Dense Hamiltonian Trajectories Near Integrable Systems

Definitions and mathematical statement

The supplied question concerns the integrable Hamiltonian \[ H_0(I)=\frac12\sum_{i=1}^{n}I_i^2. \] The number of action variables $n$, their domain, the underlying symplectic phase space, and the differentiability index $r$ are not specified in the record. Consequently, the following is a conditional formulation template, requiring these data to be supplied.

For a specified symplectic phase space $(M,\omega)$ on which the displayed $H_0$ is defined, and a specified neighborhood $\mathcal U$ of $H_0$ in a specified $C^r$ topology, ask whether there exists \[ H\in C^\infty(M,{\mathbb{R}})\cap\mathcal U \] whose unit-energy set \[ \Sigma_H:=H^{-1}(\{1\}) \] contains a dense Hamiltonian trajectory. More explicitly, define the Hamiltonian vector field by $\iota_{X_H}\omega=-{\,\mathrm{d}} H$. If $\phi_H^t(x)$ denotes its maximal trajectory through $x$, with maximal existence interval $J_x$, the density requirement is \[ \exists x\in\Sigma_H\qquad \overline{\{\phi_H^t(x):t\in J_x\}}^{\,\Sigma_H}=\Sigma_H, \] where the closure is taken in the relative topology of the entire unit-energy set.

The source does not specify whether the requested trajectory must exist for all real times or whether density is required for a forward trajectory alone. It also does not state that $1$ is a regular value of $H$; thus calling $\Sigma_H$ a surface does not supply a regularity hypothesis. Finally, if "in a $C^r$-small neighborhood" means arbitrarily close to $H_0$, the intended quantifiers would be: for every neighborhood $\mathcal U$ of $H_0$, there exists such an $H$. The record does not explicitly distinguish that interpretation from existence in a particular sufficiently small neighborhood.

Short English statement

The question asks whether a smooth Hamiltonian close to the quadratic integrable Hamiltonian can have one trajectory dense in its whole energy-one set. The phase space, dimension, regularity index, and precise meaning of closeness must be supplied before this becomes a fully specified mathematical problem. The supplied partial result concerns density in a positive-measure subset, which does not establish the requested density in the whole energy surface.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001289$, title: Dense Hamiltonian Trajectories Near Integrable Systems; source problem code: catalogue entry. The record gives proposed year $2009$ and names no proposer.
• Supplied source citation: Dynamische Systeme (2010), Oberwolfach Reports, pp. 1787--1842, DOI 10.4171/owr/2009/32. Source URL and literature-source link labeled "OWR report": https://doi.org/10.4171/owr/2009/32. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source link labeled "Herman question and partial theorem": https://ems.press/content/serial-article-files/46234. No theorem number, author list, or theorem statement is supplied for this reference.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially solved and reports that a cited theorem constructs an orbit dense in a positive-measure subset, but does not provide a trajectory dense in the whole energy surface. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; the source explicitly requests independent verification of time-sensitive status and cited claims.