| Title:
|
Solution of the inverse problem of the calculus of variations (English) |
| Author:
|
Chrastina, Jan |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
119 |
| Issue:
|
2 |
| Year:
|
1994 |
| Pages:
|
157-201 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Poincaré-Cartan form |
| Keyword:
|
Lagrange problem |
| Keyword:
|
Monge systems |
| Keyword:
|
inverse problem |
| Keyword:
|
constrained variational integrals |
| MSC:
|
49N45 |
| MSC:
|
58E30 |
| idZBL:
|
Zbl 0821.49026 |
| idMR:
|
MR1293249 |
| DOI:
|
10.21136/MB.1994.126079 |
| . |
| Date available:
|
2009-09-24T21:04:39Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126079 |
| . |
| Reference:
|
[An] Ian. M. Anderson: Aspects of the inverse problem to the calculus of variations.Archivum Mathematicum (Brno), vol. 24 (1988), 181-202. Zbl 0674.58017, MR 0983236 |
| Reference:
|
[Br] 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. Zbl 0726.58002, MR 1083148 |
| Reference:
|
[Ca] E. Cartan: Les systémes différentielles extérieurs et leurs applications.Actualités scient. et. ind. no. 944, 1945. |
| Reference:
|
[Ch] J. Chrastina: Formal calculus of variations on fibered manifolds.Folia Fac. Sclent. Nat. Univ. Purkynianae Brunensis, Mathematica 2, Brno 1989. Zbl 0696.49075, MR 1036730 |
| Reference:
|
[Da] G. Darboux: Leçons sur la théorie génerale des surfaces.Paris 1894, §§604 and 605. Zbl 0257.53001 |
| Reference:
|
[Do] J. Douglas: Solution of the inverse problem of the calculus of variations.Transactions AMS 50 (1941), 71-128. Zbl 0025.18102, MR 0004740, 10.1090/S0002-9947-1941-0004740-5 |
| Reference:
|
[Ma] : Mathematical developments arising from Hilbert problems.Proc. Symp. in Pure and Appl. Math. AMS, Vol. XXVII, 1976. Zbl 0326.00002 |
| Reference:
|
[Ol] Peter J. Olver: Application of Lie groups to differential equations.Graduate Text in Mathematics 107, Springer-Verlag 1989. MR 1240056 |
| Reference:
|
[Sa] R.M. Santilli: Foundation of theoretical mechanics I., The inverse problems in Newtonian mechanics.Texts and Monographs in Physics 18, Springer-Verlag 1978. MR 0514210 |
| . |
