On the equivalence of variational problems. II

Jan Chrastina

Archivum Mathematicum (1993)

  • Volume: 029, Issue: 3-4, page 197-220
  • ISSN: 0044-8753

Abstract

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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.

How to cite

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Chrastina, Jan. "On the equivalence of variational problems. II." Archivum Mathematicum 029.3-4 (1993): 197-220. <http://eudml.org/doc/247439>.

@article{Chrastina1993,
abstract = {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.},
author = {Chrastina, Jan},
journal = {Archivum Mathematicum},
keywords = {constrained variational integral; equivalence problem; diffiety; Poincaré-Cartan form; Frenet coframe; Poincaré-Cartan form; variational problem with differential constraints; integral functional; Cartan's moving frame method},
language = {eng},
number = {3-4},
pages = {197-220},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the equivalence of variational problems. II},
url = {http://eudml.org/doc/247439},
volume = {029},
year = {1993},
}

TY - JOUR
AU - Chrastina, Jan
TI - On the equivalence of variational problems. II
JO - Archivum Mathematicum
PY - 1993
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 029
IS - 3-4
SP - 197
EP - 220
AB - 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.
LA - eng
KW - constrained variational integral; equivalence problem; diffiety; Poincaré-Cartan form; Frenet coframe; Poincaré-Cartan form; variational problem with differential constraints; integral functional; Cartan's moving frame method
UR - http://eudml.org/doc/247439
ER -

References

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  1. Lie-Bäcklund transformations in applications, SIAM Philadelphia 1979. MR0520395
  2. Les systèmes de Pfaff a cinq variables, Oeuvres complétes II 2, Paris 1955. 
  3. On the equivalence of variational problems I, (Journal of Differential Equations, Vol. 98 (1992), 76-90. Zbl0764.49008MR1168972
  4. From elementary algebra to Bäcklund transformations, Czechoslovak Math. Journal, 40 (115) 1990, Praha. Zbl0726.58041MR1046292
  5. Solution of the inverse problem of the calculus of variations, (to appear). Zbl0821.49026MR1293249
  6. The method of equivalence and its applications, CBMS-NSF regional conference series in appl. mathematics 58, Philadelphia 1989. Zbl0694.53027MR1062197
  7. Exterior differential systems and the calculus of variations, Progress in Mathematics 25, Birkhäuser 1983. Zbl0512.49003MR0684663
  8. Mathematical developments arising from Hilbert problems, Proc. Symp. in Pure and Appl. Math. AMS, Vol. XXVII, 1976. Zbl0326.00002
  9. On the equivalence problem of Élie Cartan, Académie Royale de Belgique, Memoire de la classe de sciences, Collection in - série, T. XLV - Fascicule 7 et dernier - 1989. 
  10. On variational bicomplexes associated to differential equations, Osaka J. Math. 19 (1982), 311-363. 

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