# Poisson Brackets

1. May 4, 2009

### roeb

1. The problem statement, all variables and given/known data

Calculate the Poisson bracket [H, Lz] in Cartesian Coords. Transform your result to cylndrical coords to show that [H, Lz] = -dU/dphi (note: partial derivs), where U is the potential energy. Identify the equivalent result in the Lagrangian formulation

2. Relevant equations

3. The attempt at a solution
I was able to do the first part easily enough, after a lot of math. I am having trouble figuring out what the question means by showing the equivalent result in Lagrangian formulation.

The best I have been able to come up with is:
L = 1/2*m*(r'^2 + r^2*(phi')^2) - U(r,phi,z)
$$\frac{\partial L}{\partial \phi} = - \frac{\partial U}{\partial \phi} = L_z$$
where $$L_z$$ = angular momentum
I don't think that is what the question is asking?
So angular momentum is not conserved?

Can anyone set me on the right path?

Last edited: May 4, 2009