1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Mar 3, 2013 #1
    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
    It is conserved if dH/dt=0

    Euler-Lagrange equation

    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 [itex]\varphi[/itex]')^2
    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'
    But I don't know how to show that it's conserved, since it doesn't fit the Euler Lagrange equation...
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted