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!