{
  "abstract": "We study Brenier's relaxed least-action problem in the unit disc $D:=\\{x\\in\\mathbb{R}^2:\\ |x|<1\\}$ at the critical time $T=π$, with the endpoint pair $i_D$ (the identity on $D$) and $-i_D$. We prove that the critical energy shell \\[ S^3=\\{(x,v)\\in\\mathbb{R}^2\\times\\mathbb{R}^2: |x|^2+|v|^2=1\\} \\] supports stationary action-minimising generalised incompressible flows that are not invariant under physical rotations. This answers the question posed by Bernot, Figalli, and Santambrogio. Our construction relies on the normalised surface measure $σ$ on $S^3$ and the Hopf quotient \\[ Π=(N,M,L):S^3\\to\\mathbb S^2_{1/2}, \\qquad \\mathbb S^2_{1/2}:=\\{(n,m,\\ell)\\in\\mathbb{R}^3: n^2+m^2+\\ell^2=1/4\\}. \\] Here, \\[ N=\\frac12(x_1^2+v_1^2-x_2^2-v_2^2),\\quad M=x_1x_2+v_1v_2,\\quad L=x_1v_2-x_2v_1 . \\] This quotient is a first integral of the harmonic-oscillator flow. Let $τ:=Π_\\#σ$, let $g$ be a bounded Borel function on $\\mathbb S^2_{1/2}$ whose $L^\\infty(τ)$-class is odd under $(n,m,\\ell)\\mapsto(n,m,-\\ell)$, and let $δ\\in\\mathbb{R}$. If $1+δg\\circΠ\\ge0$ $σ$-a.e., then \\[ dμ_{δ,g}=π(1+δg\\circΠ)\\,dσ\\] has Lebesgue spatial marginal and is stationary. The induced path measure is then minimising. We also determine exactly which members of this tilted family are rotationally invariant. If $δ\\ne0$, this is equivalent to axisymmetry of the $L^\\infty(τ)$-class of $g$, modulo $τ$-null sets. In particular, taking $g(n,m,\\ell)=\\ell m$ with $0<|δ|<8$ gives strictly positive non-rotationally invariant minimisers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19008",
  "author_total": 1,
  "authors": [
    {
      "name": "Maja Gwozdz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aded7t7jqcv9b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19008",
  "fulltext_url": null,
  "id": "aded7t7jqcv9b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19008",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7213508970",
  "pdf_url": "https://arxiv.org/pdf/2609.19008",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aded7t7jqcv9b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.19008",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19008"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Non-rotationally invariant generalised geodesics in the disc",
  "title_text": "Non-rotationally invariant generalised geodesics in the disc",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/aded7t7jqcv9b",
  "venue": "arXiv",
  "versions": []
}