Hamiltonian mechanics and special relativity?

Question

Is there a relativistic version of Hamiltonian mechanics? If so, how is it formulated (what are the main equations and the form of Hamiltonian)? Is it a common framework, if not then why?

It would be nice also to provide an example --- a simple system with its Hamiltonian.

As far as I remember, in relativistic mechanics we were only taught to use conservation laws, that is integral invariants and thus I have a vague perception of relativistic dynamics.

This post imported from StackExchange Physics at 2015-11-01 18:11 (UTC), posted by SE-user Yrogirg

Comments

For a simple example, see e.g. this Wikipedia page.

This post imported from StackExchange Physics at 2015-11-01 18:11 (UTC), posted by SE-user Qmechanic

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@Qmechanic does it mean that the Hamiltonian equations themselves (the structure of phase space) doesn't change? Is the only thing that changes the allowed form of Hamiltonian? I couldn't find in Wikipedia anything about relativistic Hamiltonian mechanics itself.

This post imported from StackExchange Physics at 2015-11-01 18:11 (UTC), posted by SE-user Yrogirg

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I posted this link - icmp.lviv.ua/journal/zbirnyk.25/001/art01.pdf - in response to a similar question. Although the paper I've linked is really aimed at stat mech you might find it useful reading.

This post imported from StackExchange Physics at 2015-11-01 18:11 (UTC), posted by SE-user John Rennie

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See ncatlab.org/nlab/show/phase+space for some high maths fuelled stuff; the basic idea is however quite physical, and very helpful. Another useful line of research is motivated by canonical quantum gravity, which much deal with a much larger symmetry group. It seems that people are beginning to see the benefits of not singling out a particular coordinate to denote time.

This post imported from StackExchange Physics at 2015-11-01 18:11 (UTC), posted by SE-user genneth

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pirsa.org/displayFlash.php?id=12040021 is a nice 1 hour lecture

This post imported from StackExchange Physics at 2015-11-01 18:11 (UTC), posted by SE-user genneth

Answer 1

Relativistic Lagrangian and Hamiltonian mechanics can be formulated by means of the jet formalism which is appropriate when one deals with transformations mixing position and time.

This formalism is much advocated by G. Sardanashvily, please see his review article.

This post imported from StackExchange Physics at 2015-11-01 18:11 (UTC), posted by SE-user David Bar Moshe

Comments

+1: thanks a lot for that link; my own musing went into the direction of modeling relativistic mechanics via local contact structures on the space of geodesics parametrized by arc length (ie proper time); that's just another way to arrive at $J^1_1Q$, and now that I know where I need to end up eventually, I might revisit that idea...

This post imported from StackExchange Physics at 2015-11-01 18:11 (UTC), posted by SE-user Christoph

Answer 2

One-particle Hamiltonian mechanics is easy to make relativistic, as the 0-component of the momentum 4-vector is the Hamiltonian. For example, $H=\sqrt{\mathbb{p}^2+m^2}$ for a free particle, and by minimal substitution one can add an external electromagnetic field.

Multiparticle Hamiltonian mechanics is somewhat awkward (and hardly ever pursued) as there is a no-go theorem for the ''natural'' situation; see Jordan-Currie-Sudarshan, Rev. Mod. Phys. 35 (1963), 350-375.

Relativistic classical field theory has again a good Hamiltonian formulation; see http://count.ucsc.edu/~rmont/papers/covPBs85.PDF

This post imported from StackExchange Physics at 2015-11-01 18:11 (UTC), posted by SE-user Arnold Neumaier