Billiards and Quantitative Symplectic Geometry — Alfonso Sorrentino — Problem 1

Definitions and mathematical statement

The supplied problem is a request for examples, rather than a conjecture with a specified universal quantifier. A standard mathematical framework is the following. Let $M$ be a smooth real manifold of dimension $2n$, where $n\geq 1$. A locally conformal symplectic structure consists of a smooth nondegenerate two-form $\omega$ and a closed one-form $\theta$, called a Lee form, such that \[ {\,\mathrm{d}}\omega=\theta\wedge\omega,\qquad {\,\mathrm{d}}\theta=0. \] Locally, writing $\theta={\,\mathrm{d}} f$, the form $e^{-f}\omega$ is symplectic. If $\theta$ is globally exact, this rescaling can be made globally; the supplied record does not say whether such cases are to be included or excluded.

One possible convention for the evolution is preservation of the conformal class of $\omega$: a flow $\varphi_t$ satisfies \[ \varphi_t^*\omega=e^{g_t}\omega \] on its domain of definition, and a smooth local diffeomorphism $F$ satisfies \[ F^*\omega=e^h\omega \] for suitable smooth real-valued functions $g_t$ and $h$. These are provisional conventions for interpreting the request. The supplied excerpt does not establish that they are the contributor's intended definitions of locally conformal symplectic flows or maps, or whether a more restrictive Hamiltonian convention is intended.

Problem (Alfonso Sorrentino, Problem 1). Provide examples of real-world systems, from physics, mechanics, biology, economics, or other fields, that can be modeled as locally conformal symplectic flows or maps. A mathematical presentation of an example should identify the system, its state space and evolution, and the locally conformal symplectic structure involved.

The source announces subsequent material explaining both the global conformally symplectic case on a symplectic manifold and the locally conformal symplectic case, but that material is absent from the supplied record. Consequently, the precise intended class of admissible models remains unresolved. No particular dimension, boundary condition, completeness requirement, or exclusion of globally conformal cases is specified.

Short English statement

Find real-world systems whose evolution can be modeled by locally conformal symplectic flows or maps. The supplied excerpt omits the promised formal setting, so the definitions above provide a provisional interpretation requiring mathematical review. The registry labels the problem open; that label is not independently verified here.

Sources

• Local UnsolvedMath registry, permanent ID $800478$, exact source problem code catalogue entry; supplied title: Billiards and Quantitative Symplectic Geometry --- Alfonso Sorrentino --- Problem 1. The registry attributes the collection to Albach, Barraud et al., with proposed year 2026, and labels this entry open.
• Supplied external reference: Albach, Barraud et al., Open Problems in Billiards and Quantitative Symplectic Geometry (2026), contributor Alfonso Sorrentino, Problem 1, https://arxiv.org/abs/2602.12896. This attribution and reference are supplied provenance and have not been independently checked.
• The supplied background reports access on 2026-10-16 and extraction by arxiv_latex. This is a supplied access/extraction record, not a claim of source consultation or independent literature verification in this draft.