Hamilton-Jacobi Equation

[Piano man]

Homework Statement

Using the Hamilton-Jacobi equation find the trajectory and the motion of a particle in the
potential $$U(r)=-Fx$$

Homework Equations

Hamilton-Jacobi Equation: $$\frac{\partial S}{\partial t}+H(q_1,...,q_s;\frac{\partial S}{\partial q_1},...,\frac{\partial S}{\partial q_s};t)=0$$

The Attempt at a Solution

$$H(p_x,p_y,p_z,x,y,z)=\frac{p_x^2}{2m}-Fx+\frac{p_y^2}{2m}+\frac{p_z^2}{2m}$$

From HJE, since y and z are cyclic,
$$S(x,y,z;p_x,p_y,p_z;t)=-Et+p_yy+p_zz+S(x,p_x)$$

All this is grand, but the next step in the solutions I have say that we can now say that $$E=p_x+\frac{p_y^2}{2m}+\frac{p_z^2}{2m}$$

I don't see where this comes from.

Any ideas?
Thanks

 

[mark726]

for sharing your progress with us! It looks like you have made some good progress so far. The next step in the solutions is based on the fact that the Hamiltonian, H, is equal to the total energy, E, of the system. This means that we can equate the two expressions for H and E and solve for E, which gives us the expression E=p_x+\frac{p_y^2}{2m}+\frac{p_z^2}{2m}. This can then be substituted into the expression for S to find the trajectory and motion of the particle. I hope this helps clarify things for you. Keep up the good work!