- #1

Matthew_

- 4

- 2

Now, the conjugated momenta of the generalized coordinates are: $$p_1=m\dot{x}_1-\dfrac{qB}{2c}x_2,$$ $$p_2=m\dot{x}_2+\dfrac{qB}{2c}x_1,$$ $$p_3=m\dot{x}_3.$$

It was claimed that canonical poisson brackets hold, namely ##\left\{p_i,p_j\right\}=0##. I have no idea why this is the case tho, since the derivation of the lagrangian with respect of the generalized velocities gives a clear dependence between the coordinates and the adjoint momenta. Evaluating ##\left\{p_1,p_2\right\}## I think I should get something like (summation over j is omitted): $$\left\{p_1,p_2\right\}=\frac{\partial p_1}{\partial x_j}\frac{\partial p_2}{\partial p_j}-\frac{\partial p_1}{\partial p_j}\frac{\partial p_2}{\partial x_j}=\frac{\partial p_1}{\partial x_2}-\frac{\partial p_2}{\partial x_1}=-\frac{qB}{c}\neq 0.$$ Is there some reason why this does not work?