On Newton equations which are totally integrable at infinity

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 $n=2$.