(B) Use the Hamilton's equations to find the expression of the first and second total derivative of a function, say \(F\), which depends on generalized coordinates, corresponding conjugate momenta and time in general. Express the final results in terms of Poisson bracket and the partial derivatives of \(F\) with respect to time.
A Hamiltonian of one degree of freedom has the form
\(H = \frac{p^2}{2a} - bqpe^{-\alpha t} + \frac{ba}{2}q^2e^{-\alpha t}(a + be^{-\alpha t}) + \frac{kq^2}{2}\),
where \(a, b, \alpha,\) and \(k\) are constants.
(A) Find a Lagrangian corresponding to this Hamiltonian.
(B) Find an equivalent Lagrangian that is not explicitly dependent on time.
