A mechanical system, with generalised coordinates q_{1} and q_{2}, has Lagrangian
L(q_{1}, q_{2}, q_{1}prime, q_{2}prime). State Lagrange's equations for the system, and deduce that
q _{1}prime \frac{ delta L }{ delta q_{1}prime } (q_{1}, q_{2}, q_{1}prime, q_{2}prime) + q _{2}prime \frac{ delta L }{ delta q_{2}prime } (q_{1}, q_{2}, q_{1}prime, q_{2}prime) - L(q_{1}, q_{2}, q_{1}prime, q_{2}prime)
is constant during the motion.

Question

A mechanical system, with generalised coordinates q_{1} and q_{2}, has Lagrangian
L(q_{1}, q_{2}, q_{1}prime, q_{2}prime). State Lagrange's equations for the system, and deduce that
q _{1}prime  \frac{ delta L }{ delta q_{1}prime } (q_{1}, q_{2}, q_{1}prime, q_{2}prime) + q _{2}prime  \frac{ delta L }{ delta q_{2}prime } (q_{1}, q_{2}, q_{1}prime, q_{2}prime) - L(q_{1}, q_{2}, q_{1}prime, q_{2}prime)
is constant during the motion.