$C^0$ Hamiltonian Dynamics and Continuous Symplectic Structures

Definitions and mathematical statement

For terminology, a smooth symplectic manifold is a smooth manifold $M$ equipped with a smooth differential $2$-form $\omega$ such that $\mathrm d\omega=0$ and $\omega$ is nondegenerate at every point. A smooth symplectomorphism is a diffeomorphism $f\colon M\to M$ satisfying $f^*\omega=\omega$. For a smooth function $H\colon M\to{\mathbb{R}}$, its Hamiltonian vector field is the uniquely determined vector field $X_H$ satisfying \[ \iota_{X_H}\omega=-\mathrm dH. \] This fixes a sign convention for explanatory purposes. Smooth Hamiltonian dynamics studies the integral curves of such vector fields, including those arising from time-dependent Hamiltonians. These definitions provide background terminology, not additional hypotheses imposed on the source question.

Source question. Can one define $C^0$ counterparts of classical smooth symplectic objects, in particular a meaningful theory of $C^0$ Hamiltonian dynamics?

Unresolved formulation. Here $C^0$ indicates a continuous level of regularity, but the supplied record does not specify the objects to be extended, the ambient spaces, the topology of approximation, or the properties that make a proposed theory meaningful. It does not specify dimension, compactness, boundary conditions, regularity of Hamiltonians or trajectories, time domains, or completeness requirements. Consequently, it supplies a research question rather than a proposition with fixed hypotheses and quantifiers. No particular completion, limiting construction, or existence assertion is selected by this draft.

Short English statement

The question asks for continuous counterparts of smooth symplectic geometry, especially Hamiltonian dynamics. The supplied record leaves the desired objects and defining properties unspecified, so it does not determine a unique mathematical proposition. Its supplied open status should not be read as asserting that no $C^0$ Hamiltonian framework exists.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30002125$, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. No individual proposer is named; the supplied proposed year is 2012.
• Supplied report citation: Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2013), Oberwolfach Reports, pp. 2139--2203, DOI 10.4171/owr/2012/35. Supplied source link and literature reference labeled "OWR report": https://doi.org/10.4171/owr/2012/35. The background describes these reports as documenting workshops and their open-problem discussions.
• Supplied literature reference labeled "C0 Hamiltonian dynamics": https://arxiv.org/abs/1808.09790. The record supplies no authors or fuller bibliographic details for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open, describes it as broad rather than a precise proposition, and states that the $C^0$ Hamiltonian framework has partial results and remains an active open area. This is supplied dated curation derived from the supplied OWR curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified 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 claim, not verification performed for this draft.