quantum mechanics - Canonical momentum in different coordinate system

The canonical momentum is defined as

$p_{i} = \frac {\partial L}{\partial \dot{q_{i}}}$, where $L$ is the Lagrangian.

So actually how does $p_{i}$ transform in one coordinate system $\textbf{q}$ to another coordinate system $\textbf{Q}$ ?

http://en.wikipedia.org/wiki/Hamiltonian_mechanics#Charged_particle_in_an_electromagnetic_field

When dealing with the Hamiltonian of the electromagnetic field, the derivation of $p_{j} = m \dot{x_j} + eA_j$ on the above link is usually written as $\textbf{p} = m \textbf{v} + e\textbf{A}$

but the derivation is based on using Cartesian coordinates, does it mean that $\textbf{p}$ is really a vector? If we are using another general coordinates, say, spherical coordinates, can we still have $\textbf{p} = m \textbf{v} + e\textbf{A}$ ? If no, I think the form of Hamiltonian in electromagnetic field

$H = \frac{(\textbf{p} - e\textbf{A})^2}{2m} + e\phi$

will only be valid in Cartesian coordinates. In any other coordinates, $H$ carries a different form!

Any comments are appreciated.