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Ratpigeon
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Homework Statement
Show that for a particle in a central potential; V=f(|r|)
H is conserved.
Homework Equations
THe hamiltonian is
H=[itex]\sum[/itex](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...)
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]-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...