# 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.

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– 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
– John Rennie Jul 19 '12 at 15:04
– genneth Jul 19 '12 at 16:56
– 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.

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+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 (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

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