Solution of the inverse problem of the calculus of variations

Jan Chrastina

Mathematica Bohemica (1994)

  • Volume: 119, Issue: 2, page 157-201
  • ISSN: 0862-7959

Abstract

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Given a family of curves constituting the general solution of a system of ordinary differential equations, the natural question occurs whether the family is identical with the totality of all extremals of an appropriate variational problem. Assuming the regularity of the latter problem, effective approaches are available but they fail in the non-regular case. However, a rather unusual variant of the calculus of variations based on infinitely prolonged differential equations and systematic use of Poincaré-Cartan forms makes it possible to include even all constrained variational problems. The new method avoids the use of Lagrange multiplitiers. For this reason, it is of independent interest especially in regard to the 23rd Hilbert's problem.

How to cite

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Chrastina, Jan. "Solution of the inverse problem of the calculus of variations." Mathematica Bohemica 119.2 (1994): 157-201. <http://eudml.org/doc/29233>.

@article{Chrastina1994,
abstract = {Given a family of curves constituting the general solution of a system of ordinary differential equations, the natural question occurs whether the family is identical with the totality of all extremals of an appropriate variational problem. Assuming the regularity of the latter problem, effective approaches are available but they fail in the non-regular case. However, a rather unusual variant of the calculus of variations based on infinitely prolonged differential equations and systematic use of Poincaré-Cartan forms makes it possible to include even all constrained variational problems. The new method avoids the use of Lagrange multiplitiers. For this reason, it is of independent interest especially in regard to the 23rd Hilbert's problem.},
author = {Chrastina, Jan},
journal = {Mathematica Bohemica},
keywords = {Poincaré-Cartan form; Lagrange problem; Monge systems; inverse problem; constrained variational integrals; Poincaré-Cartan form; Lagrange problem; Monge systems; inverse problem; constrained variational integrals},
language = {eng},
number = {2},
pages = {157-201},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Solution of the inverse problem of the calculus of variations},
url = {http://eudml.org/doc/29233},
volume = {119},
year = {1994},
}

TY - JOUR
AU - Chrastina, Jan
TI - Solution of the inverse problem of the calculus of variations
JO - Mathematica Bohemica
PY - 1994
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 119
IS - 2
SP - 157
EP - 201
AB - Given a family of curves constituting the general solution of a system of ordinary differential equations, the natural question occurs whether the family is identical with the totality of all extremals of an appropriate variational problem. Assuming the regularity of the latter problem, effective approaches are available but they fail in the non-regular case. However, a rather unusual variant of the calculus of variations based on infinitely prolonged differential equations and systematic use of Poincaré-Cartan forms makes it possible to include even all constrained variational problems. The new method avoids the use of Lagrange multiplitiers. For this reason, it is of independent interest especially in regard to the 23rd Hilbert's problem.
LA - eng
KW - Poincaré-Cartan form; Lagrange problem; Monge systems; inverse problem; constrained variational integrals; Poincaré-Cartan form; Lagrange problem; Monge systems; inverse problem; constrained variational integrals
UR - http://eudml.org/doc/29233
ER -

References

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  1. Ian. M. Anderson, Aspects of the inverse problem to the calculus of variations, Archivum Mathematicum (Brno), vol. 24 (1988), 181-202. (1988) Zbl0674.58017MR0983236
  2. R.L. Bryant, S.S. Chern, R.B. Gardner, H.L. Goldschmidt, P.A. Griffiths, Exterior differential systems, Math. Sc. Research Inst. Publ. 18, Springer Verlag 1991. (1991) Zbl0726.58002MR1083148
  3. E. Cartan, Les systémes différentielles extérieurs et leurs applications, Actualités scient. et. ind. no. 944, 1945. (1945) 
  4. J. Chrastina, Formal calculus of variations on fibered manifolds, Folia Fac. Sclent. Nat. Univ. Purkynianae Brunensis, Mathematica 2, Brno 1989. (1989) Zbl0696.49075MR1036730
  5. G. Darboux, Leçons sur la théorie génerale des surfaces, Paris 1894, §§604 and 605. Zbl0257.53001
  6. J. Douglas, 10.1090/S0002-9947-1941-0004740-5, Transactions AMS 50 (1941), 71-128. (1941) Zbl0025.18102MR0004740DOI10.1090/S0002-9947-1941-0004740-5
  7. Mathematical developments arising from Hilbert problems, Proc. Symp. in Pure and Appl. Math. AMS, Vol. XXVII, 1976. (1976) Zbl0326.00002
  8. Peter J. Olver, Application of Lie groups to differential equations, Graduate Text in Mathematics 107, Springer-Verlag 1989. (1989) MR1240056
  9. R.M. Santilli, Foundation of theoretical mechanics I., The inverse problems in Newtonian mechanics, Texts and Monographs in Physics 18, Springer-Verlag 1978. (1978) MR0514210

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