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Keywords:
constrained variational integral; equivalence problem; diffiety; Poincaré-Cartan form; Frenet coframe
constrained variational integral; equivalence problem; diffiety; Poincaré-Cartan form; Frenet coframe
Summary:
Elements of general theory of infinitely prolonged underdetermined systems of ordinary differential equations are outlined and applied to the equivalence of one-dimensional constrained variational integrals. The relevant infinite-dimensional variant of Cartan’s moving frame method expressed in quite elementary terms proves to be surprisingly efficient in solution of particular equivalence problems, however, most of the principal questions of the general theory remains unanswered. New concepts of Poincaré-Cartan form and Euler-Lagrange system without Lagrange multiplies appearing as a mere by-product seem to be of independent interest in connection with the 23rd Hilbert problem.
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References:
[1] Anderson, R. S., Ibragimov, N. H.: Lie-Bäcklund transformations in applications. SIAM Philadelphia 1979. MR 0520395
[2] Cartan, E.: Les systèmes de Pfaff a cinq variables. Oeuvres complétes II 2, Paris 1955.
[3] Chrastina, J.: On the equivalence of variational problems I. (Journal of Differential Equations, Vol. 98 (1992), 76-90. MR 1168972 | Zbl 0764.49008
[4] Chrastina, J.: From elementary algebra to Bäcklund transformations. Czechoslovak Math. Journal, 40 (115) 1990, Praha. MR 1046292 | Zbl 0726.58041
[5] Chrastina, J.: Solution of the inverse problem of the calculus of variations. (to appear). MR 1293249 | Zbl 0821.49026
[6] Gardner, R. B.: The method of equivalence and its applications. CBMS-NSF regional conference series in appl. mathematics 58, Philadelphia 1989. MR 1062197 | Zbl 0694.53027
[7] Griffiths, P. A.: Exterior differential systems and the calculus of variations. Progress in Mathematics 25, Birkhäuser 1983. MR 0684663 | Zbl 0512.49003
[8]
Mathematical developments arising from Hilbert problems. Proc. Symp. in Pure and Appl. Math. AMS, Vol. XXVII, 1976. Zbl 0326.00002
[9] Kamran, N.: On the equivalence problem of Élie Cartan. Académie Royale de Belgique, Memoire de la classe de sciences, Collection in $8^0$-$2^{\text{e}}$ série, T. XLV - Fascicule 7 et dernier - 1989.
[10] Tzujishita, T.: On variational bicomplexes associated to differential equations. Osaka J. Math. 19 (1982), 311-363.
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