Elementary Poisson brackets | Classical mechanics

Elementary Poisson brackets

The Poisson brackets constructed out of the canonical coordinates themselves (co-ordinate and momenta) are called elementary Poisson brackets.

Properties of Poisson bracket

(1) [q_i, q_j] = [p_i, p_j] = 0

$\because [u,v]_{(p,q)}=\sum_{i=1}^{n}\left (\frac{\partial u}{\partial q_{i}}\frac{\partial v}{\partial p_{i}}-\frac{\partial u}{\partial p_{i}}\frac{\partial v}{\partial q_{i}} \right )$

$\therefore [q_{i},q_{j}]_{(p,q)}=\sum_{i=1}^{n}\left (\frac{\partial q_{i}}{\partial q_{i}}\frac{\partial q_{j}}{\partial p_{i}}-\frac{\partial q_{i}}{\partial p_{i}}\frac{\partial q_{j}}{\partial q_{i}} \right ) =\left ( 1\times 0 -0\times \frac{\partial q_{j}}{\partial q_{i}}\right ) =0$

or [q_i, q_j] = 0

Similarly [p_i, p_j] = 0
Thus [q_i, q_j] = [p_i, p_j] = 0

(2) [q_i, p_j] = – [p_j, q_i] = δ_ij

(3)

(4)

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