Calogero–Moser–Vlasov

Definitions and mathematical statement

The source describes an infinite-dimensional Calogero--Moser system of particles on the real line with inverse-square pair potential. For a countable-particle interpretation, formal equations with coupling $c$ are \[ \dot q_i=p_i,\qquad \dot p_i=2c\sum_{j\ne i}(q_i-q_j)^{-3}, \] corresponding in finite dimension to $\tfrac12\sum_i p_i^2+c\sum_{i<j}(q_i-q_j)^{-2}$. The infinite index set, coupling convention/sign, phase space, summability or renormalization, and collision restrictions are not supplied. Thus these formulas explain the stated interaction but do not define a convergent infinite Hamiltonian or an already-existing flow. A kinetic/continuum limit is a different possible interpretation of the title's "Vlasov" term and is not identified with the particle system without additional data.

Existence question (Oliver Knill). Does the intended infinite-dimensional dynamics exist? A precise answer would have to specify the admissible states and the sense and time interval of a flow $\Phi^t$, including uniqueness if well-posedness is intended. No such initial-data class or local/global convention is given.

Integrability question. Is it integrable in the source's stated sense that every invariant measure gives rise to almost-periodic dynamics? An invariant measure satisfies $(\Phi^t)_*\mu=\mu$. The source neither restricts to ergodic measures nor specifies whether measures are probabilities. For invariant probabilities, one possible measure-theoretic meaning is almost periodicity of the Koopman group: each orbit \[ \{h\circ\Phi^t:t\in{\mathbb{R}}\}\subset L^2(\mu) \] is relatively compact for every $h\in L^2(\mu)$. Almost periodicity of individual particle trajectories is another meaning and needs a state-space topology. These are not silently equated or substituted for the source's undefined notion.

The supplied summary claims substantial global existence and broad integrability results for continuum/kinetic Calogero--Moser dynamics, mentioning suitable spaces, mass constraints in a focusing case, conservation laws, and solitons. It gives no precise equation or citation matching the infinite inverse-square particle system. It separately says the every-invariant-measure almost-periodicity formulation is less addressed. Those existence/integrability notions are retained as distinct unverified claims, not used to settle the specific questions.

Short English statement

Construct the infinite inverse-square Calogero--Moser dynamics and ask whether every invariant measure yields almost-periodic dynamics. The particle or kinetic model, phase space, convergence, existence scope, and meaning of almost periodicity are unspecified. Reported continuum/PDE integrability is not identified with this exact measure-theoretic demand.

Sources

• Local UnsolvedMath registry, permanent ID 4200013, source code catalogue entry, "Calogero--Moser--Vlasov"; proposer: Oliver Knill, proposed year 2000. Supplied status partially solved, classification PARTIAL-PROGRESS, updated 2026-08-06.
• Knill, Open problems in Hamiltonian dynamics (2000), item "13) CALOGERO-MOSER-VLASOV," https://people.math.harvard.edu/~knill/seminars/intr/. Reported HTML extraction and access: 2026-07-29. The record says the source presents an open item and no later authoritative resolution check is recorded. Rights: NEEDS_REVIEW; public access verified but redistribution permission not established.
• The summary's continuum/kinetic well-posedness, focusing mass constraints, conserved-quantity, and soliton claims have no supplied papers or precise equations.