Solve Classical Mechanics Homework Statement

[sebb1e]

Homework Statement

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

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.

 

[paranormal]

It appears that you are on the right track with your solution. In order to show that the four quantities are constants, you will need to use the Hamilton's equations of motion, which are dxj/dt=dH/dpj and dpj/dt=-dH/dxj. These equations relate the time derivatives of the position and momentum variables to the Hamiltonian, which is a constant in time. Additionally, you can use the fact that the Hamiltonian is a function of the position and momentum variables, so dH/dxj and dH/dpj are also constants. This should help you to show that the four quantities are indeed constants.