# Hamiltonian mechanics and special relativity?

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.

• For a simple example, see e.g. this Wikipedia page. – Qmechanic Jul 19 '12 at 13:05
• @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. – Yrogirg Jul 19 '12 at 13:12
• 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. – John Rennie Jul 19 '12 at 15:04
• 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. – genneth Jul 19 '12 at 16:56
• pirsa.org/displayFlash.php?id=12040021 is a nice 1 hour lecture – genneth Jul 19 '12 at 17:03

## 2 Answers

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.

• +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... – Christoph Jul 21 '12 at 14:03

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 as there is a no-go theorem for the ''natural'' situation; see Jordan-Currie-Sudarshan, Rev. Mod. Phys. 35 (1963), 350-375. However, there is an extended literature on relativistic Hamiltonian quantum mechanics, starting with Bakamjian & Thomas 1953 and surveyed in the paper ''Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics'' by Keister and Polyzou.

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