Variational submanifolds of Euclidean spaces

TL;DR

This paper investigates conditions under which non-variational differential systems can be realized as variational systems on submanifolds within Euclidean spaces, using variational sequence theory and providing explicit examples.

Contribution

It formulates and solves the problem of existence of variational submanifolds for second-order systems using variational sequence theory.

Findings

Conditions for variationality of submanifolds are established.

Existence of variational submanifolds is proven for second-order systems.

Examples demonstrate variational submanifolds in geometry and mechanics.

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.