Classical Mechanics

  • #1

Homework Statement [Broken]

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

Homework Equations

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

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:


I think I'm fine on the last part as long as I can assume the constants.
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Answers and Replies

  • #2
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.
  • #3
Thanks, that works perfectly.
I presume my mistake lay in partial dxi/dt (and pi) not being equal to the Hamilton partial derivatives.
  • #4
Yes, exactly. The partial derivatives of the Hamiltonian give you total time derivatives, not partial time derivatives.

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