A Hamiltonian of one degree of freedom has the form where a, b, α, and k are constants.(a)...

A Hamiltonian of one degree of freedom has the form

where a, b, α, and k are constants.

(a) Find a Lagrangian corresponding to this Hamiltonian.

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(b) Find an equivalent Lagrangian that is not explicitly dependent on time.

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(c) What is the Hamiltonian corresponding to this second Lagrangian, and what is the relationship between the two Hamiltonians?