Exercise 3

The following coordinate transformations are given: \[ q_1 \rightarrow q_1 + b \delta \] \[ q_2 \rightarrow q_2 + a \delta \]

Since the potential energy \(V = V(aq_1 - bq_2)\) depends on \((aq_1 - bq_2)\), the coordinate transformation given above has no influence on the argument of \(V\) and hence no influence on the value of \(V\): \[ aq_1 - bq_2 \rightarrow a(q_1 + b\delta) - b (q_2 + a\delta) \] \[ = aq_1 + ab\delta - bq_2 - ba\delta = aq_1 -bq_2 \]

Since \(b\delta\) and \(a\delta\) are time invariant (according to the assumption), these expressions vanish if differentiated with respect to time and don’t appear in the expression for the kinetic energy. Hence the Lagrangian is invariant under the given coordinate transformation.