Hamiltonian mechanics and special relativity?

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.

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