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## Homework Statement

A system with only one degree of freedom is described by the following Hamiltonian:

[tex] H = \frac{p^2}{2A} + Bqpe^{-\alpha t} + \frac{AB}{2}q^2 e^{-\alpha t}(\alpha + Be^{-\alpha t}) + \frac{kq^2}{2} [/tex]

with A, B, alpha and k constants.

a) Find a Lagrangian corresponding to this Hamiltonian;

b) Find an equivalent Lagrangian which is not explicitly time-dependent;

c) What Hamiltonian corresponds to this second Lagrangian, and how is it related to the original Hamiltonian?

## Homework Equations

Since this system has only one degree of freedom, the Hamiltonian is

[tex] H = \dot{q}p - L [/tex]

which implies that the Lagrangian is

[tex] L = \dot{q}p - H [/tex]

Also, canonical momentum

*p*is defined as

[tex] p_i \equiv \frac{\partial L(q_j,\dot{q_j},t)}{\partial \dot{q_i}} \\

\Rightarrow p = \frac{\partial L}{\partial \dot{q}} [/tex]

## The Attempt at a Solution

I've manipulated the Hamiltonian enough now that I'm pretty sure it corresponds to a damped harmonic oscillator with mass A. I also have a pretty good idea of what I should be doing, but I stumble at every step. We've just started working on the Hamiltonian formulation in class, so nothing is automatic yet. This is also the first time homework deals with a Lagrangian that is explicitly dependent on time.

First step is obviously obtaining [tex]L(q,\dot{q},p,t)[/tex] by using the second Legendre transformation above, which is trivial. Next, I want to remove [tex]p[/tex] to obtain a proper [tex]L(q,\dot{q},t)[/tex]. This is where I hit the first obstacle. Using the definition of [tex]p_i[/tex] doesn't help here, because [tex]p = \frac{\partial L(q,\dot{q},p,t)}{\partial \dot{q}} = p[/tex]. So, I have no way to formally obtain [tex] p = p(q,\dot{q},t)[/tex] to obtain the real Lagrangian [tex]L(q,\dot{q},t)[/tex]. Nevertheless, intuition suggests [tex] p = \dot{q}[/tex], which would give the explicitly time-dependent Lagrangian

[tex]L = \frac{p^2}{2A}\ + \ Bq\dot{q}e^{-\alpha t} \ - \ \frac{AB}{2}q^2 \alpha e^{-\alpha t} \ - \ \frac{A B^2}{2}q^2e^{-2\alpha t} \ - \ \frac{kq^2}{2} [/tex]

From there on, I'm clueless as how to make the time-dependence disappear. I've tried completing the square of the 3rd and 4th terms, tried to link the 2nd and 3rd terms through a total time derivative, a combination of both these approaches, etc.. but nothing seems to be working correctly. (Except posing [tex]B = e^{\alpha t}[/tex] which is not allowed because B is a constant... but one can dream that homework should be that easy.)

And obviously, since I can't get a time-independent Lagrangian, I can't use that to obtain a time-independent Hamiltonian and compare it to the original Hamiltonian.

So to sum up, these are my questions:

1) Knowing H, how do I find [tex] p = p(q,\dot{q},t)[/tex] to get a "clean" Lagrangian?

2) Having found that time-dependent Lagrangian, how do I remove the explicit time dependency?

Any hint would be greatly appreciated. Thank you very much.