# Show that the Hamiltonian is conserved in a central Potential

1. Mar 3, 2013

### Ratpigeon

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

Show that for a particle in a central potential; V=f(|r|)
H is conserved.

2. Relevant equations
THe hamiltonian is
H=$\sum$(piq'i)-L
It is conserved if dH/dt=0

Euler-Lagrange equation
d/dt(dL/dq')=dL/dq

Noether's Theorem
For a continuous transformation, T such that
L=T(L) for all T,
T is related to a conserved quantity (Although my lecturer was sketch on how it relates...)

3. The attempt at a solution

velocity is
v=r'^2+(r θ')^2+(r sinθ $\varphi$')^2
p=mv
H=1/2m p^2 -(qA)^2 +V(r)

(Where V is the potential energy,
V(r)=q$\phi$-q A. r';
and A is the magnetic vector potential)

I tried directly differentiating wrt time and got
dH/dt =1/m (p-qA)p'
=(mv-qA)v'
=d/dt(L)
But I don't know how to show that it's conserved, since it doesn't fit the Euler Lagrange equation...