Natural boundary conditions in geometric calculus of variations

TL;DR

This paper derives natural boundary conditions for a broad class of variational problems with free boundaries, emphasizing the geometric and topological flexibility of the framework using jet bundle techniques.

Contribution

It introduces a general geometric approach to natural boundary conditions in variational calculus, applicable to various topologies and Lagrangian orders.

Findings

Derived explicit natural boundary conditions for first-order Lagrangians.

Established a geometric framework using jet bundles and horizontal forms.

Demonstrated the approach with examples involving hypersurface area.

Abstract

In this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of $Y$, the manifold of dependent and independent variables underlying a given problem, as well as the order of its Lagrangian. Our result follows from the natural behavior, under boundary-friendly transformations, of an operator, similar to the Euler map, constructed in the context of relative horizontal forms on jet bundles (or Grassmann fibrations) over $Y$. Explicit examples of natural boundary conditions are obtained when $Y$ is an $(n+1)$--dimensional domain in $\R^{n+1}$, and the Lagrangian is first-order (in particular, the hypersurface area).