Symmetries of relativistic Lagrangian and Hamiltonian systems

Question

In non-relativistic mechanics, the conserved quantities found using Noethers theorem in Lagrangian mechanics are the same as those quantities which are conserved under canonical commutation with the Hamiltonian in Hamiltonian mechanics.

Does this carry over straight-forwardly to relativistic mechanics?

Answer 1

If the Lagrangian is non-singular so that the Legendre transformation to pass from Lagrangian to Hamiltonian formalism is well defined, the answer is Yes. If a quantity is conserved in view of Noether's theorem in Lagrangian formulation, passing to the Hamiltonian formulation it turns out to be the generator of a canonical transformation that preserves in form the Hamiltonian function and thus it is conserved as well. The proof does not depend on any overall group of symmetries of the theory (Poincaré/ Galileo groups) but it only relies upon the general Lagrangian/Hamiltonian formalism.

Answer 2

Noether's theorem doesn't really discriminate between relativistic and non-relativistic theories. As long as there is an action formulation and a symmetry, she will provide a conservation law via the standard Noether procedure.

However, the question formulation (v1) touches upon other issues that are far from trivial, such as, e.g.,

1. The status of an inverse Noether's theorem, cf. e.g. this Phys.SE post and links therein.

2. The correspondence between Lagrangian and Hamiltonian theories in the case of non-singular Legendre transformations, cf. e.g. this Phys.SE post and links therein.