Open Problems in Integrable Systems — Two-dimensional case.

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.

Describe and classify the natural systems on closed two-dimensional surfaces admitting a nontrivial polynomial integral, organized by the smallest such degree: construct examples and prove nonexistence results, with any additional hypotheses made explicit. The source's table covers $S^2,T^2$ and degrees $1,2,3,4,\ge5$. Its precise subsidiary conjectures ask: (i) whether a cubic polynomial integral for a Riemannian geodesic flow on $T^2$ forces a Killing vector field; (ii) whether, on $T^2$ with nonconstant $U$, an odd-degree polynomial integral forces a linear integral and a nontrivial even-degree integral forces a nontrivial quadratic integral; and (iii) the same assertion in (ii) when the torus metric is flat. A Killing vector field generates local isometries and yields a linear integral. [S2]

Short English statement

Classify polynomially integrable natural systems on closed surfaces by their smallest integral degree, including the stated torus reductions to linear or quadratic integrals.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100301 (AMR-020-0301). 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 3.1, table, and Conjectures 3.2–3.4. https://arxiv.org/abs/1804.03737.