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Hamilton-Jacobi Problem

  1. Mar 1, 2014 #1
    1. The problem statement, all variables and given/known data

    I'm given the time-dependent potential,

    [tex] V(x,t) = -mAxe^{-\gamma t} [/tex]

    and asked to find the solution to the Hamilton-Jacobi equation,

    [tex] H(x,\frac{\partial S}{\partial x}) + \frac{ \partial S}{\partial t} = 0 [/tex]

    3. The attempt at a solution

    Without any additional information, I'm assuming the correct Hamiltonian is given simply by,

    [tex] H = \frac{p^2}{2m} -mAxe^{-\gamma t} [/tex]

    which gives me,

    [tex] \frac{1}{2m}\bigg ( \frac{\partial S}{\partial x} \bigg )^2 - mAxe^{-\gamma t} + \frac{ \partial S}{\partial t} = 0 [/tex]

    but I'm having troule separating the variables in order to solve this equation. Normally, when [itex] V = V(x) [/itex] you can use the form [itex] S(x,\alpha,t) = W(x,\alpha) - Et [/itex], but here this won't work.

    Have I somehow used the wrong Hamiltonian, or do I just need to guess correctly the right form of [itex] S [/itex]?
  2. jcsd
  3. Mar 1, 2014 #2
    I suggest using the method of characteristics to solve this PDE problem.
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