I'm "researching" on unquantised Yang-Mills theory. For that I'm studying the Dirac's method for singular constrained systems and having problems to follow the first considerations on that matter. I have, basically, two questions:
(1) I'm aware that the condition for a system to be singular and, therefore, not all velocities be uniquely written as a function of the canonical momenta - $\dot{q_i}=\dot{q_i}(q,p)$ (where $i$ runs through all the initial assumed degrees of freedom) - is that the determinant of the hessian matrix is null - $det(W_{ij})=|\frac{\partial²L}{\partial\dot{q_i}\partial\dot{q_j}}|=0$. Why is this? I know that the Legendre transformation by itself is bounded by this condition and that this is a consequence of the inverse (or implicit) function theorem, but I'd like to understand it in a somewhat more "viewable" way (even if that is the demonstration or explicit use of the theorems on this specific case).
(2) Also, I know that this is the requirement for the E-L equations to be (certainly) uniquely solvable for the accelerations. It is clear, by expanding the total time derivative on the E-L equations, that this is because if the determinant is zero (so that the matrix is not invertible), we cannot uncouple the accelerations from the hessian:
$W_{ij}\ddot{q_j}=f(q,\dot{q})$
where the here unimportant explicit form of the RHS is being neglected. But why this prevents the equation from being uniquely solvable (I'm pretty sure that this is more than obvious, even so I'm having trouble assimilating it)?
Sorry about my terrible English.
Thanks a lot!
