Velocities as function of canonical momenta

[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.

 

[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.