Exercise 3

We calculate the following Poisson Brackets \[ \{x, L_z\} \] \[ \{y, L_z\} \] \[ \{z, L_z\} \]

by using the definition and then by applying the axioms. To simplify notation we rename the variables: \[ x \to q_1, \; p_x \to p_1, \; L_x \to L_1 \] \[ y \to q_2, \; p_y \to p_2, \; L_y \to L_2 \] \[ z \to q_3, \; p_z \to p_3, \; L_z \to L_3 \]

The following holds true: \[ L_3 = q_1 p_2 - q_2 p_1 \] \[ \frac{\partial{L_3}}{\partial{q_1}} = p_2, \; \frac{\partial{L_3}}{\partial{p_1}} = -q_2\] \[ \frac{\partial{L_3}}{\partial{q_2}} = -p_1, \; \frac{\partial{L_3}}{\partial{p_2}} = q_1\] \[ \frac{\partial{L_3}}{\partial{q_3}} = 0, \; \frac{\partial{L_3}}{\partial{p_3}} = 0\] \[ \frac{\partial{q_i}}{\partial{q_j}} = \delta_{ij}, \; \frac{\partial{q_i}}{\partial{p_j}} = 0 \]

Hence \[ \left\{ x, L_z \right\} = \left\{ q_1, L_3 \right\} = \sum_i \left( \frac{\partial{q_1}}{\partial{q_i}} \frac{\partial{L_3}}{\partial{p_i}} - \frac{\partial{q_1}}{\partial{p_i}} \frac{\partial{L_3}}{\partial{q_i}} \right) = - q_2 = -y \] \[ \left\{ y, L_z \right\} = \left\{ q_2, L_3 \right\} = \sum_i \left( \frac{\partial{q_2}}{\partial{q_i}} \frac{\partial{L_3}}{\partial{p_i}} - \frac{\partial{q_2}}{\partial{p_i}} \frac{\partial{L_3}}{\partial{q_i}} \right) = q_1 = x\] \[ \left\{z, L_z \right\} = \left\{ q_3, L_3 \right\} = \sum_i \left( \frac{\partial{q_3}}{\partial{q_i}} \frac{\partial{L_3}}{\partial{p_i}} - \frac{\partial{q_3}}{\partial{p_i}} \frac{\partial{L_3}}{\partial{q_i}} \right) = 0\]

Now we use the axioms. In each step it is indicated which of the axioms (1) - (8) we are using. \[ \left\{x, L_z \right\} = \left\{x, xp_y - yp_x \right\} \] \[ \stackrel{\rm (2)}{=} - \left\{xp_y - yp_x, x\right\} \] \[ \stackrel{\rm (5)}{=} - \left[ \left\{xp_y, x \right\} - \left\{ yp_x, x \right\} \right] \] \[ \stackrel{\rm (6)}{=} - \left[ (x \left\{p_y, x \right\} + p_y \left\{ x, x \right\} ) - (y \left\{p_x, x \right\} + p_x \left\{ y, x \right\} ) \right] \] \[ \stackrel{\rm (7)(2)}{=} - \left[ (x \cdot 0 + p_y \cdot 0) - (y \cdot (-1) + p_x \cdot 0 ) \right] \] \[ = -y \]

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\[ \left\{y, L_z \right\} = \left\{y, xp_y - yp_x \right\} \] \[ \stackrel{\rm (2)}{=} - \left\{xp_y - yp_x, y\right\} \] \[ \stackrel{\rm (5)}{=} - \left[ \left\{xp_y, y \right\} - \left\{ yp_x, y \right\} \right] \] \[ \stackrel{\rm (6)}{=} - \left[ (x \left\{p_y, y \right\} + p_y \left\{ x, y \right\} ) - (y \left\{p_x, y \right\} + p_x \left\{ y, y \right\}) \right] \] \[ \stackrel{\rm (7)(2)}{=} - \left[ (x \cdot (-1) + p_y \cdot 0) - (y \cdot 0 + p_x \cdot 0 ) \right] \] \[ = x \]

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\[ \left\{z, L_z \right\} = \left\{z, xp_y - yp_x \right\} \] \[ \stackrel{\rm (2)}{=} - \left\{xp_y - yp_x, z\right\} \] \[ \stackrel{\rm (5)}{=} - \left[ \left\{xp_y, z \right\} - \left\{ yp_x, z \right\} \right] \] \[ \stackrel{\rm (6)}{=} - \left[ (x \left\{p_y, z \right\} + p_y \left\{ x, z \right\} ) - (y \left\{p_x, z \right\} + p_x \left\{ y, z \right\}) \right] \] \[ \stackrel{\rm (7)}{=} - \left[ (x \cdot 0 + p_y \cdot 0) - (y \cdot 0 + p_x \cdot 0 ) \right] \] \[ = 0 \]