# 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

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