Calculate Poisson Bracket [H,Lz] in Cartesian Coords

In summary, the question asks to calculate the Poisson bracket [H, Lz] in Cartesian coords and transform it to cylindrical coords to show that it is equivalent to -dU/dphi in the Lagrangian formulation. This is done by using the relationship between the Hamiltonian and Lagrangian and showing that the Poisson bracket is equal to -Lz. This demonstrates that angular momentum is conserved in this system.
  • #1
roeb
107
1

Homework Statement



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

Homework Equations


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)
[tex]\frac{\partial L}{\partial \phi} = - \frac{\partial U}{\partial \phi} = L_z[/tex]
where [tex]L_z[/tex] = 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:
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  • #2


Actually, your attempt is pretty close to the correct answer! The question is asking for the equivalent result in the Lagrangian formulation, which means expressing the Poisson bracket in terms of the Lagrangian instead of the Hamiltonian.

To do this, we can use the relationship between the Hamiltonian and the Lagrangian:
H = p_i*q'^i - L
where p_i is the generalized momentum and q'^i is the derivative of the generalized coordinate q^i with respect to time.

So in this case, we have:
[H, Lz] = [p_i*q'^i - L, Lz]
= [p_i*q'^i, Lz] - [L, Lz]
= p_i*[q'^i, Lz] - [L, Lz]
= p_i*[q'^i, Lz] - 0 (since L does not depend on Lz)
= p_i*[q'^i, Lz]
= p_i*phi' (since Lz = p_phi)
= -dU/dphi
= -\partial U/\partial phi
= -Lz

So we have shown that in the Lagrangian formulation, the Poisson bracket [H, Lz] is equivalent to -Lz, which is the same as the angular momentum. This means that angular momentum is conserved, as expected.

Hope this helps!
 

1. What is the Poisson Bracket?

The Poisson Bracket is a mathematical operation used in classical mechanics to calculate the time evolution of a system. It is denoted by {A,B} and is defined as the product of the partial derivatives of A and B with respect to the position and momentum variables.

2. How is the Poisson Bracket calculated?

The Poisson Bracket of two functions, A and B, can be calculated using the formula {A,B} = ∂A/∂q * ∂B/∂p - ∂A/∂p * ∂B/∂q, where q represents the position variables and p represents the momentum variables.

3. What is the significance of calculating the Poisson Bracket?

The Poisson Bracket is significant because it helps us understand the dynamics of a system and how it evolves over time. It is also used to calculate important physical quantities such as energy, angular momentum, and Hamiltonian.

4. What is the Poisson Bracket [H,Lz] in Cartesian coordinates?

In Cartesian coordinates, the Poisson Bracket [H,Lz] is calculated using the formula {H,Lz} = ∂H/∂x * ∂Lz/∂px - ∂H/∂px * ∂Lz/∂x, where x and px represent the position and momentum variables in the x-direction, and Lz is the angular momentum in the z-direction.

5. How is the Poisson Bracket related to the Heisenberg uncertainty principle?

The Poisson Bracket is related to the Heisenberg uncertainty principle because it is used to calculate the time evolution of quantum mechanical systems. It helps us understand the relationship between position and momentum variables and how their uncertainties affect the dynamics of the system.

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