Equipartition Theorem: Hamiltonian Form & Canonical Transformations

[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?

 

[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.

 

[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?