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Points of egress in problems of Hamiltonian dynamics

Published online by Cambridge University Press:  24 October 2008

C. J. Amick  and
J. F. Toland
C. J. Amick
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637, U.S.A.
J. F. Toland
Affiliation:
School of Mathematical Sciences, University of Bath, Bath, BA2 7AY
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Extract

First we consider an elementary though delicate question about the trajectory in ℝn of a particle in a conservative field of force whose dynamics are governed by the equation

Here the potential function V is supposed to have Lipschitz continuous first derivative at every point of ℝn. This is a natural assumption which ensures that the initial-value problem is well-posed. We suppose also that there is a closed convex set C with non-empty interior C° such that V ≥ 0 in C and V = 0 on the boundary ∂C of C. It is noteworthy that we make no assumptions about the degeneracy (or otherwise) of V on ∂C (i.e. whether ∇V = 0 on ∂C, or not); thus ∂C is a Lipschitz boundary because of its convexity but not necessarily any smoother in general. We remark also that there are no convexity assumptions about V and nothing is known about the behaviour of V outside C.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 2 , March 1991 , pp. 405 - 417
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

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