[QUE/CM-03008]

From symmetry considerations alone find a conserved quantity for the Lagrangian $$L={1\over2 } m \dot{\vec{r}} - V_0 x \sin\left({ 2\pi z\over R}\right) - V_0 y \cos\left( {2\pi z \over R} \right) $$ where $V_0$ and $R$ are constants. What is symmetry of $L$? Find generalized co-ordinates one of which is cyclic. Hint : It can be proved that for an unconstrained particle moving in a potential the only time independent transformations under which the Lagrangian  is invariant are combinations of rotations and translations.