Hamilton-Jacobi theory problem

[Thales Castro]

Homework Statement

A particle moves on the $xy$ plane having it's trajectory described by the Hamiltonian
$$
H = p_{x}p_{y}cos(\omega t) + \frac{1}{2}(p_{x}^{2}+p_{y}^{2})sin(\omega t)
$$

a) Find a complete integral for the Hamilton-Jacobi Equation

b) Solve for $x(t)$ and $y(t)$ with $x(0) = y(0) = 0$; $\dot{x}(0) = 0$, $\dot{y}(0) = v_{0}$.

Homework Equations

Hamilton-Jacobi equation, if $H = H\left( \{ q_{i} \} , \{ p_{i} \} ,t \right)$

$$
H\left( \{q_{i} \} , \{ \frac{\partial S}{\partial q_{I}} \} , t \right) + \frac{\partial S}{\partial t} = 0
$$

The Attempt at a Solution

a) First, because $H$ doesn't depend explicitly on $x$ and $y$, the Hamilton principal function $S$ can be written as:

$$
S = \alpha_{1} x + \alpha_{2} y + \bar{S}(\alpha,t)
$$

Now, all we need for $S$ to be a complete integral is

$$
det\left( \frac{\partial^{2}S}{\partial q_{i} \partial {\alpha_{i}}} \right) \neq 0
$$

which is immediately satisfied by our $S$.

Now, is there any way of choosing the easiest $\bar{S}(t)$ to find the equations of motion? I tried $\bar{S}(t) = 0$, but the HJ Equation got me the trivial solution $\alpha_{1} = \alpha_{2} = 0$. Any ideas? Thanks in advance

 

[Thales Castro]

Ok, I just realized that the Hamilton-Jacobi Equation was already giving me the derivative of $\bar{S}(t)$