Regularity of generating families of functions

TL;DR

This paper explores the geometric conditions under which families of functions generate immersed Lagrangian submanifolds, focusing on the role of the Hessian in the variational formulation of physical theories.

Contribution

It provides new criteria involving the Hessian for when a family of functions generates an immersed Lagrangian submanifold in geometric variational frameworks.

Findings

Conditions for generating immersed Lagrangian submanifolds are characterized by the Hessian.

The geometric structures are linked to the variational formulation of physical theories.

The paper clarifies the role of these structures in the constitutive sets of physical systems.

Abstract

We describe the geometric structures involved in the variational formulation of physical theories. In presence of these structures, the constitutive set of a physical system can be generated by a family of functions. We discuss conditions, under which a family of functions generates an immersed Lagrangian submanifold. These conditions are given in terms of the Hessian of the family.