This is a past examination question. Let n 2 1 be an integer, let to to is given by a stationary path of the Lagrangian functional C: . dt L(t, x, x) , x(0) = XO, x(t1 ) = x1, where L = T - V and T is the total kinetic energy T = ) mik k=1 Using the above first-integral, show that, if V is independent of t, the total energy E =T + V of the particle is a constant of the motion. [3] Now consider the case V (t, x1 , ..., In) = > Vuj(t, Ii -xj), 1cicjen where, for each pair i, j, the function Vy is a smooth potential function. Using Noether's theorem, show that the linear momentum Et, mir; is a constant of the motion of the particles. [4]