The class of integral variational functionals for paths in smooth manifolds, whose extremals are (nonparameterized) sets, is considered in this study. Recently, it was shown that for the functionals depending on tangent vectors, this property follows from any of the following two equivalent conditions: (a) the Lagrange function, defined on the tangent bundle, is positively homogeneous in the components of tangent vectors and (b) the Lepage differential form of the Lagrange function is projectable onto the Grassmann fibrations of rank 1 and order 1 (projective bundle); the classical Hilbert form was rediscovered this way as the projection of the Lepage form. In this paper, we extend these results to variational functionals of any order. Projectability conditions onto Grassmann fibrations of any order are found. The case of projective bundles is then studied in full generality. The proofs are based on the Euler–Zermelo conditions and the properties of higher-order Grassmann fibrations of rank 1. As an application, equations for set solutions in Riemannian geometry are derived.
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December 2020
Research Article|
December 07 2020
Variational principles: Projectability onto Grassmann fibrations Available to Purchase
Demeter Krupka
Demeter Krupka
a)
Department of Mathematics and Computer Science, Transilvania University
, Str. Iuliu Maniu 50, 500091 Brasov, Romania
and Lepage Research Institute
, 17 November St., 081 16 Presov, Slovakia
a)Author to whom correspondence should be addressed: [email protected]
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Demeter Krupka
a)
Department of Mathematics and Computer Science, Transilvania University
, Str. Iuliu Maniu 50, 500091 Brasov, Romania
and Lepage Research Institute
, 17 November St., 081 16 Presov, Slovakia
a)Author to whom correspondence should be addressed: [email protected]
J. Math. Phys. 61, 123501 (2020)
Article history
Received:
June 24 2020
Accepted:
November 18 2020
Citation
Demeter Krupka; Variational principles: Projectability onto Grassmann fibrations. J. Math. Phys. 1 December 2020; 61 (12): 123501. https://doi.org/10.1063/5.0019676
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