# 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. Jul 19, 2012 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. Jul 19, 2012 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. Jul 19, 2012 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. Jul 19, 2012 at 16:56
• pirsa.org/displayFlash.php?id=12040021 is a nice 1 hour lecture Jul 19, 2012 at 17:03

• +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... Jul 21, 2012 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.