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,

i get

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!