Hamiltonian Meandering Transitions in Infinite Dimensions

Definitions and mathematical statement

A standard framework for interpreting the question is a phase space $X$ carrying a Hamiltonian evolution $\Phi_t$, together with a Lie group $G$ acting by symmetries, so that \[ \Phi_t(g\cdot u)=g\cdot\Phi_t(u) \] whenever the evolution is defined. This framework explains the terminology; the supplied record does not specify $X$, its Hamiltonian or Poisson structure, the symmetry group, or the evolution equation.

In this framework, a solution through $u_0\in X$ is a relative equilibrium if there exists $\xi$ in the Lie algebra of $G$ such that \[ \Phi_t(u_0)=\exp(t\xi)\cdot u_0 \] for every time in its interval of existence. A solution $u(t)$ is relative periodic if there exist $T>0$ and $g\in G$ such that \[ u(t+T)=g\cdot u(t) \] whenever both sides are defined. Relative equilibria can also satisfy this relative-periodicity condition; requiring a relative periodic solution that is not a relative equilibrium is an additional distinction whose precise role must be recovered from the underlying transition results.

Source-level problem. Can results on the Hamiltonian meandering transition---from relative equilibria to relative periodic orbits---be extended to infinite-dimensional Hamiltonian systems, such as PDE models of vortex dynamics?

Here "Hamiltonian meandering transition" retains the source terminology. The supplied record does not identify the finite-dimensional results to be extended, define the transition precisely, specify a parameterized family, or provide hypotheses and a proposed conclusion for an infinite-dimensional theorem. PDE models of vortex dynamics are given as examples, without a specified equation, spatial domain, boundary conditions, or solution class. Consequently, the record determines a research question but does not determine a unique fully quantified extension theorem. These qualifications remain unresolved in this statement-only draft.

Short English statement

The problem asks whether Hamiltonian results describing transitions from relative equilibria to relative periodic motion extend to infinite-dimensional systems, including PDE models of vortex dynamics. The specific results, equations, and hypotheses are unspecified, so a precise theorem requires further source clarification. The supplied status is partially solved; it is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001126$, source problem code catalogue entry, title "Hamiltonian Meandering Transitions in Infinite Dimensions," in the collection "Oberwolfach Reports Open Problems." No individual proposer is identified in the supplied record; the recorded proposed year is 2008.
• Supplied source citation and literature reference "original OWR report": Dynamics of Patterns (2009), Oberwolfach Reports, pp. 3201--3271, DOI 10.4171/owr/2008/57. https://doi.org/10.4171/owr/2008/57. The background describes Oberwolfach Reports as reports of mathematical workshops and their open-problem discussions. The citation's 2009 label and the record's 2008 proposed year are retained without resolving their relationship.
• Supplied literature reference "KAM for vortex patches": https://doi.org/10.1134/S1560354724540013. No authors, theorem statement, or explanation of its precise implications for this transition is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports substantial KAM and quasi-periodic-solution theory for infinite-dimensional Hamiltonian PDEs, including vortex-patch results, while describing a general extension to PDE models of vortex dynamics as open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement exact and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is a supplied verification assertion; no independent source consultation is claimed here.