On invariant sets in Lagrangian graphs

TL;DR

This paper proves that Hamiltonians are constant on certain invariant sets within smooth Lagrangian graphs and discusses counterexamples when smoothness conditions are not met.

Contribution

It establishes conditions under which Hamiltonians are constant on invariant sets in Lagrangian graphs and provides counterexamples for less smooth cases.

Findings

Hamiltonians are constant on compact invariant sets in smooth Lagrangian graphs.

Counterexamples show the necessity of smoothness assumptions.

Results clarify the role of smoothness in Hamiltonian invariance.

Abstract

In this exposition, we show that a Hamiltonian is always constant on a compact invariant connected subset which lies in a Lagrangian graph provided that the Hamiltonian and the graph are smooth enough. We also provide some counterexamples for the case that the Hamiltonians are not smooth enough.