How Does the Hamilton-Jacobi Equation Describe Particle Motion in a Plane?

[CAF123]

Homework Statement

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.

Homework Equations

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

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

The Attempt at a Solution

[/B]
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.

 

[TSny]

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 $W = W_{r} + W_{\theta}.$