A multidimensional birkhoff theorem for time-dependent tonelli hamiltonians

TL;DR

This paper proves that certain invariant Lagrangian submanifolds in time-dependent Tonelli Hamiltonian systems are graphs over the base manifold, extending classical results to a multidimensional, time-dependent setting with implications for autonomous systems.

Contribution

It establishes a multidimensional Birkhoff theorem for time-dependent Tonelli Hamiltonians, showing invariant Lagrangian submanifolds are graphs over the manifold.

Findings

Invariant Lagrangian submanifolds are graphs over the base manifold.

In autonomous case, invariance extends to all time maps.

Results generalize classical Birkhoff theorems to time-dependent systems.

Abstract

Let $M$ be a closed and connected manifold, $H:T^*M\times \mathbb{R} / \mathbb{Z} \to \mathbb{R}$ a Tonelli $1$-periodic Hamiltonian and $\mathcal{L} \subset T^*M$ a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if $\mathcal{L}$ is invariant by the time-one map of $H$, then $\mathcal{L}$ is a graph over $M$. An interesting consequence in the autonomous case is that in this case, $\mathcal{L}$ is invariant by all the time $t$ maps of the Hamiltonian flow of $H$.