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.