# Homework Help: Hamilton-Jacobi Equation

1. Aug 17, 2010

### Piano man

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. 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