this doubt is related to Generalized Hamiltonian Dynamics paper by Dirac.
Consider the set of $n$ equations : $p_i$ = $∂L/∂v_i$,
(where $v_i$ is $q_i$(dot) = $dq_i/dt$, or time derivative of $q_i$)($L$ is the lagrangian, $q$ represent degrees of freedom in configuration space)
Now Dirac says : "If the $n$ quantities $∂L/∂v_i$ on the right-hand side of the given equations are NOT independent functions of the velocities, we can eliminate the $v$'s from (the above-given) set of equations and obtain one or more equations: $\phi_j(q, p) = 0$, ($j = 1, 2, ... , m$ if there are $m$ such constraints)"
Could anyone please explain how this comes about? I can't understand how the $v$'s can be necessarily eliminated, and if the $p$'s are not all independent, then we can simply obtain relations like $\sum_i a_ip_i = 0$ (where $a$'s are non-zero coefficients), not involving $q$'s at all.
Thanks (and apologies in case I'm missing something obvious).