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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
    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...)

    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]-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...
     
  2. jcsd
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