Classical Mechanics

[sebb1e]

Homework Statement

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

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.

 

[vela]

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

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

Evaluate the partial derivative and write $\dot{x_2}$ in terms of $p_2$, and you should find everything cancels.

 

[sebb1e]

Thanks, that works perfectly.
I presume my mistake lay in partial dxi/dt (and pi) not being equal to the Hamilton partial derivatives.

 

[vela]

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