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Invariant variation problems. (English) Zbl 0292.49008

49K27 Optimality conditions for problems in abstract spaces
49R50 Variational methods for eigenvalues of operators (MSC2000)
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[1] Hamel, Math. Ann pp 59–
[2] Z. F. Math. u. Phys pp 50–
[3] Herglotz, Ann. d. Phys 36 pp 511–
[4] Fokker. Jan. 27 1917. Jan. 27, Verslag d. Amsterdamer Akad. For further bibliography, compare Klein’s second Note,Göttinger Nachrichten, July 19, 1918. A recently published work by Kneser (Math.Zschr. 2) deals with the setting up of invariants by a similar method
[5] Lie, in ”Grundlagen fur die Theorie der Unendlichen kontinuierlichen Transformationsgruppen” (Foundations of the theory of infinite continuous groups of transformations),Ber. d. K. Sachs. Ges. d. Wiss. 1891 (cited asGrundlagen., defines the infinite continuous group as a group of transformations which are given by the most general solutions of a system of partial differential equations, provided these solutions do not depend only on a finite number of parameters. One of the above-mentioned types differing from the finite group will thus be thereby obtained; whereas conversely the limiting case of infinitely many parameters need not necessarily satisfy a system of differential equations
[6] I suppress the subscripts, insofar as feasible, even in summations; thus, 2u/x2for 2u{\(\alpha\)}/x{\(\beta\)}x{\(\gamma\)}, etc
[7] By way of abbreviation, I write dx, dy for dx1... dxn, dy1... dyn
[8] All arguments x, u, {\(\epsilon\)}, p(x) occurring in the transformations are to be assumed real, whereas the coefficients may be complex. But since the final results are concerned withidentitiesin the X’S, u’s, parameters and arbitrary functions, these hold also for complex values, provided only that all functions that occur are assumed analytic. A large portion of the results, incidentally, can be justified without integrals, so that here the restriction to reals is not necessary even to the arguments. On the other hand, the developments at the close of Section II and beginning of Section V do not appear to be feasible without integrals
[9] For certain trivial exceptions, compare Section II, 2nd remark
[10] Somewhat more generally, we may alternatively put {\(\psi\)}i= Ti; cf. Section III, (15)
[11] Cf. close of Section III
[12] Cf. e.g., Klein’s presentation
[13] That is, {\(\psi\)}i{\(\delta\)}uiacquires a factor upon transformation
[14] Compare Klein’s second note
[15] Cf. e.g., Lie,Grundlagen, p. 331. Where arbitrary functions are concerned, the special values a{\(\sigma\)} of the parameters are to be replaced by fixed functions p{\(\sigma\)}, p{\(\sigma\)}/x, ...; and correspondingly, the values a{\(\sigma\)}+ {\(\epsilon\)} by p + p(x), p{\(\sigma\)}/x + p/x, etc
[16] (12) goes over into 0 = 0 for the trivial case – which can occur only if {\(\delta\)}x, {\(\delta\)}u depend also on derivatives of the u’s – when Div (f{\(\delta\)}x) = 0, {\(\delta\)}u = 0; thus these infinitesimal transformations are always to be eliminated from the groups, and only the number of remaining parameters, or arbitrary functions, is to be counted in the formulation of the theorems. Whether the remaining infinitesimal transformations still form a group must be left moot
[17] That it signifies no restriction to assume the p’s free from u, u/x, is shown by the converse
[18] {\(\psi\)}i= 0, or, somewhat more generally, {\(\psi\)}i= Ti, where Tiare newly adjoined functions, are referred to in physics as ”field equations.” In the case {\(\psi\)}i= Ti, the identities (13) go over into equations Div B({\(\lambda\)})= Ti{\(\delta\)}ui({\(\lambda\)}), likewise known in physics as laws of conservation
[19] Provided f is non-linear in the {\(\kappa\)}-th derivatives
[20] Otherwise we also have u’{\(\lambda\)}= const. for every{\(\lambda\)}, corresponding to u”(u’){\(\lambda\)}-1= 1/{\(\lambda\)} d/dx (u’){\(\lambda\)}
[21] Hence it follows in particular that the groupGgenerated from the infinitesimal transformations {\(\delta\)}x, {\(\delta\)}u of aG{\(\rho\)}reduces back toG{\(\rho\)}. ForG{\(\rho\)}contains no infinitesimal transformations distinct from {\(\Delta\)}x, {\(\Delta\)}u dependent on arbitrary functions, and cannot contain any independent of them but depending on parameters, as otherwise it would be a mixed group. But according to the above, the infinitesimal transformations determine the finite ones
[22] The question whether perhaps this latter case always occurs was raised in a different formulation by Lie (Grundlagen, #7 and #13 at end)
[23] As in Section III, it here again follows from the converse that besides I, every integral I different from it by an integral over a divergence likewise admits of an infinite group, with the same {\(\delta\)}u’s, though {\(\delta\)}x and {\(\delta\)}u will in general involve derivatives of the u’s. Such an integral I was introduced by Einstein in the general theory of relativity to obtain a simpler version of the laws of conservation of energy; I specify the infinitesimal transformations that this I admits of, adhering precisely in nomenclature to Klein’s second note. The integral I = ... K dw = ... KdS admits of the group ofalltransformations of the w’s and those induced thereby for the g{\(\mu\)}v’s; to this correspond the dependencies (Klein’s (30))
[24] That is, {\(\psi\)}i{\(\delta\)}uitakes on a factor upon transformation, and this always used to be termed relative invariance in the algebraic theory of invariance
[25] These conclusions fail if y depends also on the u’s, since in that case {\(\delta\)}f(y,v, v/y, ...) also contains terms f/y {\(\delta\)}y, so that the divergence transformation does not lead to the Lagrange expressions; and similarly if derivatives of the u’s are admitted; for in that case the {\(\delta\)}v’s become linear combinations of {\(\delta\)}u, {\(\delta\)}u/x, ..., and so lead only after another divergence transformation to an identity ... ({\(\psi\)}i(u, ...)u)dx = 0, so that again the Lagrange expressions do not appear on the right. The question whether it is possible to argue from the invariance of ... ({\(\psi\)}i{\(\delta\)}ui)dx back to the subsistence of divergence relationships is synonymous, according to the converse, with the question whether one can thence infer the invariance of I with respect to a group leading not necessarily to the same {\(\delta\)}u, {\(\delta\)}x, but to the same {\(\delta\)}u’s. In the special case of the single integral and only first derivatives in f, it is possible for the finite group to argue from the invariance of the Lagrange expressions to the existence of first integrals (cf. e.g., Engel,Gott.Nachr. 1916, p. 270)
[26] It turns out again that y must be taken independent cf u in order for the conclusions to hold. As an example, consider the g{\(\mu\)}vand q{\(\sigma\)}given by Klein, which satisfy the transformations for variations provided the p’s are subjected to a vector transformation
[27] In the cases where mere invariance of ({\(\psi\)}i{\(\delta\)}ui)dx entails the existence of first integrals, these do not admit of the entire groupGp; for example, (u”{\(\delta\)}u)dx admits of the infinitesimal transformation {\(\Delta\)}x = {\(\epsilon\)}2, {\(\Delta\)}u = {\(\epsilon\)}1E + X{\(\epsilon\)}3; whereas the first integral u - u’x = const., corresponding to {\(\Delta\)}x = 0, {\(\Delta\)}u = X{\(\epsilon\)}3, does not admit of the other two infinitesimal transformations, since it explicitly contains both u and x. To this first integral, there happen to correspond infinitesimal transformations for f that contain derivatives. So we see that invariance ... ({\(\psi\)}i{\(\delta\)}ui)dx is at all events a weaker condition than invariance of I, and this should be noted as to a question raised in a previous remark
[28] The laws of conservation of energy of classical mechanics as well as those of the old ”theory of relativity” (where dx2goes over into itself) are ”proper”, since no infinite groups occur
[29] From these infinitesimal transformations, the finite ones are calculated backwards by the method given in Section IV at end
[30] Jber. d. d. Math. Vereining 19 pp 287– (1910)
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