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Hamilton Jacobi Equation

  1. May 10, 2015 #1

    CAF123

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    Gold Member

    1. The problem statement, all variables and given/known data
    The motion of a free particle on a plane has hamiltonian $$H =E = \text{const} = \frac{1}{2m} (p_r^2 + \frac{p_{\theta}^2}{r^2})$$ Set up and find a complete integral for ##W##, the time independent generating function to canonical coordinates such that new coordinates are cyclic and momenta constant. (No need to evaluate explicitly)

    Using this, find r as a function of t. Similarly find r as a function of theta.

    2. Relevant equations
    Hamilton Jacobi equation $$H(q, \frac{\partial W}{\partial q}) = \alpha_1$$

    ##W = W_r(r,\alpha) + W_{\theta}(\theta, \alpha)##

    3. The attempt at a solution

    I am just a bit confused on what 'complete integral' means. Since ##\theta## is a cyclic coordinate, ##p_{\theta} = \partial W/\partial \theta = \text{\const}## so can write the H-J equation as $$\frac{1}{2m} \left((\frac{\partial W_r}{\partial r})^2 + \frac{1}{r^2}\alpha_{\theta}^2\right) = E$$ which can be rewritten like $$W_r = \int^r \sqrt{2mE - \frac{1}{r^2} \alpha_{\theta}^2} dr'$$ Then $$\frac{\partial W_r}{\partial E} = \frac{1}{2}\int^2 \frac{1}{\sqrt{2mE - \frac{1}{r'^2}\alpha_{\theta}^2}}dr' = \frac{1}{2} \int^r \frac{r' dr'}{\sqrt{2mEr.^2 - \alpha_{\theta}^2}}$$, which can be solved using a sub, but I am not sure what I have really obtained through this calculation and how to obtain r explicitly in terms of theta.

    Thanks!

    I've been given a hint that the integral $$\int \frac{dx}{x \sqrt{x^2 - b^2}}$$ should be used somewhere and eval using x = b/cos u.
     
    Last edited: May 10, 2015
  2. jcsd
  3. May 10, 2015 #2

    TSny

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    I believe an “integral” of the H-J equation is just another name for a solution of the H-J equation. A “complete integral” is a solution that depends on ##n## independent constants, where ##n## is the number of degrees of freedom. ##n## equals 2 in your problem and you can easily identify the two constants.

    In your study of the H-J theory, you should have developed a relation between ##\frac{\partial W}{\partial E}## and the time ##t##. You found an expression for ##\frac{\partial W} {\partial E}## in terms of an integral of a function of ##r##. If you carry out the integration and use the relation between ##\frac{\partial W}{\partial E}## and ##t##, you can determine ##r## as a function of ##t##. You should find that ##r## varies with time as you would expect for a free particle.

    To find the relation between ##r## and ##\theta##, you need to consider ##\frac{\partial W}{\partial \alpha_{\theta}}## where ##W = W_{r} + W_{\theta}.##
     
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