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arXiv · 1709.07826

Variational submanifolds of Euclidean spaces

Demeter Krupka·Zbyněk Urban·Jana Volná
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Abstract

Systems of ordinary differential equations (or dynamical forms in Lagrangian mechanics), induced by embeddings of smooth fibered manifolds over one-dimensional basis, are considered in the class of variational equations. For a given non-variational system, conditions assuring variationality (the Helmholtz conditions) of the induced system with respect to a submanifold of a Euclidean space are studied, and the problem of existence of these "variational submanifolds" is formulated in general and solved for second-order systems. The variational sequence theory on sheaves of differential forms is employed as a main tool for analysis of local and global aspects (variationality and variational triviality). The theory is illustrated by examples of holonomic constraints (submanifolds of a configuration Euclidean space) which are variational submanifolds in geometry and mechanics.

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Demeter Krupka, Zbyněk Urban, Jana Volná. 2017-09-22. Variational submanifolds of Euclidean spaces. https://doi.org/10.1063/1.5010221

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