Marsden, J. E.
Where q = (q1,..., qn) and p = (p1,..., pn) and g has components gij .
Using Tg 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 = Tg + 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.