How far can one move from a potential peak with small initial speed?

Abstract:

We consider a natural Lagrangian system and show that from a point ${q_0}$ in n-space, where the potential energy V has a (weak) maximum, one can go near the boundary of any compact ball where $V(q) \leq V({q_0})$ with (arbitrarily small) nonvanishing initial speeds. The result holds true for sets which are ${C^2}$-diffeomorphic to a compact ball. This property is found as a simple consequence of the Hopf-Rinow theorem and of a theorem of Gordon. As a corollary we deduce a well-known local result, namely, a ’converse’ of the Lagrange-Dirichlet theorem, thus obtained via geometric arguments.