Equipartition theorem

1. Oct 26, 2009

Petar Mali

I have one question. If I have Hamiltonian:

$$H=\sum^{f_1}_{i=1}\alpha_iq_i^2+\sum^{f_2}_{i=1}\beta_ip_i^2$$

I can show that for the Hamiltonian of this type equipartition theorem is correct. Is there any Hamiltonian which is not a function od squares of coordinates and impulses are from which I can get this Hamiltonian some using canonical transformation. Example perhaps?

2. Oct 26, 2009

Llewlyn

What does it mean "equipartition theorem is correct" ?
I think the hamiltonian is both integrable and separable, since it's not ergodic does the ensemble averages have sense?

Ll.

3. Oct 26, 2009

Petar Mali

You can prove that every degree of fredom have the same energy - equipartition theorem only for the Hamiltonian

$$H=\sum^{f_1}_{i=1}\alpha_iq_i^2+\sum^{f_2}_{i=1}\beta_ip_i^2$$

or maybe the Hamiltonian which canonical transformation is

$$H=\sum^{f_1}_{i=1}\alpha_iq_i^2+\sum^{f_2}_{i=1}\beta_ip_i^2$$

I think that ''
mysterious '' Hamiltonian have the same form as one as I wrote! So for example

$$K=\sum^{F_1}_{i=1}\alpha_iQ_i^2+\sum^{F_2}_{i=1}\beta_iP_i^2$$

Am I right?