This has been my approach. I should have added that the additional terms are a perturbation, so I'm trying to achieve even just the simpler goal of matching terms to the next order in epsilon. Even so, I'm only close in a simple case, and in complicated cases it seems hopeless to get everything...
Imagine I have a complicated second-order differential equation that I strongly suspect can be derived from a Hamiltonian (with additional momentum dependence beyond p2/2m, so the momentum is not simply mv, but I don't know what it is).
Are there any ways to test whether or not the given...
I apologize that this is rather specific, but hopefully enough people have used Goldstein. I have a basic grasp of action-angle variables, and I'm going through the time-independent perturbation theory section in Goldstein (12.4).
In this section we seek a transformation from the unperturbed...
Classical Perturbation Theory--Time Dep. vs. Time Indep (Goldstein).
Hi,
I'm going through Goldstein, and I'm a little confused on the distinction between time dependent and time independent perturbation theory. In section 12.2, they do the case of a simple harmonic perturbation on force...
Hi Vanhees, sorry yes I've been reading all your posts carefully, I just have agreed with everything you said so I didn't comment on it.
I am completely on the same page as you that the variational principle in the Hamiltonian formalism is more general than that in the Lagrangian. But I'm...
Hm, thanks for the reply Mr. Vodka. I'd be interested to see what other people think. I'm not sure that looking at the Lagrangian problem in q and \dot{q} space is quite right. I think the variation is done in q, t space, and when you derive the condition for the action to be minimized, since...
Mr. Vodka:
If you look at how you derive the Euler-Lagrange equations in the Lagrangian formalism, you don't vary q and \dot{q} independently. What you do is you vary the path q(t), and that creates variations in q and \dot{q} that are related to each other through \delta \dot{q} =...
Right, I think where I'm getting confused is that with Lagrangians, along any path you can't vary q without varying \dot{q}. I therefore have trouble seeing how you can transform to a Hamiltonian and now vary q and p independently. p = \frac{\partial L}{\partial \dot{q}}, so since L = L(q...
Hi,
A fundamental aspect in the Hamiltonian framework of mechanics is that the q's and p's are independent. I feel like I understand the steps in the Legendre transform from Lagrangian to Hamiltonian mechanics, but I don't see how you can go from a system where only the q's are independent...