On Newton equations which are totally integrable at infinity

TL;DR

This paper proves that for certain Hamiltonian systems in three or more dimensions, if all orbits near infinity are action minimizing, then the potential must be constant, generalizing previous minimal orbit results.

Contribution

It extends rigidity results for Hamiltonian systems by showing potential constancy under broader conditions involving action-minimizing orbits near infinity.

Findings

Potential must be constant if all orbits near infinity are action minimizing in dimensions 3 and higher.

Generalizes previous results requiring all orbits to be minimal.

The statement does not hold for dimension 1 and is unknown for dimension 2.

Abstract

In this paper Hamiltonian system of time dependent periodic Newton equations is studied. It is shown that for dimensions $3$ and higher the following rigidity results holds true: If all the orbits in a neighborhood of infinity are action minimizing then the potential must be constant. This gives a generalization of the previous result \cite{B3}, where it was required all the orbits to be minimal. As a result we have the following application: Suppose that for the time-1 map of the Hamiltonian flow there exists a neighborhood of infinity which is filled by invariant Lagrangian tori homologous to the zero section. Then the potential must be constant. Remarkably, the statement is false for $n=1$ case and remains unknown to the author for $n=2$.