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Classical Mechanics

  1. May 26, 2010 #1
    1. The problem statement, all variables and given/known data

    http://img337.imageshack.us/img337/3014/classicalmechs.jpg [Broken]

    I'm fine until showing that those 4 things are constants.

    2. Relevant equations

    dxj/dt=dh/dpj and dpj/dt=-dh/dxj

    3. The attempt at a solution

    I can't show they are constant, for example, can someone show me where I'm going wrong here for p1-0.5bx2:

    d(p1-0.5Bx2)/dt=d(p1-0.5Bx2)/dxj*dh/dpj+d(p1-0.5Bx2)/dpj*(-dh/dxj)
    =-0.5B*dh/dp2+(-dh/dx1)
    =-0.5B(2p2-2eA2)+(eBp2+0.5e^2Bx1)

    I think I'm fine on the last part as long as I can assume the constants.
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. May 26, 2010 #2

    vela

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    There seems to be a mistake in the problem statement as the units don't work out. The product eA has units of momentum, yet the problem asks about p1-Bx2/2. The second term has units of momentum/charge. You should be looking at the quantity p1-eBx2/2.

    I think your problem is you're mixing up partial and total derivatives. You should have

    [tex]\frac{d}{dt}\left(p_1-\frac{1}{2}eBx_2\right) = \dot{p_1} - \frac{1}{2}eB\dot{x_2} = \frac{\partial H}{\partial x_1}-\frac{1}{2}eB\dot{x_2}[/tex]

    Evaluate the partial derivative and write [itex]\dot{x_2}[/itex] in terms of [itex]p_2[/itex], and you should find everything cancels.
     
  4. May 26, 2010 #3
    Thanks, that works perfectly.
    I presume my mistake lay in partial dxi/dt (and pi) not being equal to the Hamilton partial derivatives.
     
  5. May 26, 2010 #4

    vela

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    Yes, exactly. The partial derivatives of the Hamiltonian give you total time derivatives, not partial time derivatives.
     
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