Are you required to use the HJ equation? If you just do Euler-Lagrange, you get The function in question, , is a solution of this DE.
given : Consider
Show that the extremals are given by
Now i did a previous example as follows:Given
Long story short, hamiltonian is
Which yields the Hamilton Jacobi
Now given the hint to use S of Form S=u(t) + v(x), hence
K a constant,i assume since no t in expression. From here, we let k= and solve for t by direct integration, i.e and we do the same for and finally we get Where constant of integration. then provides me with an expression in x for which i solve to get Where E is the expression found from ,which proves the extremal to be a straight line.
Now my problem is that for this example,
Which yields the Hamilton jacobi
I have so little time, not enough to complete this question properly, but i believe i'm on the wrong track with this example, should x be here? I am asked to use the exact same hints and form for S as the first example but it just doesn't work out that way ! Because of x,I can no longer use total derivatives in place of partials for S! I get Which doesn't solve the HJ! Any words of advice? Sorry for sloppy presentation of the question! I write in a week, have so much to learn!
Are you working with the following definition of the Hamiltonian:
If so, then in your case, I get
Now, I agree with the equation
It follows that
which differs from your Hamiltonian by a minus sign.
Now then, the HJ equation, for your setup here, is
where we replace the 's by thus:
which leads to the pde
Following Landau and Lifschitz's Mechanics, page 149, let us seek an additively separated solution in the form of
Then and Dots are time derivatives, primes are spatial derivatives. If we substitute these into the pde, we obtain
Separating out yields
Therefore, For the equation, we obtain
The solutions are
Unfortunately, I have not really studied the HJ equation in depth. There appears to be more to do here, such as on pages 150-1 of Landau and Lifschitz, but I'm not really qualified to do more. Do you know what to do next?