Hamilton-Jacobi Theory: Why No Time Dependence?

[Zorba]

Regarding the Hamilton-Jacobi equation in it's usual form, I am having trouble understanding the following statement from Goldstein they say

"When the Hamiltonian does not depend explicitly upon the time, Hamilton's principal function can be written in the form

$$S(q,\alpha,t)=W(q,\alpha)-at$$

where $$W(q,\alpha)$$ is called Hamilton's characteristic function."

So why is this? I don't understand why it is required that there be no explicit dependence on the time, it's seems to be as though we should be able to do this anyways due to the form of the Hamilton-Jacobi equation...

 

[RedX]

Regarding the Hamilton-Jacobi equation in it's usual form, I am having trouble understanding the following statement from Goldstein they say

"When the Hamiltonian does not depend explicitly upon the time, Hamilton's principal function can be written in the form

$$S(q,\alpha,t)=W(q,\alpha)-at$$

where $$W(q,\alpha)$$ is called Hamilton's characteristic function."

So why is this? I don't understand why it is required that there be no explicit dependence on the time, it's seems to be as though we should be able to do this anyways due to the form of the Hamilton-Jacobi equation...

I think you meant alpha*t.

But anyways, the derivative of the action with respect to time is negative the Hamiltonian. When the Hamiltonian doesn't depend on time, energy is conserved. For a time-independent Hamiltonian, if you are following a particle on a trajectory then the Hamiltonian is constant, equal to the energy.

So that form of the Hamilton-Jacobi equation has a specific trajectory in mind, on which the energy is alpha.

 

[Matterwave]

Because the Hamiltonian is time-independent, you can assign it as a constant (alpha). Therefore, when you take the time derivative of the generating function S, you get -alpha=-H. And then H+dS/dt=0 which is what you want. It's just a way to simplify your generating function S.

 

[Zorba]

So in some ways you could describe this as a "trick"? It doesn't actually tell us anything new, but it works because after taking the partial derivative the time disappears? I see how it is useful alright, even in the case of the harmonic oscillator it makes the equations much more easier to deal with etc.

Edit: Oh, and yes I meant to put \alpha there instead of a, there's a misprint in the book. I think that probably didn't make it any easier to understand in retrospect... :)

 

[RedX]

So in some ways you could describe this as a "trick"? It doesn't actually tell us anything new, but it works because after taking the partial derivative the time disappears? I see how it is useful alright, even in the case of the harmonic oscillator it makes the equations much more easier to deal with etc.

Edit: Oh, and yes I meant to put \alpha there instead of a, there's a misprint in the book. I think that probably didn't make it any easier to understand in retrospect... :)

To be honest I don't know why you'd want to do that. I don't see the utility at all. But evidently there is some utility or else it wouldn't be in books.

 

[Matterwave]

So in some ways you could describe this as a "trick"? It doesn't actually tell us anything new, but it works because after taking the partial derivative the time disappears? I see how it is useful alright, even in the case of the harmonic oscillator it makes the equations much more easier to deal with etc.

Edit: Oh, and yes I meant to put \alpha there instead of a, there's a misprint in the book. I think that probably didn't make it any easier to understand in retrospect... :)

Splitting off your S into that form already is sort of like already doing 1 integral for you (it's trivial, but still, why mess with it every time you do a problem if they always come out the same?). Similarly, with separable problems, you want to separate out each term because you don't want to solve coupled-partial diff-eqs if you don't have to.