I have a dynamical system given by equality $\frac{d}{dt} \begin{bmatrix} q \\ p \end{bmatrix} = \begin{bmatrix} B p \\ -I(q, p) \end{bmatrix} + \begin{bmatrix} 0 \\ U(q) \end{bmatrix} u$, mechanical, from Euler-Lagrange formalism. It works fine in flat horizontal space. However, I want to migrate it to a non-flat topological space, a metric manifold $(g, \mathcal{M})$.
The two issues are imh:
- how to consider the geometrical spatial properties?
- How to add influence of potential map $\phi$ on anew system?