# Simplest model of chaos with time-independent smooth Hamiltonian and trivial topology?

What is the simplest model of chaos governed by a time-independent smooth Hamiltonian on a phase-space with trivial topology?

We know that...

• With trivial topology, the minimal number of dimension to exhibit chaos with first-order ODEs is three (e.g., the Lorentz system), but these are of course not Hamiltonian.

• Indeed, time-independent Hamiltonian systems of two phase-space dimensions (one configuration variable) cannot be chaotic because the trajectories follow the fixed-energy surfaces which foiliate the phase space. Thus we need four phase-space dimensions (two configuration variables).

• There exist time-dependent examples of Hamiltonian chaos in two-dimensional phase space, e.g., the kicked-top.

• There exist discrete-time non-smooth chaotic maps in two-dimensions that preserve area, e.g., Baker's map.

• Hadamard's billiards is a time-independent chaotic Hamiltonian system on a four-dimensional phase space. However, it has non-trivial topology (a two-holed donut), exhibiting chaos on account of constant negative curvature.

The double pendulum with equal masses and equal arm lengths has Hamiltonian

$$H(\theta_1,p_1,\theta_2,p_2)= \frac{1}{6} m l^2 \left ( {\dot \theta_2}^2 + 4 {\dot \theta_1}^2 + 3 {\dot \theta_1} {\dot \theta_2} \cos (\theta_1-\theta_2) \right ) - \frac{1}{2} m g l \left ( 3 \cos \theta_1 + \cos \theta_2 \right ).$$

where $\theta_1$ and $\theta_2$ are the angles of the top and bottom arm with respect to the vertical direction, and $p_1$ and $p_2$ are the respective conjugate momenta. This satisfies all our specific requirements except that it is not very simple.

(Note that I am merely extending this question on /r/physics, which lacked the specialization to topologically trivial phase spaces.)

## 1 Answer

1. A 1D autonomous system is always Liouville integrable – the Hamiltonian $H$ itself is an integral of motion – so we would have to consider at least 2D to find chaos.

2. A quadratic potential $V$ yields a linear system, which doesn't exhibit chaos. So the potential $V$ should contain cubic or higher terms.

3. The potential should mix coordinates (e.g., terms like $x_2^2x_1$) so that the system isn't separable into two 1D systems.

4. The Henon-Heiles system in 2D with a potential $V$ containing both quadratic and cubic terms is often given as the standard example of chaos. See e.g. this Phys.SE post.

5. The homogeneous Henon-Heiles system with a purely cubic potential $V$ is an even simpler system: the Hamiltonian $$H(x_1,p_1,x_2,p_2) = \frac{g}{3}x_1^3 + x_1 x_2^2 + \frac{1}{2}\left(p_1^2 + p_2^2\right)$$ is non-integrable for many value of $g$, including $g \in (-\infty,1)$. See Example 3 (Eq. 5.2) in Section 5.1 of Ref. 1.

References:

1. J.J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Math. 179 (1999). [free PDF].
• This is darn close to being an objective answer to "simplest", a seemingly subjective criterion. Just what I was hoping for. Stupendous! Thanks Qmechanic for showing there's still a bit of magic on Physics.SE. Commented May 18, 2018 at 18:47
• Indeed, pretty good answer! Note that, besides the citeseerx link provided by @JessRiedel 's edit, a pdf file of Ref. 1 is also available in ResearchGate. Commented May 18, 2018 at 19:19
• Solutions tend to explode to infinity. Any examples that don't? Commented Jun 15, 2021 at 3:02