Consider a particle moving in three dimensions with Lagrangian L = \((1 / 2) m\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}ight)+a \dot{x}+b\), where \(a\) and \(b\) are constants.

(a) Find the equations of motion and show that the particle moves in a straight line at constant speed, so that it must be a free particle.

(b) The result of

(a) shows that there must be another reference frame \(\left(x^{\prime}, y^{\prime}, z^{\prime}ight)\) such that the Lagrangian is just the usual free-particle Lagrangian \(L^{\prime}=(1 / 2) m\left(\dot{x} \prime^{2}+\dot{y} \prime^{2}+\dot{z}^{2}ight)\). However, \(L^{\prime}\) may also be allowed an additive constant, which cannot show up in Lagrange's equations. Find the Galilean transformation between \((x, y, z)\) and \(\left(x^{\prime}, y^{\prime}, z^{\prime}ight)\) and find the velocity of the new primed frame in terms of \(a\) and \(b\).