Hamiltonian Mechanics

Abstract

Lagrange’s equations

$$\frac{\mathrm{d}} {\mathrm{d}t}\left ( \frac{\partial L} {\partial {v}_{a}}\right ) - \frac{\partial L} {\partial {q}_{a}} = 0,\qquad \frac{\mathrm{d}{q}_{a}} {\mathrm{d}t} = {v}_{a}$$

(7.1)

determine the trajectories in the extended phase space of a holonomic system subject to conservative forces. We have seen that it is possible to simplify the dynamical analysis by making coordinate transformations of the form

$$\begin{array}{lll} \tilde{{q}}_{a}& =&\tilde{{q}}_{a}(q,t)\\ \\ \\ \tilde{{v}}_{a}& =&\tilde{{v}}_{a}(q,v,t) = \frac{\partial \tilde{{q}}_{a}} {\partial {q}_{b}}{v}_{b} + \frac{\partial \tilde{{q}}_{a}} {\partial t}\\ \\ \\ \tilde{t} & =&t,\end{array}$$

(7.2)

a technique that proved particularly useful for handling constraints.