Conservation of Angular Momentum Using the Hamiltonian

[physics2018]

Homework Statement

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.

Homework Equations

q_i-dot = dH/dp_i and p_i-dot = - dH/dq_i

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.

 

[gabbagabbahey]

Hi physics2018, welcome to PF!

Homework Statement

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.

Homework Equations

q_i-dot = dH/dp_i and p_i-dot = - dH/dq_i

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 $\frac{d\textbf{L}}{dt}=\{\textbf{L},H\}=0$, why not start by computing $\{\textbf{L},H\}$?