Classical Mechanics

  • Thread starter sebb1e
  • Start date
  • #1
36
0

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.
 
Last edited by a moderator:

Answers and Replies

  • #2
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
15,141
1,735
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
36
0
Thanks, that works perfectly.
I presume my mistake lay in partial dxi/dt (and pi) not being equal to the Hamilton partial derivatives.
 
  • #4
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
15,141
1,735
Yes, exactly. The partial derivatives of the Hamiltonian give you total time derivatives, not partial time derivatives.
 

Related Threads on Classical Mechanics

  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
12
Views
3K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
10
Views
3K
Replies
6
Views
779
  • Last Post
Replies
11
Views
2K
  • Last Post
Replies
5
Views
2K
Top