- #1
jostpuur
- 2,116
- 19
Assuming I've understood some claims correctly, having defined the canonical momenta with equation
[tex]
p_k = \frac{\partial L}{\partial \dot{q}_k},
[/tex]
we can solve the velocities as functions
[tex]
\dot{q}_k(q_1,\ldots,q_n,p_1,\ldots, p_n)
[/tex]
precisely when the determinant
[tex]
\textrm{det}\Big(\Big(\frac{\partial^2 L}{\partial \dot{q}_k\partial\dot{q}_{k'}}\Big)_{k,k'\in\{1,\ldots,n\}}\Big)
[/tex]
is non-zero. Why is this the case? The result looks reasonable, but I have difficulty seeing where this is coming from.
[tex]
p_k = \frac{\partial L}{\partial \dot{q}_k},
[/tex]
we can solve the velocities as functions
[tex]
\dot{q}_k(q_1,\ldots,q_n,p_1,\ldots, p_n)
[/tex]
precisely when the determinant
[tex]
\textrm{det}\Big(\Big(\frac{\partial^2 L}{\partial \dot{q}_k\partial\dot{q}_{k'}}\Big)_{k,k'\in\{1,\ldots,n\}}\Big)
[/tex]
is non-zero. Why is this the case? The result looks reasonable, but I have difficulty seeing where this is coming from.