Minimum-Energy $N$-Body Configurations at Fixed Angular Momentum

Definitions and mathematical statement

The supplied problem asks for the minimum-energy configurations of an $N$-body system at a prescribed angular-momentum level. It does not specify a mechanical model, so the following expresses the optimization target conditionally on model data that remain to be supplied.

For a specified model, let $\mathcal P$ denote its admissible states, let $H:\mathcal P\to{\mathbb{R}}$ denote total energy, and let $J:\mathcal P\to\mathcal A$ record the angular-momentum quantity being constrained, with $\mathcal A$ its value space. For a prescribed admissible level $\ell\in\mathcal A$, define \[ \mathcal P_\ell=\{z\in\mathcal P:J(z)=\ell\},\qquad E_\ell=\inf_{z\in\mathcal P_\ell}H(z), \] where the infimum is understood in the extended real numbers, and define the global minimizing states by \[ \mathcal M_\ell=\{z\in\mathcal P_\ell:H(z)=E_\ell\}. \] If $\mathcal Q$ is the configuration space and $\pi:\mathcal P\to\mathcal Q$ extracts a state's configuration, the requested configurations are \[ \pi(\mathcal M_\ell). \] The problem is to determine these minimum-energy configurations for the intended $N$-body model and prescribed angular-momentum level. These formulas are a conditional formulation, not a specification of the missing model or an assertion that a minimum is attained.

The record does not determine $\mathcal P,H,J,\mathcal A,\mathcal Q$, or $\pi$. In particular, it does not specify the interaction law, masses, spatial dimension, body geometry, collision or contact conditions, or whether angular momentum means a vector, a component, or its magnitude. Its background invokes finite density and a proposed classification of all minimum states, but supplies neither a definition of the finite-density model nor the proposed classification. The quantification over $N$, model parameters, and angular-momentum levels is also unstated.

Short English statement

Determine all configurations arising from global energy-minimizing states of the intended $N$-body system at a prescribed angular-momentum level. The supplied record leaves the mechanical model and the meaning of the angular-momentum constraint unspecified, so only a conditional optimization formulation is possible. Its finite-density existence claim and proposed classification require source clarification and human mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001864$, source problem code catalogue entry, title "Minimum-Energy $N$-Body Configurations at Fixed Angular Momentum." The record attributes the problem to the Oberwolfach Reports collection; no individual proposer is supplied. The proposed year is recorded as 2011.
• Supplied report citation: Applied Dynamics and Geometric Mechanics (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/39, pp. 2217--2282. https://doi.org/10.4171/owr/2011/39. This link is also supplied under the literature-source label "OWR report 39." The citation's 2012 title date and the record's 2011 proposed year are retained without resolving their relationship.
• Supplied literature-source label: "finite density three body results." https://pmc.ncbi.nlm.nih.gov/articles/PMC5477774/. The record supplies no author, article title, or theorem statement for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and states that finite density makes the constrained minimum exist and permits computations for fixed angular momentum, while the report leaves a proposed classification of all minimum states unproved. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification. The record also labels its statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. Those are supplied registry annotations, not independent mathematical verification of this draft.