[QUE/CM-02012]

For a system with Lagrangian $$L = \frac{1}{2}\left( \dot{q_1}^2 +\dot{q_2}^2\right) -\alpha q_1 -\beta q_2 $$ verify explicitly, by computing time derivative, that the following quantities are constants of motion.

• $ F_1 = \frac{1}{2} m ( \dot{q}_1^2 + \dot{q}_2^2 ) + \alpha q_1 -\beta q_2 $
• $ F_2 = m \dot{q}_2 + \beta t $
• $ F_3 = q_1 - \frac{\alpha}{2m} t^2 - \dot{q}_1 t $
• $ F_4 = m ( \beta \dot{q}_1- \alpha \dot{q}_1 )$

Which of these are associated with some symmetry transformation? Give the symmetry transformation in each case.