Open Problems in Integrable Systems — Superintegrable systems

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a smooth Riemannian metric $g$ on $M$, a natural Hamiltonian on the canonical symplectic cotangent bundle $T^*M$ is \[H(x,p)=\tfrac12\sum_{i,j}g^{ij}(x)p_ip_j+U(x),\] with smooth potential $U$ and inverse metric coefficients $g^{ij}$. A polynomial integral of degree $d$ is a smooth function $F(x,p)$ polynomial of degree $d$ in the cotangent variables $p$, satisfying $\{H,F\}=0$. Following the source, integrals which are functions of $H$ alone are trivial. A geodesic flow is the case $U=0$.

Statement recorded in the supplied source.

Construct a natural system on $S^2$ with nonconstant potential $U$ and polynomial integrals $F,G$, each of degree at least $3$, such that $dH,dF,dG$ are linearly independent almost everywhere on $T^*S^2$, while every polynomial integral of degree $1$ or $2$ is trivial. This is maximal superintegrability for a two-dimensional configuration manifold; $F,G$ need not Poisson commute with each other. [S2]

Short English statement

Find a polynomially superintegrable natural system on the sphere with nonconstant potential and no nontrivial linear or quadratic integral.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100307 (AMR-020-0307). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. Bolsinov, V. S. Matveev, E. Miranda and S. Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), arXiv:1804.03737, Problem 3.10. https://arxiv.org/abs/1804.03737.