A Hamiltonian with one degree of freedom has the form

\[H=\frac{p^{2}}{2 m}+\frac{k q^{2}}{2}-2 a q^{3} \sin \alpha t\]

where \(m, k, a\), and \(\alpha\) are constants. Find the Lagrangian corresponding to this Hamiltonian. Write out both Hamilton's equations and Lagrange's equations, and show directly that they are equivalent.