Using notions of metric derivative and local Lipschitz constant, we define action
integral
and Hamiltonian operator for a class of optimal control problems on curves in metric
spaces.
We require the space to have a geodesic property (or more generally, length space
property).
Examples of such space includes space of probability measures in R^d and Banach
spaces which 
are not necessarily separable, among others. 

A well-posedness theory is developed for first order Hamilton-Jacobi equation in
this context.

Time permitting, I will explain an application which relates to variational
formulation of a compressible 
Euler equation.  Using a metric version of tangent cone concept, I will explain why
earlier attempts failed 
on well-posedness of an associated Hamilton-Jacobi equation in space of probability
measures. Such result 
is now established as a consequence of the metric result. 

This is joint work with Luigi Ambrosio.