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. 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.


  1. J.J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Math. 179 (1999). [free PDF].
  • $\begingroup$ 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. $\endgroup$ – Jess Riedel May 18 '18 at 18:47
  • 1
    $\begingroup$ 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. $\endgroup$ – stafusa May 18 '18 at 19:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.