Consider a mechanical system of n particles, with a conservative potential consisting of t...

Consider a mechanical system of n particles, with a conservative potential consisting of terms dependent only upon the scalar distance between pairs of particles. Show explicitly that the Lagrangian for the system when expressed in coordinates derived by a Galilean transformation differs in form from the original Lagrangian only by a term that is a total time derivative of a function of the position vectors. This is a special case of invariance under a point transformation (cf. Derivation 1).

Derivation 1

Let q1, …,qn be a set of independent generalized coordinates for a system of n degrees of freedom, with a Lagrangian . Suppose we transform to another set of independent coordinates s1, …, sn by means of transformation equations

(Such a transformation is called a point transformation.) Show that if the Lagrangian function is expressed as a function of sj, , and t through the equations of transformation, then L satisfies Lagrange’s equations with respect to the s coordinates:

In other words, the form of Lagrange’s equations is invariant under a point transformation.