[QUE/CM-04001]

1. Obtain the Lagrangian for a system with  Hamiltonian given by \[ H= \frac{p^2}{2\alpha} - b pq e^{-\alpha t} + \frac{ba}{2}q^2 e^{-\alpha t}(\alpha + be^{-\alpha t} ) + \frac{1}{2} k q^2\]     
2. Find condition(s) relating parameters \(a\) and \(\alpha\) such that  the Lagrangian \(L\) of the system can be written as another Lagrangian \(L_1\) up to a total time derivative  of some function \(\Omega\), {\it i.e.}, \[ L-L_1= \frac{d\Omega}{dt}.\] and where \(\Omega\ne0\).

Using the new Lagrangian \(L_1\) find a constant of motion  for the system.