Question

Momentum conservation using lagrangian formulation

Answer 1

The conservation of momentum is a special case of Noether's theorem, which says that for any symmetry of the action, describing a dynamical system, there's a conserved quantity. In Hamiltonian formulation the conserved quantity is given by the generator of the canonical transformation, describing the symmetry transformation. Thus also the other way works: Each conserved quantity defines a symmetry transformation of the action. Now, Newtonian space-time has the Galilei group as its symmetry group, which is a ten-dimensional Lie group, and thus you have 10 conserved quantities for any closed system: Time-translation invariance: Energy Space-translation invariance (3 independent parameters, namely the translation in three linearly independent directions of space): momentum vector Rotations (3 dimensional group SO(3)): angular-momentum vector Boosts (3 dimensional group): velocity of the center of mass

Reference https://www.physicsforums.com/threads/conservation-of-momentum-from-the-lagrangian-formulation.502534/