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Non smooth geodesic flows and classical mechanics

Marsden, Jerrold E. (1969) Non smooth geodesic flows and classical mechanics. Canadian Mathematical Bulletin, 12 . pp. 209-212. ISSN 0008-4395. http://resolver.caltech.edu/CaltechAUTHORS:20100824-105109180

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Abstract

As is well known, there is an intimate connection between geodesic flows and Hamiltonian systems. In face, if g is a Riemannian, or pseudo-Riemannian metric on a manifold M (we think of M as q -space or the configuration space), we may define a smooth function T_g on the cotangent bundle T^*M (q -p -space, or phase space). This function is the kinetic energy of q , and locally is given by T_g(q, p) = gij(q) P_i P_j Where q = (q^1,..., q^n) and p = (p_1,..., p_n) and g has components g^(ij). Using T_g as a Hamiltonian function, the associated flow (that is, the global solution of Hamiltonian's equations) is exactly the geodesic flow; geodesics are obtained by projection to M . Conversely, Hamiltonian motion in a potential V and metric g , that is, H = T_g + V , may be thought of as geodesic motion using the metric (e - V)g if e > V(q) . This new metric is called the Jacobi metric. Traditionally, the theory of classical mechanics and Riemannian geometry always assumes g and V are smooth functions. However, the most elementary examples in fact are not smooth. One of the main reasons for the smoothness assumption was to guarantee existence of the flow (geodesics). This objection has now been removed. The purpose of this note is to explain in an expository fashion what changes are necessary in the above theory to cover the non-smooth case. This new situation is quite different, although some interesting observations can be made.


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http://www.cds.caltech.edu/~marsden/bib/1969/02-Ma1969/AuthorUNSPECIFIED
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Additional Information:© 1969 CMS.
Subject Keywords:geodesic flows
Record Number:CaltechAUTHORS:20100824-105109180
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100824-105109180
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19629
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:14 Sep 2010 16:34
Last Modified:26 Dec 2012 12:21

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