# Generalized Coordinates

by Tac-Tics
Tags: coordinates, generalized
 Sci Advisor P: 1,588 Generalized Coordinates If the coordinates are $q_i$ and their corresponding canonical momenta are $p_i$, then the Poisson bracket of two functions f(q,p), g(q,p) is given by $$\{f, g\}_{PB} = \sum_i \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \sum_i \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i}$$ Then, the time evolution of any given function $\phi(q, p, t)$ is given by $$\frac{d\phi}{dt} = \frac{\partial \phi}{\partial t} + \{\phi, H\}_{PB}$$ where H is the Hamiltonian. It will be easiest to understand if you apply it to a simple one-dimensional system that you know already, like a harmonic oscillator or something. Just walk through the steps and see what happens.