(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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?

2. Relevant 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 momentumpis 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]

3. 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.

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# Homework Help: Classical mechanics - Time dependent Hamiltonian and Lagrangian

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