Invariance of global solutions of the Hamilton-Jacobi equation

TL;DR

This paper proves that all global viscosity solutions of certain Hamilton-Jacobi equations are invariant under the symmetry group of the Hamiltonian, highlighting the symmetry properties of solutions on closed manifolds.

Contribution

It establishes the invariance of global solutions under the identity component of the Hamiltonian's symmetry group, which is shown to be a compact Lie group.

Findings

Global viscosity solutions are invariant under Hamiltonian symmetry group

The symmetry group of the Hamiltonian is a compact Lie group

Invariant Lagrangian sections are also symmetric

Abstract

We show that every global viscosity solution of the Hamilton-Jacobi equation associated with a convex and superlinear Hamiltonian on the cotangent bundle of a closed manifold is necessarily invariant under the identity component of the group of symmetries of the Hamiltonian (We prove that this group is a compact Lie group). In particular, every Lagrangian section invariant under the Hamiltonian flow is also invariant under this group.