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Fooling around with Hamilton-Jacobi theory: can this be right?

  1. Jan 28, 2009 #1
    A system with one particle in one dimension x, momentum p, and hamiltonin H(x,p). Hamilton's principal function S(x,t) is a function satisfying.

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

    Now when we say that

    [tex]p=\frac{\partial S}{\partial x}[/tex]

    this is somewhat nonsensical. The LHS, p, is a dynamical variable, independent of x, whereas the RHS is a function of x and t. It makes sense to refer to x(t) and p(t) separately, but not generally of p(x,t).


    What [tex]p=\frac{\partial S}{\partial x}[/tex] must mean is that along any particular solution (x(t), p(t)) in the flow of solutions in phase space, it is the case that


    [tex]p(t)=\frac{\partial S}{\partial x}|_{x=x(t)}[/tex]


    That is, we can speak of a function [tex]\hat{p}(x,t)\equiv\frac{\partial S}{\partial x}[/tex] and then it is true that if x(t),p(t) are solutions to the equations of motions then for all t


    [tex]p(t)=\hat{p}(x(t),t)[/tex].


    Let me pause here before proceeding to the question proper. Is what I say so far correct?
     
    Last edited: Jan 28, 2009
  2. jcsd
  3. Jan 28, 2009 #2
    You are correct that partial derivative notation is ambiguous (cf. the additional notation which accompanies thermodynamic identities), although it should not normally cause confusion.
     
  4. Jan 29, 2009 #3
    Thanks, confinement.

    Actually, I don't need to post the actual question now. I figured out where I was going wrong. However, I would be interested in any comments about the OP.
     
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