  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 $n=2$.
