Particles and fields: theoretical physics

3) Conservation Quantities and Poisson Brackets: Consider a particle in the central potential V (r). a) What is the Lagrangian function? Show using the Noether's Theorem that the z- component of the angular momentum is conserved. Consider infinitesimal rotations around the z-axis, r = r' - ae_z x r' (i.e. only linear terms in a). If the Lagrangian function is invariant under the transformation, then according to Noether's Theorem the conserved quantity I is obtained. b) Give the Hamiltonian function. Determine the Poisson bracket {L_z, H}. Now show using the Poisson bracket that the z-component of the angular momentum is conserved. 3 aL ari I = IM