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Math Help - Hamiltonian Equation

  1. #1
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    Hamiltonian Equation

    Just wondering if you have
    dx/dt = y^3 - y
    dy/dt = -x

    then you can find the Hamiltonian which is H(x,y) = (x^2)/2 +(y^4)/2 -(y^2)/2

    My question is what does the hamiltonian say about the nature of the steady states i know that (0,0) is a saddle and (0,1) is a centre

    My second question is how can you describe the nature of the critical points of H(x,y). Im assuming it might help to draw a phase plane

    thanks
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  2. #2
    Junior Member
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    May 2010
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    I believe the Hamiltonian is actually x^2/2+y^2/2-y4/4. The steady states (i.e. critical points) occur at (x,y) = (0,0), (0,1), and (0,-1). Isn't (0,0) the center and (0,1) and (0,-1) the saddles?

    H = x^2/2 + V(y), with V(y) =y^2/2-y^4/4

    The potential function, V(y) has the following properties:

    V'(y) = 2y -y^3
    V''(y) = 2 - 3y^2

    Therefore V(0) is a minimum, and hence a stable point in the system, and V(1) and V(-1) are maxima in the potential function, and hence unstable.

    Not sure this is getting at your question...
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