The conjugate Hamiltonian can be defined from the Lagrangian as,
$$ p_i ~=~ \frac{\partial L}{\partial \dot{q}^i}$$
Typically the momenta components are given in spherical polars $(r, \theta, \phi)$.
How would one find the momenta in Cartesian coordinates?
I suspect that it will just transform as the spatial coordinates do, but am unable to show this. Example attempt at the x-component:
$$ p_x = \frac{\partial L}{\partial \dot{x}} = \frac{\partial L}{\partial \dot{r}} \frac{\partial \dot{r}}{\partial \dot{x}} = p_r \frac{\partial \dot{r}}{\partial \dot{x}}$$
But cannot see where to go from here.
Edits in response to comments
Some context: I am writing some code that takes a dataset of momentum vectors in Cartesian coordinates (not produced by me) and another set of vectors in spherical coordinates (also not produced by me), and calculates the angle between any two vectors.
To do this I need to first convert the momentum vectors in spherical coordinates, into Cartesian coordinates. Therefore I want a general transformation from momenta expressed in spherical coordinates, to Cartesian coordinates.