Conservation of Angular Momentum Using the Hamiltonian


by physics2018
Tags: angular momentum, hamiltonian, invariant, lagrangian, particle
physics2018
physics2018 is offline
#1
Oct15-09, 09:00 AM
P: 1
1. The problem statement, all variables and given/known data
The Hamiltonian for a particle mass m, moving in a central force field is given as: H = 1/(2m) * |p^2| - V(r). Take the Hamiltonian to be invariant, such that it can be shown that L = r x p the angular momentum vector is a conserved quantity: dL/dt = {L,H} = 0.


2. Relevant equations
q_i-dot = dH/dp_i and p_i-dot = - dH/dq_i


3. The attempt at a solution
I do not understand how to go about solving the following problem ( I think I understand what the Hamiltonian is, but I do not understand how to from it to what needs to be proven) To solve the problem I believe I need to get from Hamiltonian's equations to Lagrange's p_i = dL/dx_i-dot and then from there use p-dot * dr + p * dr-dot = 0 where dr is defined as the distance between two vectors r and r+dr.
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gabbagabbahey
gabbagabbahey is offline
#2
Oct15-09, 05:36 PM
HW Helper
gabbagabbahey's Avatar
P: 5,004
Hi physics2018, welcome to PF!

Quote Quote by physics2018 View Post
1. The problem statement, all variables and given/known data
The Hamiltonian for a particle mass m, moving in a central force field is given as: H = 1/(2m) * |p^2| - V(r). Take the Hamiltonian to be invariant, such that it can be shown that L = r x p the angular momentum vector is a conserved quantity: dL/dt = {L,H} = 0.


2. Relevant equations
q_i-dot = dH/dp_i and p_i-dot = - dH/dq_i


3. The attempt at a solution
I do not understand how to go about solving the following problem ( I think I understand what the Hamiltonian is, but I do not understand how to from it to what needs to be proven) To solve the problem I believe I need to get from Hamiltonian's equations to Lagrange's p_i = dL/dx_i-dot and then from there use p-dot * dr + p * dr-dot = 0 where dr is defined as the distance between two vectors r and r+dr.
Well, since you are asked to show that [itex]\frac{d\textbf{L}}{dt}=\{\textbf{L},H\}=0[/itex], why not start by computing [itex]\{\textbf{L},H\}[/itex]?


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