Hamiltonian vs. Total Energy: When Do They Differ in Analytic Mechanics?

[pmb_phy]

I was wondering if anyone knows of systems for which the Hamiltonian is not equall to the total energy? This is an interesting problem in analytic mechanics (e.g. Lagrangian and Hamiltonian dynamics) but is rarely, if ever, mentioned in forums and newsgroups. I'd love to see a large set of examples for which this is true. I'd like to get an intuitive feeling for when the Hamiltonian equals the energy. I'm also very interested in whether there are quantum mechanical systems for which the Hamiltonian is not the energy. All input, references, thoughts and comments are welcome. There is an example of this in Classical Mechanics - Third Ed., by Goldstein, Safko and Poole page 345-346. Thank you.

Best wishes

Pete

 

[Andy Resnick]

My understanding is that all nonconservative (dissipative) forces are outside of the Hamiltonian. Friction, for example.

 

[pmb_phy]

My understanding is that all nonconservative (dissipative) forces are outside of the Hamiltonian. Friction, for example.

I neglected to say that I'm interested only in monogentic systems. Such systems have only forces which are the gradients of a potetial energy function. This does not mean that the Hamiltonian quals the energy though, hence the post, i.e. I'm seeking more examples than that in Goldstein's text. Thanks.

Best wishes

Pete