(adsbygoogle = window.adsbygoogle || []).push({}); 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=[itex]\sum[/itex](p_{i}q'_{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 onhowit relates...)

3. The attempt at a solution

velocity is

v=r'^2+(r θ')^2+(r sinθ [itex]\varphi[/itex]')^2

p=mv

H=1/2m p^2 -(qA)^2 +V(r)

(Where V is the potential energy,

V(r)=q[itex]\phi[/itex]-qA. 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...

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# Homework Help: Show that the Hamiltonian is conserved in a central Potential

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