(a) Show that the relativistic Lagrangian \(L=-(1 / \gamma) m c^{2} e^{U / c^{2}}\) reduces to the Newtonian Lagrangian for a particle of mass \(m\) in a gravitational potential \(U\) in the slow motion, weak field limit \(v \ll c,|U| \ll c^{2}\).

(b) Obtain the Euler-Lagrange equations of motion, and show that they can be written in the relativistic form \(d V^{\mu} / d \tau=\left(g^{\mu u}-\left(1 / c^{2}\right) V^{\mu} V^{u}\right) \partial_{u} U\), where \(V\) is the four-velocity and \(\tau\) is the proper time of the particle. Verify that not all four-component equations are independent.