Open Problems in Integrable Systems — Polynomial in momenta integrals in higher dimensions.

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 and describe natural Hamiltonian systems on manifolds of dimension $n>2$ that have $n$ integrals with linearly independent differentials almost everywhere, pairwise Poisson commuting $F_1=H,F_2,\ldots,F_n$, each polynomial in momenta. The source asks for further examples, particularly systems on closed manifolds or arising in mathematical physics. [S2]

Short English statement

Construct further higher-dimensional natural Hamiltonian systems with a full commuting family of polynomial integrals.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100303 (AMR-020-0303). 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 3.6. https://arxiv.org/abs/1804.03737.