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Hamilton-Jacobi Equation

  1. Aug 17, 2010 #1
    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 [tex]U(r)=-Fx [/tex]

    2. Relevant equations

    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]

    3. The attempt at a solution
    [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
     
  2. jcsd
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