Prove that (1, 0, 0, 0) and (√2, 0, 0. √3/c) are unit vectors in the V4 with the metric ds² == (dx¹)² – (dx²)² -- (dx³)² + - 2² (d.x¹)² Show also that the angle between these vectors is not real.

Let n 1 be an integer, let to to is given by a stationary path of the Lagrangian functional C: L[x] = 1 dt L(t,x,x), x(0) =x0, x(t)=X1, where LT - V and T is the total kinetic energy T = n 1 k=1 2 mark. 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.