| Title:
|
Some concepts of regularity for parametric multiple-integral problems in the calculus of variations (English) |
| Author:
|
Crampin, M. |
| Author:
|
Saunders, D. J. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
59 |
| Issue:
|
3 |
| Year:
|
2009 |
| Pages:
|
741-758 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show that asserting the regularity (in the sense of Rund) of a first-order parametric multiple-integral variational problem is equivalent to asserting that the differential of the projection of its Hilbert-Carathéodory form is multisymplectic, and is also equivalent to asserting that Dedecker extremals of the latter $(m+1)$-form are holonomic. (English) |
| Keyword:
|
parametric variational problem |
| Keyword:
|
regularity |
| Keyword:
|
multisymplectic |
| MSC:
|
37Jxx |
| MSC:
|
49K10 |
| MSC:
|
49N60 |
| MSC:
|
53Cxx |
| MSC:
|
58E15 |
| MSC:
|
70Gxx |
| idZBL:
|
Zbl 1224.58012 |
| idMR:
|
MR2545653 |
| . |
| Date available:
|
2010-07-20T15:37:49Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140513 |
| . |
| Reference:
|
[1] Cantrijn, F., Ibort, A., Léon, M. de: On the geometry of multisymplectic manifolds.J. Australian Math. Soc. 66 (1999), 303-330. MR 1694063, 10.1017/S1446788700036636 |
| Reference:
|
[2] Cariñena, J. F., Crampin, M., Ibort, L. A.: On the multisymplectic formalism for first order field theories.Diff. Geom. Appl. 1 (1991), 345-374. MR 1244450, 10.1016/0926-2245(91)90013-Y |
| Reference:
|
[3] Crampin, M., Saunders, D. J.: The Hilbert-Carathéodory form for parametric multiple integral problems in the calculus of variations.Acta Applicandae Math. 76 (2003), 37-55. Zbl 1031.53106, MR 1967453, 10.1023/A:1022862117662 |
| Reference:
|
[4] Crampin, M., Saunders, D. J.: The Hilbert-Carathéodory and Poincaré-Cartan forms for higher-order multiple-integral variational problems.Houston J. Math. 30 (2004), 657-689. Zbl 1057.58008, MR 2083869 |
| Reference:
|
[5] Crampin, M., Saunders, D. J.: On null Lagrangians.Diff. Geom. Appl. 22 (2005), 131-146. Zbl 1073.70023, MR 2122738, 10.1016/j.difgeo.2004.10.002 |
| Reference:
|
[6] Dedecker, P. M.: On the generalization of symplectic geometry to multiple integrals in the calculus of variations.Lecture Notes in Mathematics, Springer 570 (1977), 395-456. Zbl 0352.49018, MR 0458478, 10.1007/BFb0087794 |
| Reference:
|
[7] Giaquinta, M., Hildenbrandt, S.: Calculus of Variations II.Springer (1996). MR 1385926 |
| Reference:
|
[8] Krupková, O.: Hamiltonian field theory.J. Geom. Phys. 43 (2002), 93-132. MR 1919207, 10.1016/S0393-0440(01)00087-0 |
| Reference:
|
[9] Rund, H.: The Hamilton-Jacobi Equation in the Calculus of Variations.Van Nostrand (1966). MR 0230189 |
| Reference:
|
[10] Rund, H.: A geometrical theory of multiple integral problems in the calculus of variations.Canadian J. Math. 20 (1968), 639-657. Zbl 0155.44301, MR 0238243, 10.4153/CJM-1968-062-1 |
| . |
