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

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 $V(q)\le V(q_0)$ 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.