Euler Equations of Problems of the Calculus of Variations with Prescribed Transversality Conditions

VOL. 17, 1931

MA THEMA TICS: L. LA PAZ

then we find readily that p i1(f) . [l(fi) (f)] P+1, 1 I () _< I(,,) (fO.

f p
the quantities rmk in (3) are expressible in the form = (Rmnk - Y,TkR,m - Y,sTmRk,,)/(1 + y,T,), (28) and that in view of (252) the quantities Fj are the unique solutions obtained by Cramer's Rule from the system of equations (29) (bTj/1yO,)F, = Nj (j = 1, .. ., n), the statements of the last paragraph may be formulated in the following theorem: THEOREM 3. A necessary and sufficient condition for the equations (24) to be a system of normal Euler equations of a non-singular problem (4) of the calculus of variations with integrand function f of class C"' is that there exist n functions Ti(x, yi, ..*. yn, Yi' . . . Yn) of class C" satisfying the inequalities (1) and a function H(x, yi, .... Yn,y1I.I.. y') such that the following 2n equations H/Jby' = Tk/(l + y'T,), (Tkl/by')Fa = Nk (k = 1, ..., n), (30) where Nk is the function defined in (16), are satisfied identically in x, yi, .. Yn, Y1l

** Yn-

VOL. 17, 1931

MA THEMA TICS: G. A. MILLER

463

Since

Rmk

=

(1 + y,"Tp)rmk + y'sTk"rm + yiTmr,

(31)

the compatibility of the sub-system (301) is insured by the relations (2) which n functions T, must satisfy if they are to be transversality coefficients, but the determination of the integrability conditions for the entire system (30) appears to be difficult and will be considered in a later paper. As an immediate corollary to Theorem 3 we have THEOREM 4. A necessary and sufficient condition for the integral curves of the system (24) to be the extremals of a problem (4) for which transversality is orthogonality is that there exists a single function l(x, yi, ...yYn) satisfying the system of partial differential equations (32) Fk = (l/ by, - yIl/ x)(I + y,y,) (k = 1, ..., n). 1 The term class is used in the sense of Bolza. Compare Vorlesungen uber Variationsrechnung, p. 13. 2 La Paz, L., Bull. Amer. Math. Soc., 36, p. 674, 1930. 3 In this connection a function f of the form (5) was first obtained by Rawles, Trans. Amer. Math. Soc., 30, p. 778, 1928. A proof that the formula (5) actually furnishes the most general integrand function of a non-singular problem with a specified transversality was first given by the writer, loc. cit., p. 680. 4Kasner, B., "The Theorem of Thompson and Tait and Natural Families of Trajectories," Trans. Amer. Math. Soc., 11, pp. 121-140, 1910; "Differential-Geometric Aspects of Dynamics," Princeton Colloquium Lectures, § 38, 1912.

THEOREMIS RELATING TO THE HISTORY OF MA THEMA TICS By G. A. MILLER DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS

Communicated June 22, 1931

Attention has frequently been directed to evidences of the dominance of the group concept in the development of mathematics, and in the second edition (1927) of his well-known Theorie der Gruppen von endlicher Ordnung A. Speiser emphasized symmetry in ancient ornaments as an evidence of a prehistoric group theory extending back to about 1500 B. C. It may be desirable to note here a few evidences of a lack of dominance of the group concept in developments closely related thereto in order to obtain a truer picture of the actual influence of this concept. The very late explicit formulation of the group concept is a striking instance in the history of mathematics of extreme slowness in coordinating related developments and it deserves emphasis on this account. Special arithmetic and geometric series appear in the writings of the