# Velocities as function of canonical momenta

1. Jul 5, 2008

### jostpuur

Assuming I've understood some claims correctly, having defined the canonical momenta with equation

$$p_k = \frac{\partial L}{\partial \dot{q}_k},$$

we can solve the velocities as functions

$$\dot{q}_k(q_1,\ldots,q_n,p_1,\ldots, p_n)$$

precisely when the determinant

$$\textrm{det}\Big(\Big(\frac{\partial^2 L}{\partial \dot{q}_k\partial\dot{q}_{k'}}\Big)_{k,k'\in\{1,\ldots,n\}}\Big)$$

is non-zero. Why is this the case? The result looks reasonable, but I have difficulty seeing where this is coming from.

2. Jul 7, 2008

### CompuChip

Basically, to solve the velocities, you will want to use the implicit function theorem, from which it follows that the Hessian must be invertible. Also see the exact statement in the link.