Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Authors: Ângelo Barone Netto and Gaetano Zampieri
Journal: Proc. Amer. Math. Soc. 121 (1994), 711-713
MSC: Primary 70H35; Secondary 58E99, 70K20
MathSciNet review: 1181156
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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.

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Keywords: Natural Lagrangian systems, Hopf-Rinow theorem
Article copyright: © Copyright 1994 American Mathematical Society