Re: Hamiltonian vs. Energy

Pmb schrieb:
I was wondering if anyone knows of systems for which the Hamiltonian is not equal to the total energy?

This is the case in any diffeomorphism invariant theory; then the Hamiltonian vanishes identically. Note that any theory can be rewritten
in diffeomorphism invariant form by introducing an additional quantity
that specifies how the time depends on the arbitrary parameter t in the action.

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 interested in the Hamiltonian for a
object sliding on plane with only gravity and friction acting on the body.

If you have friction, the problem is no longer conservative, and the
standard Lagrangian or Hamiltonian approach is no longer applicable.

The correct quantization of dissipative systems is in terms of
Lindblad master equations, or, in more mathematical terms, completely positive semigroups.


One can rewrite dissipative systems in a way that they look conservative, at the expense of destroying the physical interpretation of the Hamiltonian. There are quite a number of publications in
this direction. But I haven't seen any treatment where this
pseudo-conservative reformulation produced anything of value beyond
what was already put into the description.


Arnold Neumaier



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