zbMATH — the first resource for mathematics

Some concepts of regularity for parametric multiple-integral problems in the calculus of variations. (English) Zbl 1224.58012
Summary: We show that asserting 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.

58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
49N60 Regularity of solutions in optimal control
Full Text: DOI EuDML
[1] F. Cantrijn, A. Ibort and M. de Léon: On the geometry of multisymplectic manifolds. J. Australian Math. Soc. 66 (1999), 303–330. · doi:10.1017/S1446788700036636
[2] J. F. Cariñena, M. Crampin and L. A. Ibort: On the multisymplectic formalism for first order field theories. Diff. Geom. Appl. 1 (1991), 345–374. · Zbl 0782.58057 · doi:10.1016/0926-2245(91)90013-Y
[3] M. Crampin and D. J. Saunders: The Hilbert-Carathéodory form for parametric multiple integral problems in the calculus of variations. Acta Applicandae Math. 76 (2003), 37–55. · Zbl 1031.53106 · doi:10.1023/A:1022862117662
[4] M. Crampin and D. J. Saunders: The Hilbert-Carathéodory and Poincaré-Cartan forms for higher-order multiple-integral variational problems. Houston J. Math. 30 (2004), 657–689. · Zbl 1057.58008
[5] M. Crampin and D. J. Saunders: On null Lagrangians. Diff. Geom. Appl. 22 (2005), 131–146. · Zbl 1073.70023 · doi:10.1016/j.difgeo.2004.10.002
[6] P. M. Dedecker: On the generalization of symplectic geometry to multiple integrals in the calculus of variations. Lecture Notes in Mathematics, Springer 570 (1977), 395–456. · doi:10.1007/BFb0087794
[7] M. Giaquinta and S. Hildenbrandt: Calculus of Variations II. Springer, 1996.
[8] O. Krupková: Hamiltonian field theory. J. Geom. Phys. 43 (2002), 93–132. · Zbl 1016.37033 · doi:10.1016/S0393-0440(01)00087-0
[9] H. Rund: The Hamilton-Jacobi Equation in the Calculus of Variations. Van Nostrand, 1966. · Zbl 0141.10602
[10] H. Rund: A geometrical theory of multiple integral problems in the calculus of variations. Canadian J. Math. 20 (1968), 639–657. · Zbl 0155.44301 · doi:10.4153/CJM-1968-062-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.