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Hamilton - Jacobi method for a particle in a magnetic field

  1. Sep 1, 2017 #1
    1. The problem statement, all variables and given/known data
    Hamiltonian of charged particle in magnetic field in 2D is ##H(x,y,p_x,p_y)=\frac{(p_x-ky)^2+(p_y+kx)^2}{2m}## where ##k## and ##m## are constant parameters. For separation of this system use ##S=U(x)+W(y)+kxy+S_t(t)##. Solve Hamilton - Jacobi equation to get ##x(t), y(t)## .

    2. Relevant equations


    3. The attempt at a solution
    When I substitute ##p_i=\frac{dS}{dx_i}## in the assignment and assumptation ##S_t(t)t=a_0t## (hamiltonian don't depend on time) I get
    ##U_x^2+(W_y+2kx)^2=a (+b-b)##,
    where ##a=2ma_0##, ##b## is separation constant and subscript means derivative. I separate it and I get
    ##W_y=b##
    ##U_x=\sqrt{a-(2kx+b)^2}##...my first question is, if it is good procedure and is it right?
    Then I integrate but second terms is quite...complex, so I denote solution of this integral as ##I(x;a;b)+d##, ##c## and ##d## are constants.
    ##W=by+c##
    ##U=I+d##
    I suppose that constants ##c## and ##d## are unimportant because of they are just additive. So
    ##S=I+by+kxy-\frac{a}{2m}t##
    and finally ##x(t), y(t)## I get from
    ##\frac{dS}{da}=e## and ##\frac{dS}{db}=f## where ##e,f## are constants.
    Please comment my procedure and also if exist some "better" prcedure when I could get exact ##x(t),y(t)## because from this integral ##I## it is impossible.
    Thank you for an advice.
     
  2. jcsd
  3. Sep 1, 2017 #2

    TSny

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    Gold Member

    Did you mean to have the factor of ##t## on the left side? Do you need a negative sign in front of the right side?
    Doesn't the factor of (b-b) make the right side zero? Maybe just a typo.
    This looks correct to me.
    Yes
    This looks good. It should not be hard to evaluate ##\partial I / \partial a## and ##\partial I / \partial b##. [Edit: Differentiate "under the integral sign" and then do the integral.]
     
    Last edited: Sep 1, 2017
  4. Sep 5, 2017 #3
    Ok, thank you very much for comments.
     
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