Noether's theorem asserts a connection between invariance principles and conservation laws. In Section 7.8 we saw that translational invariance of the Lagrangian implies conservation of total linear momentum. Here you will prove that rotational invariance of ,C implies conservation of total angular momentum. Suppose that the Lagrangian of an N-particle system is unchanged by rotations about a certain symmetry axis.
(a) Without loss of generality, take this axis to be the z axis, and show that the Lagrangian is unchanged when all of the particles are simultaneously moved from (rα, θα, ϕα) to(rα, θα, ϕα + ϵ) (same ϵ for all particles). Hence show that ∑ N α = 1 ∂ L ∂ ϕ α = 0 ∑α=1N∂L∂ϕα=0
(b) Use Lagrange's equations to show that this implies that the total angular momentum Lz about the symmetry axis is constant. In particular, if the Lagrangian is invariant under rotations about all axes, then all components of L are conserved.