momentum map p: G x T,G + (T,G)* given by p ( A , B)(C) = h,(B, A d ( K ' ) ( C ) )satisfying If A ( t ) is a space motion, then p(A(t),A ( t ) - ' k ( t ) )= p o is constant. po is called the conserved momentum of the trajectory.
(4) Let p o E (T,G)* be fixed. Suppose (A(& B(t))is a Hamiltonian trajectory having momentum po. The equations h,(B(t), Ad(A(t)- I)( )) = p o , B(t) = A(t)-'k(t) lead to two reductions of the differential equations on G x T,G. We use #:(T,G)* 4 T,G by h,(v*, B ) = v(B) and b = # - I . (a) Let Z,,(A) = TL,((Ad*(A-')(p,))#). Then A(t) is a geodesic with conserved momentum po iff A(t) is an integral curve of Z p 0 . (b) Consider p(t) = B(t)b = Ad*(A(t)-')(po) = p o . Ad(A(t)) in (T,G)*. For any C E T,G, p(t)(C)= p(t)([p(t)*,C ] ) so that p(t) is an integral curve of the vector field Y , on (T,)* given by Y,(v)(C) = v([v#, C ] ) . Such integral curves are called body motions. (5) Given a body motion p ( t ) there is a unique space motion with the property A(0) = e associated with it. For then po = p(0) and A(t) is determined by Z,, and initial condition A(0) = e. Such a space motion is called a special space motion. If A(t) is any space motion then A(t) = A(O)A"(t)where A(t) is a special space motion. (6) The vector field Y , is tangent to each Ad* orbit in (T,G)*. Thus once po is chosen we are reduced to X , the Ad* orbit through p o . Such an orbit carries a natural symplectic structure o by o(v)(tB(v), tc(v)) = v([B, C ] ) , where tB(v)( ) = v([B, ( )I) V B E T,G. Defining T(v)= $h,(v#, P),we get a Hamiltonian system ( X ,o,7')and Y , is the Hamiltonian vector field.
262
13. MECHANICS OF RIGID BODY MOTION
(7) Special properties of SO(3).We have a: R 3 + T,G by a(u)(w) = u x w. Then [cT(u),a(w)] = a(u x w ) and Ad(A)(au) = ~ ( A u )Define . Z (T,G)* + R 3 by ( 6 ( p ) , u ) = r*(p)(u) = p(au). Then 6(Ad*(A)p) = A6(p). (a) Define a symmetric positive definite matrix R by h,(ou, aw) = (Ru, w) Vu, w E R 3 , and let eigenvectors be u,, u,, u, with eigenvalues I,, I,, I,. Let S = R - ' . Then b = 6 - ' R K 1 , # = oS6. (b) X, = {p E (T,G)*l(a"(p),&)) = r 2 } is an Ad* orbit. Let TE = {P E (TeG)*I T(P)= E } .
Then C?(TE) = { u E R31(u, Su) = 2 E ) = {&Ui IX I ; ( U i ) 2 = 2 E } is an ellipsoid. Every body motion is contained in some T , n .X,, where E is the total energy and r is the total momentum. (c) p ( t ) is a body motion iff m(t) = 6 ( p ( t ) ) is an integral curve of Y(u)= u x Su. Thus, eigenvectors of S give trivial body motions which give stationary rotations of the rigid body. If m(t) = rui V t then A(t)ui = ui V t . (d) Let v ( t ) be a body motion and A(t) be the special space motion determined by v(t). Define W(t)= Ad(A(t))(v(r)$)
and
w(t) = a- '(W(t));
o(t)is
the usual angular velocity Vector (why?). (8) Interpretation: (a) The Kinetic energy metric h on G , when represented on G x ( R 3 x R 3 ) using p (1 x a): G x R 3 + TG, becomes 0
so I(A)(n,n) gives the moment of inertia of rigid body about n, n a unit vector. (b) I(A)(u, w ) = (R,u, w ) defines R , . Its eigenvectors u,(A) diagonalize I(A) and are the principal axes for ( A t , , . . . , AtN);its eigenvalues are the moments of inertia about these axes. (c) I(A)(u,w ) = I(e)(A-'u, K ' w ) gives R , = A R A - I . (Note that Re = R.) Thus u,(A) = Au,; these are called body axes. (d) 6p(A(t),A ( t ) - ' k ( t ) )= C m i ( A ( t ) t ix k(t)ti)= 6(po); m(t)= A(t)-'(C?p,,) gives the motion of the angular momentum vector as seen by an observer moving with the body. With these remarks in hand we now resume the main flow of the development. Since h is left-invariant it is useful to use the body representation 1:G x TeG -,TG in studying the geodesics of h. Let & , be the metric on the trivial
263
LEFT-INVARIANT 1-FORM
vector bundle G x T,G
+G
determined by h and 1.Thus, by definition,
@ ( A , B),( A , C ) )= h(TLA(B),TLA(C))= h,(B, C),
where the second equality follows from left-invariance of h. Using the metric h on G, we can construct, in the usual way, a Hamiltonian system (TG, Q h , h), where the function h: TG -+ R is the kinetic energy function for the metric h, h(v) = ih(u, v). Recall the metric h induces a Legendre transformation 9: TG + T*G, and Qh = 9 * R , where R is the canonical 2-form on T*G. The trajectories of this Hamiltonian system are the curves c’(t)in TG where c ( t ) is a geodesic of h in G. Now using I., we construct an equivalent model, an isomorphic Hamiltonian system on the manifold G x T,G. The symplectic form on G x T,G is j”*Rh= fi and H : G x T,G + R is H = h 1, where h: TG + R is the kinetic energy as above. We have the formula 0
(1 3.6)
H ( A , B) = +h,(B, B).
We consider the form fi in more detail. Recall on T*G there is the canonical 1-form 8 given by @a) = CI Tat*:Ta(T*G)+ R, for a E T*G. Let 8 = ;i*2?*8. Then -d8 = lb*9*( - do) = 1*9*(R) = i*Rh= fi. For the form we have the formula 0
e
&A, B ) = [ 6 p ]*(A,B)] 0 Tn,
where z: G x T,G
+G
(1 3.7)
is projection on the first factor. Now
= h(TeLA(B), [T A(A>B)](w) = Y(T,LA(B))(w)
w, = he(B,
TALA-l(W)).
LEFT-INVARIANT 1 -FORM DEFINITION 13.8 For A E G let O(A):TAG T,G be defined by 8(A) = TALA-,. 8 is called the cunonicul left-invariunt TeG-valued 1-form on G . Note that we then have --f
$(A, B) = h,(B, B(A) Tz( -)).
(13.8)
0
The blank is to be filled by a vector in T(A,B)(Gx T,G). We now have the Hamiltonian system X = (C x T,G, fi, H ) . This is isomorphic to the Hamiltonian system (T*G,R, h) under the mapping 3’i. 0
PROPOSITION 13.9 The trajectories of the Hamiltonian system X correspond under A: G x TeG + TG to the curves A’(t),where A(t) is a geodesic of the metric h on G.
PROOF: This is left as an easy exercise using previous results.
I
264
13. MECHANICS OF RIGID BODY MOTION
SYMMETRY GROUP We now exploit the fact that G acts on the Hamiltonian system 2 as a group of symmetries to obtain conservation laws which we shall see determine the trajectories of %. PROPOSITION 13.10 For A , E G define C,,: G x 7',G [,,(A, B) = (A,A, B). Then we have
+G
x T,G by
(a) the diagram
commutes, (b) L'fo@= @, (c) H L', = H . 0
The proof is left to the reader. = so (.,, is an isomorphism of X onto 8; A , + t,, It follows that defines a left action of G on 8as a group of isomorphisms. We call this the symmetry action of G on 2. Let B , E T,G. Define the vector field tBoon G x T,G by
{?,,a a,
(13.9)
(13.10)
ADJOl NT REPRESENTATION DEFINITION 13.11 The adjoint representation of G is the homomorphism Ad: G + Aut(T,G) given by
Ad(A)(B)= A B A - ' . This definition makes sense since G is a matrix group. An equivalent definition is Ad(A)(B)= TL,TR,- ,(I?). LEMMA 13.12
vB,(A, B) = h,(B, Ad(A-')(B,))
(13.11)
265
MOMENTUM MAPPING
exp(tB,)A
dr
=
TR,(B,).
*=,
MOMENTUM MAPPING DEFINITION 13.13 The momentum map associated with the symmetry
action of G on 2 is p: G x T,G
+
TZG,
defined by p(A, B)(C)= h,(B, Ad(A-')(C)). THEOREM 13.14 If ( A ( t ) ,B ( t ) )is a trajectory of the Hamiltonian system 2,then B(t) = A(t)-'k(t)and
d p ( A ( t ) ,B ( t ) )= 0 dt
-
(conseraation of momentum). PROOF: B(t) = A - l ( t ) k ( t ) , since ( A ( t ) ,B ( t ) )= X ' ( d A / d t ) by Proposition 13.9. Let X , = ( d H ) # , where # is with respect to a. We want to show dp(X,) = 0, which will prove the theorem, since the trajectories of 2 are the integral curves of X,. Since p takes values in TZG, we view p as a TZGvalued 1-form on G x T,G and hence, dp(X,) takes values in TZC. Let B, E T,G. We must show ( d p ( X I f ) ) ( B , )= 0. Now the following diagram commutes:
G x T,G-T,TG
by Lemma 13.12, where euB,(u)= cc(B,). Also, euBo is linear so that
d(eu,,,
7
p ) = euBo dp. 0
13. MECHANICS OF RIGID BODY MOTION
266
COADJOI NT REPRESENTATION There is another description of the momentum map. DEFINITION 13.1 5 The coadjoint representation of G on TZG is given by Ad*(A)(4) = 4 Ad(A-'). Note that, because of the A-' on the right side, the coadjoint representation gives a left action of G on TZG. Using the i4 and b mappings corresponding to the metric h, on T,G we then have 0
LEMMA 13.16 p(A, B) = Ad*(A)(B,) for all ( A , B) E G x T,G. The proof is left as an exercise.
SPACE MOTIONS WITH SPECIFIED MOMENTUM DEFINITION 13.17 Let A(t) be a space motion of the rigid body. The element po = p(A(t), A ( t ) - ' k ( t ) )in T,*G (which is independent oft) is called the conserved momentum of the motion. REMARK: po can be identified as the conserved total angular momentum of the system (see Eq. (13.30)).
We now show that if the conserved momentum pLois specified, the equations of motion reduce to a first-order differential equation. THEOREM 13.18 Let po E TZG. Suppose that A(t) is a curve in G which satisfies the differential equation
dA/dt
=
TLA((Ad*(A-')(po))').
(13.12)
Then p(A(t),A - '(t)k(t))= po for all t in the interval I on which A is defined and A(t) is a geodesic of h. PROOF: The differential equation gives
A-'(t)k(t) = (Ad*(A-'(t))po)'
267
COADJOINT ORBITS AND BODY MOTIONS
so that A d * ( A ( t ) ) [ ( A - ' ( t ) k ( t ) ) = , ] p o . But then, by Lemma 13.16, we have
p ( ~ ( t )A, - ' ( t ) k ( t ) )= p o Now let
A
for all t i n I .
be a geodesic of G such that
(a) ! ( t o ) = W O ) , (b) A'(t0) = T~,,,,,((Ad*(A(tO)- l)Po)Y? where to is some given point in I . By Theorem 13.14, vo = p(a(t), k ' ( t ) A " ( t ) ) is independent oft. Now vo = Ad*(A(t))((A-'a(t)),) by Lemma 13.16 so dA"/dt
=
T L J ((Ad*(A-')vo)').
Choose t = to. Put t = to to get v o = po. The uniqueness theorem for differential equations now shows A = A" locally, so A is a geodesic. I COROLLARY 13.19 A curve A ( t ) in G is a geodesic of h if and only if (d/dt)p(A(t),A - ' ( t ) k ( t ) )= 0 for all t .
The proof is left as an exercise. DEFINITION 13.20
For p o E T,*G the vector field Z,, on G is defined by Z,,(4
=
TL,((Ad*(A- l)Po)fl).
We then have COROLLARY 13.21 A curve A ( t )in G is a space motion of the rigid body with conserved momentum po if and only if A ( t ) is an integral curve of Zp,.
COADJOINT ORBITS AND BODY MOTIONS The preceding results, in particular the form of the vector field Z p 0 ,suggests the importance of the orbits in T,*G under the right representation of G given by A + Ad*(A-I). Of course, as point sets, these are the same as the orbits of the coadjoint representation. Let X be such an orbit. We will show that X can be given the structure of a Hamiltonian system in such a way that the Hamiltonian flow determines space motions of the body with conserved momenta in X . There is an operation on T,G called the bracket. This will be discussed in more general terms in the next chapter. For now, recalling that we are viewing T,G as the set of skew-symmetric 3 x 3 matrices, we define [B, C]
=
BC
-
CB.
It is easily checked that [B, C ] lies in T,G if B and C do.
(13.13)
268
13. MECHANICS OF RIGID BODY MOTION
DEFINITION 13.22 ad: T,G -+ L(T,G, T,G) is given by ad(B)C = [ B , C]. We have Ad: G -+ L(T,G, T,G) and, since L(T,G, T,G) is a vector space, we may review T , Ad as a map T , Ad: T,G -+ L(T,G, T,G). Then we have
PROPOSITION 13.23 T , Ad@) = ad(B). The proof is left as a straightforward exercise. Now we define, for each B E T,G, a vector field tBon TZG. We shall view this vector field as a map tB:TQG -+ TZG. DEFINITION 13.24 For B
E
T,G let
tBbe given by
t B ( v )= v ad(B). 0
LEMMA 13.25 t B ( v )= d/dt(,=,,[Ad*(exp(-tB))v]
Again, the proof is left as a simple exercise. LEMMA 13.26 If v E T,*G and X is the Ad*-orbit containing v, then every vector in T v X is of the form (,(v) for some B. Also, if tB(v) = tc(v), then v ( [ B - C, D]) = 0 for all D E T,G. PROOF: The first assertion will be proved in greater generality in the next chapter so we omit it here. If rB(v) = tc(v), then v([B, D]) = v([C, D]) for all D so v ( [ B - C,D]) = 0 for all D. I
DEFINITION 13.27 T:T,*G -+ R is defined by
T ( v )= ;h,(v', v'). If we define an inner product h^, on TZG by &?(/A 4 = he(@, v'),
then we have
T ( v )= ~ v ( v ' = ) f h , ( ~ ' ,v')
= $,(v,
v).
DEFINITION 13.28 Denote by Y, the vector field on TZG given by
YdV) = t"'(V). PROPOSITION 13.29 Y, is a smooth vector field and if X c T,*G is an orbit of the coadjoint representation, then, for v E X , Y,(v) is tangent to X . We leave the proof as an exercise.
269
COADJOINT ORBITS AND BODY MOTIONS
DEFINITION 13.30 Let a,p E T V X .Define o(v)(a,p) = v([B, A ] ) , where
A, B
E
T,G are such that CA(v) = a, = + ( r 0 ~ 3 I, T ' T ~ U ~ = ) +($I,).
Thus, ro = (213E0)1'2,that is, the sphere ( u , u ) = rg and the ellipsoid given by (v, Sv) = 2E0 are tangent at the points f r , ~ , . Now consider the body
278
13. MECHANICS OF RIGID BODY MOTION
principal a x i s determined by I,, \3V I /
FIGURE 13.3
motion v(t) and let T(v(0))= E , (C(v(O)), C(v(0))) = r2. Then v(t) always lies in the intersection of the sphere ( u , u ) = r2 and the ellipsoid ( u , Su) = 2E. Now since 1;’ is the smallest eigenvalue of S, we get (u, S u ) 2 Z;’ ( u , u ) so 2E 2 1; l r 2 . Thus, (2EZ3)”’ is just slightly greater than r (or (2E13)1/2= r, but never do we get (2EZ,)’I2 < r). Thus C(v(t)) lies on the intersection of a sphere of radius r and an ellipsoid having largest principal half-axis (2EZ3)’/’ with 0 < (2EZ3)’/2- r ( 2 ) = ~ ( x+ ,AXH, X + PXH) + dH(Xt + AXH)p
+
+ p x , ) i a,(X, + A X H )-~ a,( + ~ X H ) L Y,) + A d H ( Y ) - /.diH ( X , ) d H ( x r ) p - dH( x ) A + l b x H ) P - a,(y pxH)A = w ( x , , T) a , ( x , + 2XH)P - ‘%,(k; pxH)A. - dH(
= w(X,,
+
+
+
+
+
Suppose 2 = X , + Z lies in ker 4.Then
+ xH)p - a t ( k ; + pxH)
+
w(xt,
=
for all p, Y,. Taking p = 0, we get w ( X , , Y,) = at(Y,)for all Y,, so ix,o= a,. Suppose, conversely, that iXlw= a,for all t . Taking A = 1, we get
cn(2,y + pz)
= w(x,>
X) +
+ xH)p
-
I:+ pxH),
Taking p = 0 gives d(2, y) = w ( X , , Y) - a,(Y,) = 0, since ix,w = a,.Taking y = 0, p = 1 gives &(Z,2)= a,(X, X , ) - a,(XH)= a,(X,) = 0 again, because ixtw= a,.Since any vector is y + pZ for some y, p, we are done. I A
+
Consider ( R 2 ,dx A d p , H ) ,
EXAMPLE (One-dimensional harmonic oscillator)
+ k 2 x 2 ) .The associated GMS is ( R 3 ,(3), where 6 = dx d p + ( p d p + k ’ x d x ) dt.
where H ( x , p ) = + ( p 2
A
A
Let us perturb
(3
by adding the term f ( x ) p dx A dt, so that
6 = d x A d p + ( p d p + k2x d x ) A d t
+ f ( x ) p dx
A
dt.
We compute the kernel of 6,using Theorem 15.13. We have
ker 6 is, as shown before, spanned by p-
a
ax
-
a
a
ap + -.at
k2x -
According to Theorem 15.13, we have ker 4 spanned by p
a --
ax
a k2x ap
+ atd + X , , -
where iXtw= f ( x ) pdx. It is easily seen that X , = - f ( x ) p d/dp, and we get a 2 a a 2=p - ( k x + f ( x ) p )- + -.
ax
ap
at
304
15. GEOMETRICAL MODELS
Any trajectory of the GMS (N, h)can be parametrized by t , i.e., as (x(t),p(t), t). Then @ = - k 2 X -f(x)p.
i= p ,
Thus, trajectories of the perturbed GMS are solutions of X
+f(x)i
+ k2x = 0.
FIXED ENERGY SYSTEMS In certain important situations there is no natural choice of a global time coordinate. The following result says that surfaces of constant energy in a Hamiltonian system have a natural GMS structure. THEOREM 15.14 Let ( M , o,H ) be a Hamiltonian system, E a constant, and S, = { p E MIH(p) = E}. Assume dH never vanishes on S, so S, is a smooth codimension-one submanifold. Then (S,, o)is a GMS. PROOF: Let X , be the Hamiltonian vector field on M . Then dH(X,) = w(X,, X H )= 0, SO X , is tangent to S,. We now show that if X E T,SE, then o ( X , Y) = 0 for all Y E TpSEif and only if X is a multiple of X,(p). For if Y E T,S,, then w(X,(p), Y) = dH(p)Y = 0, which proves half of the assertion. For the converse, suppose w ( X , Y) = 0 for all Y E TpSE.Choose W E T,M with dH(p)W# 0. Then w(X,(p), W ) # 0. Let y = -o(W, X ) / w(l/t:X,) and Z = yX, X . Then if YE TpSE, we get w(Z, Y) = ~ o ( X , , Y) w ( X , Y) = 0, while o(Z, W ) = yo(X,, W ) o(X, W ) = 0 by choice of y. So w(Z, Y) = 0 for all Y E T p M so that nondegeneracy of o gives Z = 0. Hence X = -yX,, as asserted. I
+
+
+
A very nice example of the preceding theorem is the following application to pseudo-Riemannian manifolds. EXAMPLE 15.15 Let ( M , y) be a pseudo-Riemannian manifold and w = Q, the canonical 2-form on T*M. Let E # 0 be a constant, SE = { p E T * M ( g(p, p ) = E}. Then (S,, w ) is a GMS. The trajectories of this GMS cover (not necessarily affinely parametrized) geodesics on M . This last assertion follows from the fact that the trajectories are tangent to the Hamiltonian vector field X,, where the Hamiltonian is H ( p ) = g(p, p ) (see Example 7.9 and Exercise 15.8).
305
CONFIGURATION PROJECTIONS
CONFIGURATION PROJ ECTIONS In cases such as the preceding, we can no longer use Theorem 15.13 to consider the effect of perturbing the 2-form in a GMS. We introduce a new concept relevant to this. DEFINITION 15.16 Let ( M , o ) be a GMS. A smooth mapping onto n: M -r X is said to be a configuration projection if, for every 2-form 6 on X , ( M , w + n*6) is a GMS. REMARK: See Exercise 15.10
EXAMPLE 15.17 Let ( M , y) be a pseudo-Riemannian manifold. We have the GMS (S,, w ) as in Example 15.15. Then z*: S, + M is a configuration projection. To see this let 6 be a 2-form on M . First we argue that I5 = w + (r*)*S is nondegenerate on T,(T*M). This is easily seen by looking at the matrix of the form in a natural cotangent bundle chart. If (qi,pi) are coordinates, then o = dq' A dp,, (z*)*6 = fiij dq' A dq', so w (t*)*6 has matrix
+
which is clearly nonsingular. Now Y -r A( Y ) = (z*)*6(X,, Y ) is a linear functional on T,(T*M) for each p so there is a unique 2,such that
6(2,, Y ) = -A(Y).
(15.7)
Then we have 6 ( X H + 2,Y ) = w ( X H ,Y ) + (z*)*6(X,, Y ) + &(Z,Y ) = w(X,, Y), by Eq. (15.7). Thus, for any Y E T,T*M, we have shown d(X,
+ z,Y ) = W ( X H , Y ) .
(1 5.8)
From (15.8) we conclude first that 2,E T,S,; for this we need w(X,, 2,)= 0. But from Eqs. (15.7) and (15.8), cu(X,, 2,)= 6 ( X H+ Z,, 2,)= 6(X,, 2,) = -d(Z,, X,) = (r*)*6(X,, X,) = 0. Next, 2, X , is in ker d on S,, for if YE T,S,, then & ( X , + 2,Y ) = a,(X,,, Y) = 0. To prove that I5 restricted to T , S , has a one-dimensional kernel, we show that if S(z, Y ) = 0 for all Y E T,,S,, then 2 = A(X, + 2) for some A. This is true because Y+ B ( X , + Z , Y ) and Y-r G(2, Y ) are both linear functionals on T,(T*M) and both vanish on the codimension-one subspace T p S E .Therefore, there is a y such that y 6 ( X I f + 2, Y ) = d(Z, Y ) for all Y E T,T*M. So
+
306
15. GEOMETRICAL MODELS
+ Z ) , Y ) = 0 for all YE T,(T*M). Since ii, is nondegenerate on T,(T*M), we are done.
6(2- y ( X ,
LORENTZ FORCE LAW EXAMPLE 15.18 (Continue the preceding example.) T*: S, --+ M is a configuration projection. If M = R4 and g is the usual Lorentz metric of special relativity, then let
6 = -(q/2)F,, dx'
A
dx',
where ( F J are the components of the usual electromagnetic field tensor. Then, for m > 0, ( S m Z , o+ (z*)*6) is a closed GMS which gives the motion of a particle of charge q and mass m moving according to the Lorentz force law. The proofs of these facts are left as an exercise.
PSEUDOMECHANICAL SYSTEMS We now introduce a more general type of system which is useful in constructing geometrical mechanical systems to describe interactions. DEFINITION 15.19 A pseudomechanical system is a triple ( M , o,H ) , where o is a 2-form on M and H is a smooth, real-valued function on M such that
ker(o(m)) c ker(dH(m))
A trajectory of ( M , o,H ) is a curve y: I
--+
for all m in M . M such that
o(y(t))(j(t),-) = dH(y(t))
for t in I .
If ker(o) is an integrable subbundle, we say ( M , o,H ) is integrable. If d o = 0, we say that ( M , o,H ) is closed. REMARKS 15.20 (a) For any trajectory y(t), H(y(t)) is constant. This follows from ( d l d w w ) ) )= dH(Y(t))(j@)) = w(y(O)(j(t),W ) )= 0. (b) By Theorem 9.79, if d o = 0 and ker(w) has constant dimension, then ( M , w, H ) is integrable. If ( M , w, H ) is integrable, then there is a trajectory through every point. (c) If ker(o) # (0) on M , then there may be many trajectories through a point m E M . If dH(m) # 0 none of these trajectories is stationary at m. A system for which dH = 0 everywhere admits the constant trajectory at every point. If dH = 0 and dim ker w(m) = 1 Vm E M , then the nowhere stationary trajectories of ( M , w, H ) are exactly the trajectories of the GMS, ( M , o).
RESTRICTION MAPPINGS
307
NOTATION:We shall use PMS to mean pseudomechanical system. An IPMS is an integrable PMS.
R ESTR l CTl ON MAP PI NGS A PMS is often only an initial step in the geometrical description of a physical system. A simpler (or a related) PMS, and eventually a GMS, can be obtained using (or breaking) symmetries which occur in the system. Such constructions are determined by a restriction mapping, which we now describe. DEFINITION 15.21 Letf: M + N be a smooth mapping and rn E M . We say that f ( m ) is a proper value off i f f - '(f(rn)) is a submanifold of M . REMARK: By Theorem 3.19, we know that if T,fis surjective for each f(rn) is proper value o f f . A famous theorem of Sard
x ~ f - ' ( f ( r n ) )then ,
implies that iff is onto, then the set of points on N which are not proper values is a set of measure zero. See [32, pp. 45-50]. DEFINITION 15.22 A restriction mapping for an IPMS, ( M , w, H ) is a smooth map p: M + N , where N is a smooth manifold, so that for each trajectory rn: [a, b] + M there is a trajectory % [ a , b] + M satisfying
(a) G ( 4 = m(4, (b) Pu(Gi(t))= P ( r n ( 4 ) V t E [a, bl, (c) z ( f i ( t ) )= n(rn(t)),where n: M + M/ker(w) (see Lemma 9.90 and remark following Theorem 9.91). If n E image(p) is a proper value of p, the system (p- '(n),o l p - ' ( n ) , H lp- ' ( n ) ) is called a p-restricted system of ( M , w , H ) . This is a PMS (see the following lemma). LEMMA 15.23 Suppose p: M + N is a restriction mapping for the IPMS ( M , w, H ) and n E image(p) is a proper value of p. Then for each rn E p-'(n), ker(o Ip- '(n))(m)c ker dH(rn). PROOF: Let rn E p-'(n). If dH(rn) = 0, then the result is clear. Suppose dH(rn) # 0. Choose a trajectory rn: [0, 61 + M such that m(0) = m (Exercise 5.8). There is a trajectot-y K(t) such that G(0) =.m and p(G(t))= p(m) = n for 0I tI 6. Now w(m)(Gi(O), - ) = dH(rn(t)), so G(0) # 0 and, since G(0) E T,,, ( p - ' ( n ) ) , we see that ker(wIp-'(n))(rn) c ker(dH(m)). I REMARKS 15.24 As indicated above, restriction mappings occur in different contexts. In each case, however, the structure of the p-restricted
308
15. GEOMETRICAL MODELS
systems is important. Examples are: (a) If ( M , a,H ) is a Hamiltonian system, then ker(o(m)) = (0) for all m E M , so that k ( t ) = m(t) and thus, p gives conserved quantities. The construction of such a p from a Lie group of symmetries has been described in [l] and [27]. We did this in Chapter 13 for the rigid body and we will describe the p-restricted systems below. (b) If ( M , w, H ) is a Hamiltonian system such that dH(m) # 0 for all m E M , then H : M -+ R is a restriction mapping, each H(m) is a proper value, and the p-restricted systems determine geometrical mechanical systems which give the trajectories of the original system (see Theorem 15.14). (c) If ( M , o,H ) is such that d H = 0 everywhere, then every trajectory m: [a, b] + M is ker(o)-related to the stationary trajectory, fi(t) = m(a) for t E [a, b], and thus, any smooth mapping p: M + N is a restriction mapping. This will arise in the case of gauge symmetries. There, the initial dynamics is only that of symmetry, but a particle will determine a restriction mapping p: M + N so that the p-restricted systems are geometrical mechanical systems which determine the motion of the particle interacting with the gauge fields.
RIGID BODY AND TORQUE We now want to consider the rigid body in relation to our discussion of forces and restriction mappings. The model we used (in Chapter 13) for the rigid body was (G x T,G, w, H ) where G = S0(3), w = - dg, &A, B)(k,B) = h,(B, K'k) and H ( A , B ) = $he(& B). See Chapter 13 for details.
where p: G x T,G + T,*G, given by p ( A , B)C = (Ad*(A)B,)C, is the momentum mapping (see Chapter 13). We leave this proof and many other calculations in this discussion as exercises. The momentum mapping p: G x T,G + T,*G is a restriction map. For each p o E TZG, consider the system ( M p o ,W I M , ~ )where , M,,
= P-YPo) =
((4 B)JB= (Ad*(A-%oY}.
Let Go = { H E GIAd*(H-')pO = po}. If ( A , B) E M,, and H E Go, then &,(A, B) = ( H A , B) E M,, ,since Ad*(A - ' H - )po = Ad*(A- ') Ad*(H- ' ) p o =
'
309
RIGID BODY AND TORQUE
Ad*(A-')po
=
B,. Thus, we have a group action Go x M,,
--$
Mp,.
LEMMA 15.26 On MWowe have that
ker
( A , B ) = q,&G0(A, B)),
where Go(A,B) is the orbit of ( A , B ) under Go in M P o . The proof is left as an exercise. We know that each group G o is a circle group consisting of rotations about some fixed axis. Thus the restricted systems (M,,, oIM,,,, HIM,,) have a one-dimensional degeneracy along the orbits of Go. One can show that the quotient structures (MI,,,/Go,0, H), which are again Hamiltonian systems, are those giving the body motions as described in Chapter 13. Finally, we wunt to consider an external force (torque) acting on the rigid body. We take the G M S associated to the Hamiltonian system (G x T,G, 0 , H). Thus we form G x T,G x R , and use the 2-form cij = w + dH A dt. Let G, be a smooth time-dependent l-form on G. Perturb cij by adding on - G, A dt. Thus we have, on G x T,G x R, the 2-form Q = cij
-
G,~dt.
We have Q(A, B, t ) ( k ,B, $(A", B, f)= - d p ( k , @ ( A , B)(A"A-')+h,(B, A - ' k ) + he(& B ) T - he(& B)t - G,(A)kT G,(A)Ai, where we have used Lemma 15.25. Fix (A, B, t ) and (A, B, t') with t # 0 and suppose d(A, B, t)(k,B, t) (A", B, f) = 0 for all (A", B, f). Taking, successively, B = f = 0, A" = f = 0, A" = B = 0, gives -dp(A, B)(k,B)(A"A"-') G,(A)A"t= 0, (15.9)
+
+
h,(B, A -
'k)- h,(B, B)t = 0,
( 15.10)
h,(B,
B)f - G,(A)Af= 0.
(15.11)
Now p(A(t),B(t))C = h,(B(t), A (t)- ' C A ( t ) ) ,so
(g)
p(A, B)C = he(& A - l C A )
+ h,(B, - A - ' k A - ' C A ) + he@, A - l C A ) ,
which gives
Thus, dp(A, B)(k,E)(A"A-') = he(& A-'A") + he(& [ K I A " , K'k]), so that dp(A, B)(k,B)(kA-') = he(& A - l k ) . Therefore, (15.11) follows from (15.9) and (15.10). If we assume t= 1, then (15.10) gives B = A - ' A and (15.9)
310
15. GEOMETRICAL MODELS
becomes dp(A, B)(& B)(AA-') = G,(A)A
for all A.
(15.12)
Let F,(A) = G,(A)TR, be the force form pulled back to T,G using the space representation (see Chapter 13). Then (15.10) and (15.12) combine to give
d
- p ( ~ ( t )A, -
dt
' ( t ) k ( t ) )= F , ( A ( ~ ) ) for all t.
(15.13)
Thus, (15.13) is the equation of motion for a rigid body with torque (compare with Theorem 13.14).Note that (15.13) is just the statement that the rate of change of angular momentum equals the total external torque.
GAUGE GROUP ACTIONS We now introduce the concept of a gauge symmetry which was referred to in Remark 15.24(c). DEFINITION 15.27 By a gauge group action on a PMS, ( M , w, H), we mean a smooth left action of a Lie group G on M such mat
(a) each orbit is tangent to ker(w), (b) H is invariant under the action of G, (c) azw = o for all g E G, where a,: M + M is the action of g on M ; that is, w is a G-invariant form. LEMMA 15.28 Let ( M , w, H ) be a PMS with a gauge group action by G. If m: I + M is a trajectory and g is any smooth curve in G defined on I , then k(t)= g(t)m(t)is also a trajectory of ( M , o,H).
PROOF: Let a:G x M + M be the group action. Then f i ( t ) = a(g(t), m(t)), so that dfi/dt = T,a(g(t),m(t))g'(t) T,a(g(t),m(t))m'(t).But T,a(g(t), m(t)). g'(t) is tangent to an orbit and, thus, is in ker(o(fi(t))). Therefore, for any v E T,,,,M, we have w(&(t))(&'(t),v) =w(&(t))(T,a(g(t), m(t))m'(t),v)= o(m(t)). (m'(t),(Ta,~,,)-'u)=dH(m(t))(Ta,~,,)-'v=dH(fi(t))u. I
+
THEOREM 15.29 Let ( M , o,H ) be an IPMS with a gauge group action by a Lie group G . Suppose G acts on the left on the manifold F and assume each orbit is a submanifold of F. Suppose v: M + F is a smooth map such that
(a) v(gm)= gv(m)for all g E G, m E M ; (b) if m: I -+ M is a trajectory of ( M , o,H ) , then {v(m(t))lt E I } is contained in some orbit. Then v is a restriction mapping.
31 1
MOVING FRAMES AND GEODESIC MOTION
PROOF: Let m[a, b] --* M be a trajectory of ( M , o,H ) and u, We have the subgroup
G,,
=
(g E G l p ,
=
v(m(a)).
= u0>.
If Go, is the orbit of u,, then there is a commutative triangle
GIG
!I(,
where a&) = gu,, Oi, is the induced map on the quotient and p is the natural projection. Now Gu, is a submanifold of F and B,, is a diffeomorphism, so that t -+ Oi,;'v(m(t)) is a smooth curve in G/G,,. By Exercise 14.9 there is a smooth curve y: I -+ G such that y(t)uo = v(m(t))and y(a) = e, e being the identity element in G. Then f i ( t ) = y ( t ) - 'rn(t)is a trajectory by Lemma 15.28. Since y ( t ) can always be joined to e by a smooth curve and since G acts as a gauge group, we see n(m(t))= n(G(t)),where n: M -+ M/ker(w) is the projection. Finally v ( f i ( t ) )= v(y(t)-'rn(t)) = y(t)-'v(m(t))= u,. [
MOVING FRAMES A N D GEODESIC MOTION We now use the ideas developed previously to study, first the properties of geodesic motion on a pseudo-Riemannian manifold, and then more general particle motions. We have previously seen that geodesic motion can be described in terms of geometrical mechanical systems (Example 15.15). The development we now give obtains these systems as the final product of more basic physical-geometric objects, since it is our first example of a particle interacting with a gauge field. Recall that in our study of the rigid body we used an equivalent model in which the metric is "constant." For a general pseudo-Riemannian manifold this can be done only locally, using the notion of moving frames. Let ( M , g) be a pseudo-Riemannian manifold of dimension n. Let ( e l , . . . , e,) be the standard basis of R" and let h be the metric on R" for which h(ei, e j ) =
i
0 1
-1
Thus h is a metric of index n index of the metric g on M . For rn E M we consider Lfrn =
{g:R" + TrnM(a is
-
ifi#j, if i = j , 1 5 i 5 p , if i = j , p + l < i < n . p (see Definition 6.6). We take n-p to be the
a linear isomorphism and o*g(rn)= h } . (15.14)
31 2
15. GEOMETRICAL MODELS
Define Y =
u Ym,
(15.15)
msM
n : Y + M,
.(a)
if
=m
OE
9,.
(1 5.16 )
We now introduce a construction by which we make 9 into a smooth manifold in a special way. This is analogous to our construction of the tangent and cotangent bundles in Chapter 5. The result of our efforts will be our first example of what is known as a principal jiber bundle. LEMMA 15.30 For each mo E M there is an open set V containing m,
and a diffeomorphism
p: V x R" + TVsuch that
(a) for each mE I/, Pm:R" + TmV defined by isomorphism, (b) B;g(m) = h, for all m E V .
pm(t)= P(m, 5) is a
linear
PROOF: Construction of p is equivalent to construction of n (2"'-vector fields tl, . . . , t,, on V such that at each m e V we have
.I i
g(m)(ti(m),S j m ) ) =
0, 1, -1,
i #j, i =j , i=j,
The construction is outlined in Exercise 15.3.
l l i l p , p 1 I i I n.
+
I
A basis ( u l , . . . ,u,) for Tm0Msuch that
0,
g(mo)(ui,u j ) =
1,
-1,
i #j, i =j, l l i l p , i=j, p+l 0, 2, projects under +,:2 + T*M. to the Hamiltonian vector field on S,, c T * M . PROOF: This follows from the preceding theorem and the fact that M(Z,) = A(2) = 2E,. I REMARK 15.46 The integral curves of the vector fields Z , can be used to describe the motion of the tangent space along a geodesic under parallel with .(ao) = m(0) transport. Suppose m(t) is a geodesic in M . Choose oo E 9. and A E (R")*such that 2' = a; '(rh(0)). Let o(t)be the integral curve of Z , in 2 with a(0) = oo. Then o;'(rh(O)) =' A = O(d(t))= o(t)-'m(t). So if we define T(t):Tm,,,M -, Tm,,,M by T(t)= a(t)a;', we have k(t)= T(t)(m(O))." The transformations T(t)give parallel transport along the geodesic. The general definition of parallel transport determined by a connection, along any curve, will be given in Chapter 16.
321
THE SOURIAU EQUATIONS
THE SOURIAU EQUATIONS According to the general theory of relativity, the motion of a particle interacting with a gravitational field may be described as geodesic motion in a pseudo-Riemannian manifold of dimension four and index three (the spacetime manifold). We now obtain the equations of Souriau [28] for the motion of a charged particle with spin in a gravitational and electromagnetic field. This is an extension of the preceding discussion of geodesics, with h being the Minkowski metric on R4. In relativity theory, it has become standard to write the standard basis of R4 as e,, e l , e 2 , e3 so that k(e,, e,) = 1, h(e,, ei)= - 1, i = 1, 2, 3, and k(e,, e,,) = 0 for a # p. The group O(4, h) is called the Lorentz group and will, in the following discussion, be denoted by G. Also n : P + M is the Lorentz frame bundle for ( M , 9). We will use the components ha, and hap of the Minkowski metric where, as usual, ha,hDV= 8;. These will be used to raise or lower indices on tensors on R4. Let 9 be the Lie algebra of G. Either G or 9 can be viewed as matrices which operate on R4. That is, if A E G and T E 9, we have, for u E R4, Au = A(u"e,) = (AapuS)ea and Tu = T(u"e,)= (Ta,uB)e,. NOTE: When we write a (:)-tensor we offset the indices, writing T", rather than T;. This is done so that we can keep track of indices when raising or lowering is used. In the case of 8; there is no problem since if we lower or raise an index we get ha, or hap which are symmetric. LEMMA 15.47
Let A
= (A",) E
G. Then (A-')",
=
A,".
PROOF: Since A E G, we have h(Au, A w ) = h(u, w )for all u, w E
R4.Thus h,,
= k ATpApu, so that ha, = 8: = h A',hP"ABu= A,"Aou. This shows that the ? r! matrix C,where C",= A,", is the inverse of A , as desired. I
The action of G on R4 induces an action on (R4)*so that contraction is invariant. This action is the usual one given by (Ap)u = p(A- 'u).
To summarize, the actions of G are given as follows: let A
E
G, u E R4,
P E (R4)*; (Au)" = A",ufi,
( A p ) , = A,Bp,.
(15.28)
These actions induce actions on tensors of arbitrary type. For example, if L has components Lapy,then (ALyBy= AapApuAYrLPur. Such actions commute with contraction and with raising and lowering indices.
322
15. GEOMETRICAL MODELS
STRUCTURE OF THE LIE ALGEBRA OF THE LORENTZ GROUP Now G is a group of matrices, a subgroup of GL(4, R). G is determined by the requirement that it preserve the Minkowski metric h. Then the Lie algebra Y is naturally identified as a subalgebra of the Lie algebra of all 4 x 4 real matrices. This was discussed in Chapter 14. The condition that a matrix T be in Y is that, for all u, w E R4, h( Tv, W)
=
-
h(0, Tw).
(15.29)
In component form (15.29) becomes
+
ha8TapuPWa h,,VPTuSWB
= 0.
Then (To,+ T,,) = 0. In fact we have LEMMA 15.48 (T",) -+ (Tu,) is a vector space isomorphism @:Y -+
A2(R4). It satisfies @(ATA-') = A@(T)for A
E
G.
PROOF: (ATA-I)", = A",TP,(A-')", = A',TP,A," so that @ ( A T A - l ) u ,= A,,TP,A," = AaPT,,A,". But (A@(T)),,= AaPABaTpU.I
CONSTRUCTION OF A GAUGE INVARIANT 2-FORM REMARK 15.49 The isomorphism of Lemma (15.48) allows us to define, for S, T E A2(R4), ST, TS, and [S, TI = ST - TS. Let
N,
=
{ ( p , S) 10 # p
E
(R4)*, S E A2(R4), Spvpv= 0, and SpvSpv= s2>,
where s is a fixed number. For ( p , S ) E N,, A E G, A(p, S) = (Ap, AS) is again in N , , since the group action commutes with contractions. Thus, if A s= 2 x N , c 9 x (R4)* x A2(R4), we have a left action of G on Asby A(a, p , S) = (crA- ', Ap, AS). We now want to define a 2-form o on A,which will lead to the motions of a charged particle with spin in an electromagnetic and gravitational field, and is such that the action of G is a gauge action. We know that o should contain a term - d ( p e ) as already described for the case of zero charge and zero spin (here we are using p for the identity mapping on (R4)* instead of A). Also, recalling Example 15.18, there will be a term -qp( , ), where q is the charge of the particle, F = fi*(F), ii is given by fi(o,p , S) = n(a), and F is the 2-form on M describing the electromagnetic field (see Chapter 12). Now for (a,p , S) E As, 6(a) takes values in Y which is isomorphic to A2(R4) (Lemma 15.48), and S E A2(R4) N (A2(R4))* by This is the same contraction, so that S . 6 is a real-valued l-form on A,.
323
CONSTRUCTION OF A GAUGE INVARIANT 2-FORM
type of term as p .O. In fact, the use of the Poincare group (the Lorentz group together with the translation group of R4)would give p . 8 + S . 6 (see Chapter 16). We thus include a term -d(S . 6) in o.So far we have w = -d(pH) - qF( , ) - d(S .6). For this o,however, G is not a gauge action. We will show that there is a number CI such that
s)(cf,P, S)(6,6,S ) = - d ( m c f ,
Pw,p") - q & w , 6) - d(S . 6)(0, S)(cf, $(6, g) CIS.
o(~ P,,
+
[s,$1
(15.30)
defines a close 2-form on A,such that the action of G is a gauge action. First we need some preliminary lemmas. LEMMA 15.50 (a) Let o be a 2-form on a three-dimensional vector space E . Then o = CI A fl for some 1-forms CI and fi. (b) If w is a 2-form on a four-dimensional vector space E and there is a nonzero vector u E E so that i,w = 0, then o = C I Afl for 1-forms CI and 8. PROOF: (a) If w = 0 then o = 0 A fi. Suppose o # 0. Choose el, e2 so that w(e,, e2) = I. Choose f 3 so that {el, e 2 , f 3 } form a basis for E . Let e3 = f 3 - o(f3,e,)e1 + 4 f 3 , e,)e2. Then w(el, e3)
= o( e1,f3)
+ w(f33
el)w(el, e2)
=
and 4 e 2 , e3) = 4 e 2 , f 3 )
- w
3
,
e,)o(e,, el) = 0.
Let {el, e2, e 3 } be the dual basis to {el, e 2 , e 3 } .Then o = e l of (b) follows easily from (a) and we leave it as an exercise. COROLLARY 15.51 -$(SPY
A
e2. The proof
I
If (p, S ) E N , and Tap= - TBa,then S""TvpSPu=
Tp,)SY
PROOF: Since ( p , S) E N , , i,S = 0 and p # 0. Thus by Lemma 15.50 (b), Spy= L"Mv - L V M p .Writing out both sides of the equation gives the result. I
LEMMA 15.52 If (8, p", 9) E T ~ g , p . s )and A s Tab= - TDa,then FPTrw= (~/S~)T,,~S~~S~~~"~". PROOF: There is a curve (a(t), p(t), S ( t ) ) in A so that s" = (d/dt)l,=,S(t). By Corollary 15.51, we have that S7ySvpSp'= -s2Sp for each t, so that s2S.Tp(t)T,,, = S V ( t ) S y p ( t ) S P ~ ( tfor ) T ~each r t. Thus, s2FpTzp = ~yS,pSpBTpr ST"&SPWTwr + S"S,ps"PPTflr. But S~"Sv,Spo= -$SuDs".Ta,)S" by Corollary = 0 since SUBSu,= 2s2. Thus, the second term is zero. Anti15.51, and Sapgap symmetry shows that the first and third terms are the same, which gives the result.
+
324
15. GEOMETRICAL MODELS
PROOF: First note that for A, B, C in
[A,B]
A2(R4),
. c = ( [ A , B-yy)CaY= {A",B$
B",A$}C,Y = A,,Bfi,C"Y - B"PA,,C,Y = AapBPyCaY B?4,,CaY = A,,BByCaY ApyBB,Cay= 2A,,BflYCaY. -
+
+
Thus, [ [ T ,S], S ] . s" = 2 { 7 - 3 , S $ P - SadT~,S~),SfiySny} = 2T,dsd~S,YsbrY- 2(Sd"~aaySY~)T6p.
But the second term is zero by Corollary 15.51 and the fact that 15.52 now gives the result. [ Lemma 15.52 For a E 9 and [ E T U B we have O(a)t;, O(CT)t;, = 0.
t = th t h+ tu, t u ,where
s". S = 0
6(a)th 6(a)th = 0 and
5 E T u 9 ,v] E T u 9 we have that dS(o)((, q ) = LEMMA 15.54 For B E 9, d6(a)(th,q h ) - [ S ( 0 ) [ u , 6(cr)qul. PROOF: Let a. E 9 and rn, = .(ao). Choose a section B: V -, 9 so that mo E V and a(rn,) = 0,; a gives 0:V x G + 2' by 0(m,g) = o(rn)g an isomorphism. Let s'= @*6 and s^ = a*6. Then using the equation R1;6 = ad(A-')6 from Theorem 15.39, we obtain
+ A-~A",
&rn, A)(G,A) = A - ' ( & ~ ) G ) A
LLEMMA E M M A 15.55 15.55
For For AA,, B, B, CC in in A2(R4), A2(R4), [[AA,,BB]] .C .C = =A A .. [[BB,, C C].] .
325
CONSTRUCTION OF A GAUGE INVARIANT 2-FORM
PROOF: The calculation at the beginning of the proof of Corollary 15.53 shows that [ A , B] . C = 2AorpBByCay. Thus, [B, C ] . A = 2B,,CB,Aay = 2A,,B",CaY = +2A,,Ba,CYB = [ A , B] . C I
Now w, given by (15.30), can be written w(o, p , s)(6,p, S)(6,@,
S) = -PO(+
+ @q0)6 + p6(o)6e(o)6
p6(u)a(a)6 - Sh(o)6 Sd(a)6- s . d6(a)(cf,5) (15.31) - qF(a)(6,6)+ a s . [S, $1. -
+
For T E 9, < T ( a ) = ( d / d t ) l , = oo exp(tT) is a vector field on 2, and XT(a,
p , s, = ( ( d / d t ) l C = O exp tT)(a,p , s,
is a vector field on As.A simple calculation shows that XAa, p , S) =
(6,@,S) = (-tT(cr), -pT, [T, S ] ) . Since O(o)((,(o)) = 0 and 6(o)((T(a)) = T,
we get 6?6,$1 p , s)(xT(a, p , = pTO(o)6 0 (-p)TO(o)6 - 0 - [ T , S] . 6(c~)6- s". T
+ +
+ Sd6(o)((&),
6)
+ a s . [ [ T ,S ] , $1.
By Lemma 15.54, d6(a)(tT(o),6) = - [ T , 6(0)6] so we get w = - [ T , S ] . 6(0)6 - S . [ T , 6(0)6] - 5 * T a s . [[IT; S ] , q. Using Lemma 15.55 we get w = - S . T - as". [ [ T ,S], S ] so that, by Corollary 15.53, w = - 3 . T as2s". T = -(1 - as2)s". T. Thus, if we take c( = l/s2, we get then iXTm=
-
+
+
O V T E ~ . THEOREM 15.56 For each A E G, Lzw = w where L, is the action of A on A,. If we take a = l/s2 then for each (o,p , S ) E A, = 9 x N , we have that T(,,p,s,(G(o,P? S ) ) c ker 0. PROOF: The second part has been established by the preceeding calculations. Now L,(o, p , S ) = (OR,-I , p A - ', A S ) so that TL,(o, p , S)(6,p, 3) = (TR,-,d, pA-', AS). Invariance of the form w follows from the relations
T I T TRA-1 . = Tx,
~19 -+ M ,
O(aA-1) = Aa-lTn,
d(oA-')TR,-I( ) = A6(a)( )K1, dh(aA-')(TR,-i( ), TR,-i( )) = Ad 6(a)( , )K' ( A S ) . [AS, A
q = s . [S, q.
I
326
15. GEOMETRICAL MODELS
Thus o is invariant under the action of G and the G-orbits are tangent to ker o.We will now assume that our particles have positive mass so we restrict ourselves to A T = {(a,p , S ) E A s l p . p > 0). Let p , = e0 and So = sel A e2. LEMMA 15.57 A: = {(a, zp,A-', AS,)la E 9,A E Go, and z where Go is the connected component of the identity in G . PROOF: Clear.
ER+},
I
Writing p = zpoA-' and S = AS, gives 6 = ( i / z ) p - pT and S = [T, S ] , where T = A A - Thus a tangent vector at (a,p , S ) can be written as
'.
(6,Li, 9) = (6,(i/Z)P,0) + (0, -PT, [ T , S]) = (6+ 5Aah ( i l z ) ~ 0) , + (-tr(a), -PT,[T, S ] ) = ( 6 1 9 ( ~ / z ) P ,0) + X r . But we have seen that X , E ker w. Suppose that Y, = (6, ( i / z ) p ,0) + X , , E ker o(a, p , S). Then (6, ( i / z ) p , 0) E ker o.Let Y, = (6,(.?/z)p, 0) + X , , be any tangent vector. Then
0 = W V ' , Y2) = 4 6 , ( i / Z ) P , o w , ( W P . 0) = - (i/z)pe(a)6 + (qz)pe(a)6+ pqapqa)z - p s ( o ) ~ e (-o )s~. dqa)(8., 6 ) - qP(a)(6,6). Write 6 = 6 h + 5" (as in Lemma 15.54)and 6 = 6, + 6". Then we must have (if Y, is in ker w ) that
(l) p8(a)6h = O, (2) - p ( 6 ( a ) d " ) ( e ( o ) 6 h ) (3)
-(i/z)po(a)6h
+s
'
[d(a)6",6(0)6u]= 0 vcu,
+ p6(a)6"8(o)6h = s
'
d6(a) ( 6 h y 6 h ) f qP(:(a)( 8 h j Gh) V 6 h .
Here we have used Lemma 15.54.We will consider p = zp, and S = So. The group invariance gives the validity of our conclusions at all points in 4:. Write R = 6(o)17,,T = d(a)c?",v = B(o)bh.Equation (2)is p(Ru) = [T ,R] . S = - R . [ T , S ] VR. If pR = 0, then R . [ S , TI = 0. This shows that T = aieo A e' as so that T EA ~ p' + as. Now p S = 0 and [ S , S ] = 0, so that x.-s(a,p,S)=(5s(g),090). Thus Y1 =X,, +c~X-,+(~,+~,-~~S(O),(~/Z)P, 0). Since ax-,E ker o,the last term must be in ker w. Thus, we can assume T = aieo A e'. Then, using [el A e 2 ,eo A e'] = e0 A e2, [e' A e2,eo A e2] = -eO A el, [e' A e2, eo A e3] = 0 we get, using R = yieo A e' + R,, R, E A2(p'), that zyivi = 2sa1y2 - 2sa2y1,Vy,, y 2 , y 3 . Thus, v1 = -(2/z)cr2s,
xi
+
327
CURVATURE FORM
v2 = (2/z)a,s, and v3 = 0. Note that Eq. (1) implies uo = 0. Thus, 6, is given in terms of 6,. We now turn to Eq. (3). Define F: Y -+A2(R4) by F(a)(tl, t2)= F(z(o)). ( O < l ? a O let F E = { s ~ F I R ( s ) = Eand } We know that is a GMS which determines the trajectories (see Theorem 15.14)(FE, with A = E. The map n3:4 -+ (R")*, given by n3(6,u, A,_sGo) = A is a restriction mapping, since each trajectory of the PMS (A, C,H ) has constant and thus n3 takes it into an O(n, h) orbit in (R")*. Let ,? E (R")* and E , = *h*(,?,A). Then A,= n;'(,?) = (9 0 P ) x {A} x G/G, N (9 0 P) x G/G, and we get the extension of Theorem 15.44. ,?6 A 0 - d(pc1) - Q, En). Then S , is a PMS For ,? E (R")* let S, = (AA, with a gauge action by G,. Furthermore, SJG, is isomorphic with the GMS (FE, Here G, = G, x G. But now we can do a further reduction. The = projection n4(cr, u, ,?,gG,) = gC, is a restriction map for S,. Let An,v, n'i '(Go)n A,.
cE)
eE).
THEOREM 16.20 For ,? E (R")* let S,,vo = (A,,vo, Ah A 0 - d(v,cc)). Then S,,vo is a PMS with a gauge action by G,. Furthermore, S,,vo/G,is isomor-
phic with the GMS (FE, &). Notice that, in terms of Theorem 16.20, the Hamiltonian function is reduced to a constant corresponding to a certain orbit. In this approach, the Hamiltonian is a labeling for the orbits corresponding to the rest mass of the particle. The constructions leading to Theorem 16.19 used the principal bundle 9 0 P with group O(n, h) x G. We have connections 6 and c1 on 9 and P, respectively. However, 2 was treated in a special way, since - d(prx) appears in C but there is no such term for 6. Instead we have w = --d(I* 0) in C, where 0 is the R"-valued l-form on 2 defined by 0(o) = 6' Trio. Such a In order to construct form 6' is special to a tangent frame bundle such as 9. 0
0
348
16. PRINCIPAL BUNDLES AND CONNECTIONS
the interaction entirely from connections it would be desirable to incorporate 6 into a connection on some principal bundle. This can be done by replacing linear frames by affine frames (e.g., enlarging the Lorentz group to the Poincare group). Thus, we replace 9 by a larger bundle 9 as follows: Let A(n, k ) be the semidirect product of O(n, k) with R". That is, we can write
with
[" '][b ;I=[; "y] 0 1 0 1
and, identifying u E R" with [;I,
[" 'lcu] ;'1, 0 1
1
=
+
so that [g f ] gives the transformation v + au 5. Of course, O(n, k ) c A(n, k ) as a subgroup so that O(n, k ) acts on A(n, h) through left translation. Let 9 = 9 x ~ ( ~A(n, , ~h)) be the associated bundle. For y o , r E A(n, k), defining [a, r]ro = [a, rro] gives 9'the structure of a principal A(n, h)-bundle. We thus have j : 9.+9 by j ( u ) = [u, id] and j ( 9 ) gives a reduction of the group of 9 to O(n, k). The Lie algebra of A(n, k ) is denoted by d ( n , k ) and can be represented by matrices [;f z], where A E O(n, k), u E R", and we see (16.8)
The adjoint representation of A(n, k ) on d ( n , k ) is given as follows. If
then A
u
A
v
+ uv
Ad(a)(A) -Ad(a)(A)' 0 0
=[ Thus
i'A5
+ a-'v 0
11.
(16.9)
349
AFFINE GROUP MODEL
Let A be a connection on 9'. Then we can write
where 6 is an o(n, h)-valued l-form on 9 and $ is an R"-valued 1-form on 9.Let
(a) 6, is a connection on 9, RB(4,) = a4, Va E O(n, h). (b) We leave the calculations as an exercise. Of course, A, determines A uniquely because of the A(n, h) action on 9. LEMMA 16.21
'
0
AFFINE GROUP MODEL DEFINITION 16.22 A connection A on B for which afine connection.
4,
=0
is called an
Now given 6, a connection on Y , define
which determines a unique affine connection A on 2'. Thus, there is a natural correspondence between connections on Y and affine connections on 9. Now ( d ( n ,h))* = O(n, h)* @ (R")*. Choose V, = ((0,A,), v o ) E d ( n , h)* x 9*.In the following we will write G for A(n, h) x G and
Go = {g We wish to calculate
so that
With
E
G1Ad*(J)(?,)
= go}.
on G/G,. Using Eq. (16.9), we see that
350
16. PRINCIPAL BUNDLES AND CONNECTIONS
where
so that U is the O(n, h) x G orbit of Vo. We then have
(2@ P ) x u c ( 9 0 P ) x G/G,. When we restrict Z = -d(P E ) - fi to (9'0P ) x U we get 0
Q(GoK(t1,l l h
since
t = 0,
v 1 = u2
= 0;
and G,
= W G O ) ( i l , 12),
(52,tJ) 0
&(a,u, gG,) = M ( o )
5 = 0, A = IZ0u-l, 4 = 8 on 2.We thus get THEOREM 16.23
Let
=
-d(p
o
E)
-
+p
0
a(u, gG,), since
fi on (9@P ) x G/Go. We have
embeddings I , J
2-k
A! = (2 @ P ) x (R")* x G/Go such that J * @ ) = Z*(X). Furthermore, if E FE,where pr: A + 9.
(9 0 P ) x G/Go = *h*
(Ao, Lo), then pr(image I ) =
REMARK 16.24 Theorem 16.23 shows that we can construct the GMS (FE, zE)entirely from a connection on the A(n, h) x G bundle 9 x P , the coadjoint representation of A(n, h) x G, and the reduction 9 ' x P of the group to O(n, h) x G. This construction is also due to Sternberg and co-workers [33, 341. We could take a more general Go; Go = ((So,Ao),vo) in d(n,h)* x 9*.This would give a spin-curvature coupling term.
351
EXERCISES
EXERCISES 16.1 Complete the proof of Lemma 16.6. 16.2 Give the proof of Theorem 16.12. 16.3 Give the proof of Theorem 16.16. 16.4 Suppose that a is a connection on the principal G-bundle 71: P + M and that Da is its curvature form. Show that R,*(Da) = Ad(9-l) 0 Da Vg E G. 16.5 Let S' = SO(2) = { z l z E @, Z Z = 1). We consider S' as a Lie group. Writing
gives two coordinate charts for S ' , say -7112 < 4 < 3x12 and 0 < 4 < 271. For each z E S' we have T,S' = {irzlr E R}. Thus T,S' = firlr E R } . Of course S' is abelian, so that the bracket is zero on TeS1and Ad(z) = Id V z E S'. (a) For either of the two coordinate systems, using ei4,show that d irzE T,S' is equal to r -. Thus, d l d 4 is the unit tangent vector d4 field pointing counterclockwise. (b) Show that id4 is canonical T,S'-valued 1-form on S' (recall that the Lie algebra has been identified with the set of purely imaginary complex numbers). 16.6 Let n: P + M be a principal S'-bundle and a a connection on P . Show that: (a) We can write a = ib, where fi is a 1-form on P . (b) Da = dcc = idp. (c) There exists a unique 2-form x on M such that TC*(X)= dp. It is defined by x(m)(u, w) = d p ( z ) (f,(u), L',(w)), where ~ ( z=) m and L', is horizontal lift. (d) If 0 : I/ + P is a section of 7c: P + M and = o*p, then x = dp. Thus dX = 0. (e) For M = R4 with the Minkowski metric, work through the construction of Theorem 16.19 in this case. Here Y R so get that 3* = R and the orbits of the coadjoint representation are points. Write v0(d/d4(,) = q. Show that the GMS ( S E Z E with ) E = m2 is the GMS (Sm2.w + (~*)*6)of Example 15.18.
p
-
352
16. PRINCIPAL BUNDLES AND CONNECTIONS
16.7 Derive Eq. (16.5). 16.8 Derive Eq. (16.8).
16.9 Suppose that n:P + M is a principal G-bundle over a manifold M and that f :Y + M is a smooth mapping. Define a set
f * P )= {(Y?4 l Y E
y and u E P f ( J
and mappings E : f * ( P )+ Y
by Z(y, u) = y
f l : f * ( P )4 P
by f"(y, u) = U.
We thus obtain a commutative diagram f * ( P )-5 P
Suppose that {(K, Cpi)liE I } is a system of local trivializations for P . That is, each K c M is open, M qli E Z}, and &i: n-l (q)+ q x G is a local trivialization. Define $i:iY1( f - ' ( K ) ) 4 f - l (F) x G by where 4i(u) = (nu,4i2u).Show that: Icli(y, u) = (y,
u{
(a) there is a unique differential structure onf*(P) so that each $i is a diffeomorphism between open sets; (b) defining (y, u)g = (y, ug) gives f * ( P ) the structure of a principal G-bundle; (c) if CI is a connection on P , then f;c(a)is a connection f * ( P ) . ii.:f*(P)+ Y is called the pullback of n:P
+M
under f .
16.10 In the constructions leading up to Theorem 16.19, we had
4= (2'@ P ) x ((R")*x G/Go), where 9 is the Lorentz frame bundle over spacetime M and n: P + M is a principal G-bundle with a connection. We also had defined the mapping p: GIG, + G*. (a) Take P = 2' with the connection 6 constructed in Chapter 15. Thus G is the Lorentz group. Recall that we have isomorphisms 3 N A2(R4) N %*.
For So in A, (R4), let Go = { A E GI Ad*(A)So = S o } so we have p: G/Go + 9?*. Then the form X is given by X = w - d(S .6) - Q.
353
EXERCISES
(b) Let A' = {(@,p,A,S)E cdlcr= u and AS = 0 ) (where the spin and momentum are given in the same Lorentz frame) and let
A s ={(o,A,S)I;1S=OandS.S=s2}, which was used in Chapter 15. If s2 = So . So, we have an imbedding of eA'in As, given by
1,S )
(0, 0,
+
( 0 , A S).
Show that C equals the form given in Eq. (15.30) (with F = 0) on A'. (c) To include that F term, we would take P = 2 63 P , , where 71: PI + M is a principal S1-bundle with a connection such that iqF is the curvature form on M (see Exercise 16.6). (d) Thus, (A1,X) is imbedded into ( A S , m )of Chapter 15. There is obtained using the diagonal map a group action of G on A', G + G x G, given by A + ( A , A ) . The map H(O,Eb,S) = +h*(A,
A) - f ( 0 , S )
is invariant under this action and factors to 9 = A 1 / G . Show 2) can be identified with the system (9, K * (a))/ that (5E, G(A0, So)of Chapter 15, where FEc 9 is given by l? = E .
77 Quantum Effects, Line Bundles, and Holonomy Groups
QUANTUM EFFECTS The quantum behavior of particles occurs in experiments in which particles exhibit a wavelike nature manifested by certain interference effects, a simple case of which we now describe. A beam of identical particles is split into two parts at a point S. Or we can consider S as a source point for beams of particles, all of which are reflected back from a barrier except for the beams which reach the points A and B, where there are slits in the barrier. The points A and B then act as secondary sources allowing particles to go off in all directions (see Fig. 17.1). We now place a detection screen at a distance d from the barrier (see Fig. 17.2). Let us coordinatize the screen by x. We are interested in the following question.
FIGURE 17.1
354
355
PROBABILITY AMPLITUDE PHASE FACTORS
PROBABI LlTY AMPLITUDES Of the total number of particles which reach the screen in a certain time interval, what fraction arrive at x? This fraction can be interpreted as the probability that a particular particle will arrive at x. A model which correctly describes this is based on three principles written down by Feynman [ 113: (a) There is a complex function @(x)so that lCD(x)lzis the probability that a particle will arrive at x. The complex number @(x)is called the probability amplitude for the recording of the arrival of the particle at x. (b) For each path p there is a probability amplitude CD, so that lCD,12 represents the probability that the particle traveled along p. If pi is the path li followed by ti,then @(x) = CD,, Thus, if an event can result from several different processes, you add the probability amplitudes to obtain the probability amplitude for the event. This is in contrast to classical probability theory in which you would add the probabilities of the possible processes which produce the event. (c) If the path consists of two parts such as p = A u t,then CD, = On@(.
+
PROBABILITY AMPLITUDE PHASE FACTORS Thus we obtain interference effects. If CD,,(x) A , ( ~ ) d @then ~(~),
(apl+
@,2)(@,,
+ 6,J
=
A:
=
Al(x)ei@l(x) and CD,(x)
=
+ 4 + 2A1’42 cos (41 - 4 2 )
+
so that \O(x)[’ varies between ( A , A,)’ and ( A , - A,), depending upon - $z. If 6 = 2nn, we get a maximum probability the phase diflerence 6 =
FIGURE 17.2
356
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
point and if S = (2n + 1)7c, we get a minimum probability point. Of course, A , and A , must be such that f @ ( x ) m d x= 1, but A , and A , have no effect on the locations along the screen of the max and min probability points. The location of these points relates to the properties of atomic particles and fields interacting with them. For example, suppose we know the locations of the max-min points for certain particles in an inertial frame subject to no force. When a field is turned on which interacts with the particles, the locations of the max-min points will shift. This is what we want to calculate. Thus we will not worry about A in @ = Ae'4 but will see how to determine the 4. We will refer to ei@as the probability amplitude phase factor.
DeBROGLIE AND FEYNMAN The idea of associating a wave disturbance with a particle motion was developed by Louis deBroglie in 1924. Later Richard Feynman gave us the principle that the probability amplitude phase factor of a path q(t) in the configuration space of a classical mechanical system with kinetic energy function T(q, q) and potential energy function U(q)is exp(i 1(T(q,4) - U(q))dt). The deBroglie wave for a particle of mass m in its rest frame is @(t,x,y, z ) = @oe-2"i", where v = mc2/h, c being the speed of light and h Planck's constant. We include the minus sign in the exponent to be in agreement with Feynman's prescription for particles moving with a velocity much smaller than c as we shall see. Using the Lorentz transformation (11.14) we see that the deBroglie wave of a particle of mass m in an inertial frame in which the particle is moving with velocity u parallel to the x-axis is @(t,x,y , z) = exp [-2niv{yt - y(u/c2)x}],
where y = l / , / l _ O . Now let p be any classical path of such a particle, so that p is given by t = t, x = xo + vt, t, < t < t g . Then @(B)= @ ( A ) . exp [ -2niv(t, - t , ) , / m ] , so that the phase change of @ from point A = (t,, xo + ut,) to B = (t,, xo + ut,) is S, = - 2 n v ~ , ~ ,where T, is the elapsed proper time along p from A to B. This is the phase change of the deBroglie wave along the classical path of the particle. Following Feynman, we shall take the probability amplitude phase factor of any future pointing timelike path p to be exp( - 2niv AT).
PHASE FACTORS AND 1 -FORMS Now let 6 be the curve in phase space obtained from using the relativistic energy momentum relations. Recall that
357
PHASE FACTORS AND 1 -FORMS
where v that
= dp/dt.
Since the metric is ds2 = c2 dt2 - dx2 - d y 2 - dz2, we get
If 8 is the canonical 1-form on phase space, then
Thus, j,- 8 = r n c 2 , / m ( A t ) ,so that 6,, = -(27c/h) culation is valid on any Lorentzian space-time.
Jp
8. In fact, this cal-
LEMMA 17.1 Let A4 be any four-dimensional manifold with a Lorentz metric g . If p(t) is a timelike curve in M and p(t) in T*M is defined by g(p(t))(md p / d q ) = fi(t), then, for a particle of rest mass m, the probability amplitude phase factor of p is exp( (- i/h) jo 0). If p is future pointing, 0 = mc2(A.rh where
so
is the elapsed proper time along p (h = (1/27c)h). REMARK 17.2 The lemma gives the phase factor if there are no external fields influencing the particles, otherwise it must be modified. We will see how to do this in the case of the electromagnetic field. Note that the gravitational field is included in the lemma since it is represented by replacing the space time of special relativity with a general Lorentzian manifold.
We now return to the situation of Fig. 17.2. With pi = ,Ii u Li, the elapsed time of p 1 equals the elapsed time p2 in our reference frame. Since we are assuming that the initial speeds are the same, we get that the proper time of 2, = the proper time of 1,. Now letting L, and L2 also denote the length of the paths, and writing v1 = t ,j A t , u2 = [ , / A t , where At is the elapsed time of path el and e,, the phase difference is
If L, = e,, then let u = u1 = u2 and we get 6 = 0 so 0 is a maximum probability point. Let us find the closest minimum probability point above 0
358
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
assuming that u 5) = d&(m)(t,i)+ [a^(m)t7a^(m)CI. Furthermore, R,
= a*(Da).
PROOF: This all follows from Eqs. (18.1) and (18.2).
I
374
18. PHYSICAL LAWS FOR THE GAUGE FIELDS
LEMMA 18.2 Using the same notation as in Lemma 18.1 suppose that y:V+ G is smooth and (il: V - + P by al(m) = a(m)y(m). Then R,,(m) = Ad(y(m)- ')R,(m) Vm E V . PROOF:
R(m)(t, i)= [o(m),R,(m)(t,
01 and
R(m)(t, i)= [adm), R u m ( < ?01 = [o(m)y(m),R,,(mXt, 01 = [o(m),Ad(y(m))(R,,(m)(S,011. I The quantity R which we have constructed is our first example of a bundlevalued differential form. Previously we had considered vector-valued differential forms. We saw that we could take the exterior derivative of vector-valued forms componentwise (see the end of Chapter 9). But bundlevalued forms take values in different vector spaces at different points. However, if € is a vector bundle on M which is the associated bundle to the principal G-bundle z: P -+ M obtained from a linear representation of G on some vector space E, then &-valued differential forms on M correspond to certain E-valued differential forms on P in the way that R corresponds to Da. More precisely, suppose F(m):T,M x T,M x . . . x T,M -+ c?, is an €,-valued k-form on T , M . For u E n-'(m) define F(u)(tl, t 2 , .. . , t k ) E E by the condition F(m)(Tz(ti),Tn( x k ) ) = $-'(F(m)(xlj . . . xk))> We summarize the preceding constructions with the following proposition: 9
PROPOSITION (1 8.3) (Trilogy). Let z: P -+ M be a principal G-bundle and € an associated vector bundle obtained from a linear representation of G on a vector space E . Then, a smooth €-valued differential k-form on M
375
COVARIANT DERIVATIVE OF SECTIONS
is determined by any one of the following three structures: (1) a smooth section F of the bundle L:(TM, 6);thus F(m):T,M x . . . x T,M -+ B, is an alternating multilinear mapping for each m E M , (2) a smooth E-valued k-form F on P satisfying Eq. (18.3), (3) a collection {( oi,F U i )I i E I } where (a) each r/: c M is open and M = (b) each ci: -+ P is a smooth section of P; (c) each FUiis a smooth E-valued k-form on r/:; (d) on r/: n V;. we have F,J(m) = y(m)- 'FUi(m),where ojm) = oi(m)dm).
L'
386
18. PHYSICAL LAWS FOR THE GAUGE FIELDS
(c) If image T = image L and image D n kernel t = (0), then each B in A(M, d)can be written uniquely as B = D"A + C, where C is in ker L, by choosing A so that T(A)= L(B). Thus we obtain a tangent plane transverse to the action of Gauge(P). This plane is tangent to the set F(P) = constant, passing through tl.
18.15 Suppose M = R4 is the spacetime of special relativity. Let P = M x G and tl be a connection on P with d-valued curvature form R. For each ~ ( m=) (m,s(m))we have the o-current of Definition (18.15) J,. Define T , = J R 3 *( J,)
= total
%charge.
Suppose that al(m) = (m,s(m)g(m))and that g(m) = h for m outside of some large bounded set. Now R,,(m) = Ad(g(m)-')(R,(m)), so that T,, = lim r+
=
a,
lim r+m
S,, *(J,,) = lim j *R,, r-tm
Sr
Jsr Ad(g(m)- ')(*R,)
= Ad(h-
')( T,).
Thus the total charge is invariant under the action of Gauge,(P).
-
Bibliography
Abraham, R., and Marsden, J. “Foundations of Mechanics,” second edition. BenjaminCummings, New York. (1979). [2] Arnold, V. I. “Mathematical Methods of Classical Mechanics.” Springer-Verlag, New York. (1979). [3] Boothby, W. M. On two classical theorems of algebraic topology. Amer. Math. Monthly, 78, No. 3. (1971). [4] Chern, S. S., and Wolfson, J. G. A simple prool of the Frobenius theorem, I n “Manifolds and Lie Groups.” Birkhauser, Boston. (1981). [S] Chevalley, C. “Theory of Lie Groups.” Princeton University Press, Princeton, New Jersey. (1946). [6] Curtis, W. D. The automorphism group of a compact group action. Trans. Amer. Math. SOC.,203.(1975). [7] Curtis, W. D., and Miller, F. R. Gauge Groups and classification of bundles with simple structural group. Pacific J . Math., 68. (1977). [8] Curtis, W. D., Lee, Y. L., and Miller, F. R. A class of infinite dimensional subgroups of Diff‘(x) which are Banach Lie groups. Pacific J . Math., 47. (1973). [9] De Rham, G. Varietes differentiables. A S I . No. 1222. (1955). [lo] Feynman, Leighton, and Sands. “The Feynman Lectures on Physics,” Vol. 11. Addison Wesley, Reading, Massachusetts. (1 964). [ll] Feynman, Leighton, and Sands. “The Feynman Lectures on Physics,” Vol. 111. Addison Wesley, Reading, Massachusetts. (1965). 1121 Goldstein, H. “Classical Mechanics,” Second edition. Addison Wesley, Reading, Massachusetts. (1980). [13] Greenberger, D. M., and Overhauser, A. W. Coherence effects in neutron diffraction and gravity experiments, Rev. Modern Phys. 51, No. 1. (1979). [I41 Guillemin, V., and Sternberg, S. Geometric asymptotics, Math. Suroeys, No. 14, Amer. Math. SOC.,Providence, Rhods Island. (1977). [151 Helgason, S. “Differential Geometry of Symmetric Spaces.” Academic Press, New York. (1962). [161 Hirzebruch, F. Topological methods in algebraic geometry, Grundlehren Math. Wiss. 131, Springer-Verlag, New York. (1966). [17] Kobayashi, S., and Nomizu, K. “Foundations of Differential Geometry,” Volume I. Interscience, New York. (1963). [181 Lang, S. “Analysis 11.” Addison Wesley, Reading, Massachusetts. (1969). [191 Lang, S. “Differential Manifolds.” Addison Wesley, Reading, Massachusetts. (1973). [20] Lorraine, P., and Corson, D. “Electromagnetic Fields and Waves.” Freeman, San Francisco, California. (1962). [21] Markus, L. Line element fields and Lorentz structures on differential manifolds. Ann. of Math., Series 2, 62. (1955). [22] Milnor, J. W. Analytic proofs of the hairy ball theorem and the Brouwer fixed point theorem. Amer. Math. Monthly, 85. (1978). [23] Misner, C. W., Thorne, K. S., and Wheeler, J. A. “Gravitation.” Freeman, San Francisco, California. (1973). [24] Quigg, C. “Gauge Theories of the Strong, Weak, and Electromagnetic Interactions.” Benjamin-Cummings, New York. (1983).
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Sard, R. D. “Relativistic Mechanics.” Benjamin, New York. (1970). Simmons, G . F. “Introduction to Topology and Modern Analysis.” McGraw-Hill, New York. (1963). [27] Souriau, J. M. “Structure des System Dynamiques.” Dunod, Paris. (1970). [28] Souriau, J . M. Modele de particule a spin dans le champs electromagnetic et gravitationnel, Ann. Inst. H . Poincarh., XX, No. 4. (1974). [29] Spank, E. H. “Algebraic Topology.” McGraw Hill, New York. (1966). [30] Spivak, M. “Calculus on Manifolds. ”Benjamin, New York. (1965). [31] Steenrod, N. “The Topology of Fiber Bundles,” Princeton University Press, Princeton, New Jersey. (1951) 1321 Sternberg, S. “Lectures on Differential Geometry.” Prentice-Hall, Englewood Cliffs, New Jersey. 1964. [33] Sternberg, S. On the role of field theories in our physical conception of geometry. Lecture Notes in Math., No. 676. Springer-Verlag, Berlin. (1978). [34] Sternberg, S., and Ungar, T., Classical and prequantized mechanics without Lagrangians or Hamiltonians. Hadronic J., No. I . (1978). [35] Wu, T. T., and Yang, C. N. Concept of nonintegrable phase factors and global formulation of gauge fields. Phys. Reo. D,12, No. 12, 1975. [36] Yang, C. N., and Mills, R. L. Conservation of isotopic spin and isotopic gauge invariance. Phy. Reo. 96, No. 1 (1954).
[25] [26]
- Index
A Action of Gauge(P), 378, 383-386 Adjoint representation, 264, 291 Affine frame bundle, 336 Affine group model, 349-350 Ampere's law, 37 1-372 Antisymrnetry operator, 111, 141 Area 2-forms, 169 Associated bundle, 341 Atlas, 24 Atwood's machine, 12, 101
B Bilinear map bundle, 71 Bohm-Aharanov effect, 360 Boundary of a manifold, 162 Bracket of vector fields, 135 B-slice, 185 Bundle of horizontal vectors, 337 Bundle of linear maps, 70 Bundle of states, 343 Bundle of vertical vectors, 337 Bundle-valued differential form. 374-375
C Canonical coordinates, 289 Canonical left-invariant form, 263, 292 Canonical I-form, 68-69 Canonical 2-form, 114 Chain rule, 18, 35 Change of variable formula, 197 Charge conservation, 372, 382
Chart, 24 Choice of gauge, 38 1 Christoffel symbols, 103 C'-function, 16 Closed subgroup, 290 Coadjoint action, 343 Coadjoint representation, 266 Complex line bundle, 360-361 Configuration projection, 305, 333 Configuration space, 3, 5 , 95 Connection, 318, 337 Conservation of energy, 105, 235 Conservation laws, 329-332 Conservation of momentum, 235 Conservative forces, 99 Constraint, 11, 96 Continuum assumption, 213 Contravariant degree, 127 Coset spaces, 293-294, 296 Cotangent bundle, 66 Covariant degree, 127 derivative, 376 exterior derivative, 375-376 tensor, 72,127 vector, 67 COW experiment, 359 Critical points of a function, 90 Curl of a vector field, 169 Current defined by Yang and Mills, 382 determined by a gauge, 381 in electromagnetic theory, 243-245 induced by an infinitesimal gauge transformation, 380
389
INDEX Curvature form, 327, 329 &valued, 376 integral conditions for, 362-363
D D’Alembertian, 171 De Broglie, Louis, 356 De Broglie wave, 356 Decomposable form, 145 De Rham cohomology group, 161 Derivation, 135 Derivative as a linear map, 18, 21 Determinant, 146 Diffeomorphism, 18, 34 Differential form, 150-151 bundle-valued, 374 exterior derivative of, 156-159 integral of, 196-198 support of, 196 vector-valued, 189-190 Differential of a function, 35 Differential manifold, 25 Differential I-form, 68 Differential structure, 24 Differential 2-form, 112 Displacement current, 371 Divergence of a vector field, 169, 170, 203 Divergence operator, 168 on bundle-valued forms, 380 Doppler effect, 245
E Electromagnetic field tensor, 242 Electromagnetic plane wave, 248 Energy function, 105 Equation of state, 328 Equation of structure, 339 Equivalent model, 256 Euler equations, 282-283 Exponential mapping, 289 Exterior derivative, 112, 156-159 covariant, 375-376 Exterior form, 141
F Faraday tensor, 242 Feynman, Richard, 356
Field of lines, 89 First integral, 9 Fixed energy system, 304 Flat and sharp mappings, 114 Flow of a vector field global, 54 local, 42-52 Force free motion with constraints, 102-103 Forces, 95, 301-302 Forces of constraint, 4, 96 Four-acceleration, 229 Four-current, 243-245 Four-momentum, 233 Four-potential, 250 Four-vector, 226 Four-velocity, 227 Frobenius integrability theorem, 178 Function of class Ck,16 critical points of, 90 differential of, 35 energy, 105 gradient of, 84, 169, 170 integration, 207-208 potential, 99
G Galilean transformation, 21 8 Gauge field, 342, 371-386 Gauge group action on a PMS, 310 Gauge invariant 2-form, 322-325 Gauss’s divergence theorem, 208-209 Gauss’s law, 253, 371, 380 Geodesic, 103 Geodesic motion, 102-103, 304-305, 311-320 Geometrical mechanical system, 298 for particles in a gauge field, 347 Geometric quantization, 366-367 Global flow, 54 Gradient of a function, 84, 169, 170 Green’s theorem, 205, 210 Group of gauge transformations, 377-378
H Hairy ball theorem, 65, 212 Hamiltonian function, 10, 106, 255 system, 255-256
INDEX
391
vector field, 115, 122, 255-256 Hamilton-Jacobi equation, 126 Hamilton’s equations, 107, 117, 255 Harmonic differential form, 211 Harmonic oscillator, 303 Hodge decomposition theorem, 2 12 Hodge *-operator, 165- 168 Holonomic constraints, 95 system, 11, 95 Holonomy group, 361 operator, 361 and reduction of structural group, 367 restricted group, 367 Homomorphism continuous, 296 smooth, 291 Hopf fibration, 336 Horizontal lift, 3 18 of curves, 341 of vectors, 338 Hyperbolic motion, 232 Hypersurface null, spacelike and timelike, 77 orientation of, 154-155
1
Imbedding, 64 Implicit function theorem, 23 Index of a metric, 74 Inertial reference frame, 214 Inertia tensor of a rigid body, 275 Infinitesimal gauge transformation, 380 current induced by, 380 Infinitesimal generator, 7 Inaccessability theorem, 187 Integrable subbundle, 176 Integral cohomology class, 364 Integral curve, 2, 9 , 40 Integral of a differential form, 196-199 Integral of a function, 207-208 Integral manifold, 184 maximal, 185-186 Interior of a manifold with boundary, 162 Interior product, 156 Invariant measure, 292-293, 296 Invariant spacetime interval, 223 Invariant subbundle, 195
Inverse function theorem, 19
J Jacobi identity, 135
K Kernel of a differential form, 173 Killing form, 379 Kinetic energy function, 94 metric, 94 relativistic, 236 Kronecker delta, 128
L Lagrange multipliers, 38 Lagrange:s equations, 7, 99-100, 121, 255 Lagrangian function, 7, 99, 121, 255 time dependent, 121 two-form, 118, 255 vector field, 118 Laplace-Beltrami operator, 168 Left-invariant I-form, 263, 292 Left-invariant vector field, 287 Legendre transformation, 8, 103, 125, 255 Lie algebra, 135, 287 Lie bracket, 135 Lie derivative, 132 moving frame description of, 134-135 Lie group, 286 adjoint representation of, 29 1 canonical coordinates on, 289 canonical left-invariant I-form on, 292 closed subgroup of, 290-291 exponential mapping of, 289 homomorphism, 291, 296 invariant measures on, 292-293, 296 left action of, 294-295 Lie algebra of, 287 right action of, 335 Lightlike vector, 75 Linear map bundle, 70 Liouville’s theorem, 299 Lipschitz condition, 41 Lorentz-Fitzgerald contraction, 222
392
INDEX
Lorentz force law, 239, 306 Lorentz gauge condition, 250 Lorentz group, 224 Lorentz transformation, 2 19
M Magnetic field of a line current, 252 of a monopole, 252 of a moving point charge, 242 Manifold with boundary, 162 Maximal integral curve, 53 Maximal integral manifold, 186 Maxwell’s equations, 246-248 Meter of time, 228 Metric existence of, 87 index of, 74 left-invariant, 260 kinetic energy, 94 Lorentzian, 74, 76 Riemannian, 74, 76 volume element of, 149-150 Minkowski spacetime, 170- 171 Mobius strip, 153 Momentum mapping, 265 Moving frames, 311-320, 332-333
N Nonholonomic constraints, 189 Nonintegrable subbundles, 188 Null hypersurface, 77 subspace, 75 vector, 75
0 Orientation of a hypersurface, 154 induced on the boundary, 164 of a manifold, 151 of a vector space, 147 Orthochronous Lorentz transformation, 225 Orthogonal group, 256 Orthonormal frame, 312 Orthonormal frame bundle, 313, 336
P Parallel frames, 341 Parallel transport, 342, 376-377 determined by a foliation, 365 Partition of unity, 86 Phase factor calculations, 357, 359, 365-366, 368 Phase plane analysis, 283 Phase space, 5, 8, 106, 255 Physical process, 213 Poincare group, 224 Poincark lemma, 161 Poinsot construction, 280-281 Polarization, 366 Positively oriented basis, 147 Positive unit normal, 155 Potential function, 99 Principal axes, 276 Principal fiber bundle, 312, 335 Probability amplitude, 355 Probability amplitude phase factor, 356 Projective plane, 154 Projective spaces, 191 Proper Lorentz group, 224 Proper time axiom, 213 Proper value, 307 Pseudomechanical system, 306 closed, 306 integrable, 306 for particles in a gauge field, 345 Pseudo-Riemannian manifold, 75 Pullback bundle, 352 commutes with exterior derivative, 159 of a covariant tensor, 78 of a differential form, 151 of a vector field by a diffeomorphism, 77 Pushforward, 77
Q Quantum effects, 354 Quantum field theory, 382
R Reference frame, 2 13 Relativistic correction to Newtonian mechanics, 234
INDEX
393
energy, 233 4-momentum, 233 kinetic energy, 236 law of velocity addition, 220-221 length contraction, 222 mass, 236 time dilation, 223 time units, 227-228 3-momentum, 234 velocity. 227 Relativity group, 213 Rest frame, 229 Rest mass, 236 Restricted system, 307 Restriction mapping, 307 and particles, 308 Rigid body, 308-310 angular velocity vector of, 262, 279 body motion of, 267-268, 271 body representation of, 259 body stable theorem for, 277 conserved momentum of, 261, 266 Euler equations for, 282 inertial ellipsoid of, 280 inertia tensor of, 275 invariable plane of, 281 left invariant metric for, 260 moment of inertia of, 262, 276 outline of analysis of, 261-262 phase plane analysis for motion of, 283 Poinsot construction for, 280-281 principal axes of, 276 space motion, 260, 266-267 space representation of, 259 space stable theorem for, 280 special space motion, 261, 266-267 stationary rotation of, 274 Riemannian metric, 76 Rotation group, 256-258, 271 -274
S Sard’s theorem, 307 Scalar invariants in electromagnetism, 253 Sharp and flat mappings, 114 Simple pendulum, 100 Simultaneity in special relativity, 222 Skew symmetric tensor, 110, 141 Souriau equations, 32 1 Souriau GMS, 328-329, 352-353
Spacelike hypersurface, 77 subspace, 75 vector, 75 Space odyssey, 229-232 Spacetime, 2 I3 manifold, 225 metric, 226 Special relativity, 214 Standard topology on the cotangent bundle, 67 on the tangent bundle, 61 Starlike set, 159 State, I , 5 State space, I , 4,95, 255 Stationary rotation of a rigid body, 274 stability of, 277-280 Sternberg’s theorem, 346 Stokes’s theorem classical version, 209 for a generalized cylinder, 205-206 for a manifold with boundary, 201 for rectangles, 205 Straightening out theorem, 177 Structural group, 335 Subbundle of a vector bundle, 172 integrable, 176 Submanifold, 26 Submanifold chart, 26 Symmetric product, 81 Symmetry groups, 330-331, 333-334 Symplectic structure, 110
T Tangent bundle, 61 Tangent space, 30 Tangent vector, 29 Tensor, 127 Tensor bundles, 140 Tensor field, 129 Time dependent mechanical system, 121-124 vector field, 41 Timelike hypersurface, 77 subspace, 75 vector, 75 Topology and critical points, 90 Torque, 309-310
INDEX Torsion form, 383 Total charge for electromagnetism, 372 for gauge theory, 386 Total force mapping, 95 Trajectory, 2, 4 , 9 , 106, 255 Transformation law for contravariant vector components, 29 for covariant components, 67 for tensor components, 130 Trilogy, 374-375 Two-form on coadjoint orbits, 343-344
U Uncertainty principle, 358
v Variational principle, 300 Vector bundle, 63, 172 Vector field, 39 complete, 55 curl of, 169
as a differential operator, 137 divergence of, 169-170 global flow of, 54 Hamiltonian, 115, 255-256 integral curve of, 2, 40 Lagrangian, 118, 255 left invariant, 287 local flow of, 42-52 local representative of, 39 maximal integral curve of, 53 o-divergence of, 203 time dependent, 41 Vector subbundles, 172 Vector-valued differential form, 189-190 Velocity boost, 220 Velocity parameter, 22 I Velocity vector, 32 Virtual work, 96 Volume element, 149 Volume 3-forms. 171 W Wedge product, 111, 142, 190 World line, 213, 226
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