[QUE/CM-03005]

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

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

Which of these are associated with some symmetry transformation? Are there
any more constants of motion independent of $F_1,..,F_4$? WHY?