# Eliminating velocities from Lagrange E.o.M

1. Jun 15, 2012

### sphyrch

Hello, this doubt is related to Generalized Hamiltonian Dynamics paper by Dirac.

Consider the set of n equations : pi = $\partial$L/$\partial$vi,

(where vi is qi(dot) = dqi/dt, or time derivative of qi)(L is the lagrangian, q represent degrees of freedom in configuration space)

Now Dirac says : "If the n quantities $\partial$L/$\partial$vn 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:
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 Ʃaipi = 0 (where a's are non-zero coefficients), not involving q's at all.

Thanks.