0

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?

UserB
  • 124
  • Possible duplicates: http://physics.stackexchange.com/q/45545/, http://physics.stackexchange.com/q/38286/ – Kyle Kanos Dec 18 '13 at 22:00
  • A clarification to the question (v1): By 'canonical commutation' do you mean Poisson bracket and classical mechanics rather than commutator and quantum theory? – Qmechanic Dec 19 '13 at 09:08
  • Yes, Poisson bracket, I only mean in the classical sense. – UserB Dec 19 '13 at 18:11

2 Answers2

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.

Valter Moretti
  • 71,578
  • Thanks, so this should work when we look at mechanics of a single particle and the mechanics of a continuous field? – UserB Dec 19 '13 at 18:15
  • I never tried to prove it for fields, but I do not see any obstruction, since the formalism is identical, and I am confident that it works anyway. – Valter Moretti Dec 19 '13 at 18:18
0

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.

Community
  • 1
Qmechanic
  • 201,751