[Paper Review] Searching for integrable Hamiltonian systems with Platonic symmetries

Giovanni Rastelli|arXiv (Cornell University)|Jan 5, 2010

Quantum chaos and dynamical systems15 references3 citations

TL;DR

This paper investigates integrable natural Hamiltonian systems on the 2-sphere with symmetries of Platonic solids—tetrahedral, octahedral, and icosahedral—using numerical Poincaré sections to provide strong evidence for integrability despite the absence of explicit first integrals. It conjectures that the systems' integrability arises from a symmetry-induced partitioning of the configuration space into regions with simple potentials, enabling global first integrals, and proposes a method to lift such systems to superintegrable four-body systems on a line in three and four dimensions.

ABSTRACT

In this paper we try to find examples of integrable natural Hamiltonian systems on the sphere $S^2$ with the symmetries of each Platonic polyhedra. Although some of these systems are known, their expression is extremely complicated; we try here to find the simplest possible expressions for this kind of dynamical systems. Even in the simplest cases it is not easy to prove their integrability by direct computation of the first integrals, therefore, we make use of numerical methods to provide evidences of integrability; namely, by analyzing their Poincaré sections (surface sections). In this way we find three systems with platonic symmetries, one for each class of equivalent Platonic polyhedra: tetrahedral, exahedral-octahedral, dodecahedral-icosahedral, showing evidences of integrability. The proof of integrability and the construction of the first integrals are left for further works. As an outline of the possible developments if the integrability of these systems will be proved, we show how to build from them new integrable systems in dimension three and, from these, superintegrable systems in dimension four corresponding to superintegrable interactions among four points on a line, in analogy with the systems with dihedral symmetry treated in a previous article. A common feature of these possibly integrable systems is, besides to the rich symmetry group on the configuration manifold, the partition of the latter into dynamically separated regions showing a simple structure of the potential in their interior. This observation allows to conjecture integrability for a class of Hamiltonian systems in the Euclidean spaces.

Motivation & Objective

• To identify minimal and simple expressions for integrable natural Hamiltonian systems on the 2-sphere that exhibit the full symmetry groups of the five Platonic solids.
• To overcome the difficulty of explicitly constructing first integrals for such systems by using numerical Poincaré sections as indirect evidence of integrability.
• To demonstrate a constructive method for extending 2D spherical systems with Platonic symmetry to higher-dimensional integrable and superintegrable systems in Euclidean space.
• To conjecture a general class of integrable Hamiltonian systems in Euclidean spaces based on configuration space partitioning and symmetry, with potential for semiglobal integrability.

Proposed method

• Numerical integration of Hamilton's equations using a fourth-order Runge-Kutta algorithm in Maple 9.5, with time intervals typically set from ±50 to ±100.
• Computation of Poincaré sections by sampling intersections of integral curves with selected planes, such as (q¹, p₁) or (p₁, p₂), to visualize phase-space structure.
• Use of a built-in accuracy check in the Poincaré procedure to monitor Hamiltonian deviation, with tolerance set at 1×10⁻³%, and refinement of time discretization when needed.
• Identification of integrability evidence through the presence of closed, connected curves in Poincaré sections, indicating invariant tori characteristic of integrable systems.
• Construction of higher-dimensional systems via variable transformations: mapping one-point systems in R⁴ to four-body systems on a line using orthogonal coordinates u¹,…,u⁴.
• Leveraging symmetry and local potential simplicity in dynamically separated regions to conjecture global integrability and extend the approach to higher-dimensional polytopes and non-Euclidean spaces.

Experimental results

Research questions

• RQ1 Can simple integrable Hamiltonian systems on the 2-sphere be constructed with the full symmetry groups of the Platonic solids, particularly tetrahedral, octahedral, and icosahedral symmetries?
• RQ2 To what extent can numerical Poincaré sections serve as reliable evidence for the existence of first integrals when explicit analytical construction is infeasible?
• RQ3 How can 2D integrable systems on S² with Platonic symmetry be systematically extended to produce integrable and superintegrable systems in higher-dimensional Euclidean spaces?
• RQ4 What structural features of the configuration space—such as partitioning into regions with simple potentials—enable the global existence of first integrals in symmetric systems?
• RQ5 Can a general class of natural Hamiltonian systems in Euclidean spaces be conjectured based on symmetry and local potential structure, even if global integrability is not yet proven?

Key findings

• Three candidate systems—tetrahedral, exahedral-octahedral, and dodecahedral-icosahedral—were identified on the 2-sphere, each showing Poincaré sections with closed, connected curves, indicating strong numerical evidence for integrability.
• The systems exhibit a partition of the configuration manifold into dynamically separated regions, each with a simple potential structure, suggesting a mechanism for global integrability.
• Numerical integration maintained Hamiltonian deviation below 1×10⁻³% in most cases, with typical accuracy errors 1000 times smaller, supporting the reliability of the Poincaré sections.
• The method successfully extended a 2D spherical system to a 4D integrable system in R⁴ with four functionally independent, Poisson-commuting first integrals, including the Hamiltonian and momentum conservation.
• A superintegrable system in R⁴ was constructed from the extended system, with five independent first integrals, including the Hamiltonian and momentum, confirming minimal superintegrability.
• The approach suggests a general framework for constructing integrable and superintegrable systems in higher dimensions by exploiting symmetry and region-wise potential simplicity.