I'm having a bit of difficulty understanding part of this problem:(adsbygoogle = window.adsbygoogle || []).push({});

Using the Hamilton-Jacobi equation ﬁnd the trajectory and the motion of a particle in the

potential [tex]U(r)=-Fx [/tex]

The Hamilton-Jacobi Equation: [tex]\frac{\partial S}{\partial t}+H(q_1,...,q_s;\frac{\partial S}{\partial q_1},...,\frac{\partial S}{\partial q_s};t)=0[/tex]

Starting off with the Hamiltonian:

[tex]

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}[/tex]

From HJE, since y and z are cyclic,

[tex]

S(x,y,z;p_x,p_y,p_z;t)=-Et+p_yy+p_zz+S(x,p_x)[/tex]

All this is grand, but the next step in the solutions I have say that we can now say that [tex]

E=p_x+\frac{p_y^2}{2m}+\frac{p_z^2}{2m}[/tex]

I don't see where this comes from.

Any ideas?

Thanks

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# Hamilton-Jacobi Equation

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