### Hamiltonian system

Let us consider the hamiltonian of two harmonic oscillators [DangVu and Delcarte (2000), Perko (2001)]:

\begin{equation*} H = \frac{1}{2} \big( p_1 ^2 + \omega_1 ^2 q_1 ^2 \big) + \frac{1}{2} \big( p_2 ^2 + \omega_2 ^2 q_2 ^2 \big) . \end{equation*}
In action and angular variables, the hamiltonian reads:
\begin{equation*} \mathcal{H} = \omega_1 J_1 + \omega_2 J_2 . \end{equation*}
The orbit lies on a torus $T^2$. It is periodic if $\frac{\omega_1}{\omega_2}$ is rational, dense if not. You can visualize it : choose values of $\omega_1$ and $\omega_2$ (decimal numbers between $0$ and $10$).

### Simulation

$\omega_1 =$
$\omega_2 =$ You must enter decimal numbers between $0$ and $10$.

Orbit of an hamitonian system on a torus

## Bibliography

H. Dang-Vu and C. Delcarte. Bifurcations et chaos: une introduction à la dynamique contemporaine avec des programmes en Pascal, Fortran et Mathematica. Universités. Mécanique. Ellipses, 2000.

L. Perko. Differential equations and dynamical systems. Springer, New York, 3rd edition, 2001.