{
  "abstract": "Let $G$ be a Lie group equipped with commuting left and right linear representations on a vector space $V$, and let $G\\Join V$ be the associated centered semidirect product. We develop a reduction scheme for covariant field theories on principal bundles with structure group $G\\Join V$ from both Lagrangian and Hamiltonian perspectives, obtaining the reduced Euler-Poincaré and Lie-Poisson field equations decomposed into their $\\mathfrak g$- and $V$-components, where $\\mathfrak g$ is the Lie algebra of $G$. We also derive the reduced multisymplectic form formula, as well as the Noether and energy-momentum balance laws. Our reduction results are compared with those for principal bundles with standard semidirect-product structure groups, and the role of the centered variables is investigated: they retain the original two-sided geometric data and separate the left and right contributions via the heart and diamond operators. The framework is then applied to a two-sided matrix microstructural model and to the second-order jet group on $\\mathbb R^r$, $G_r^2$. For the matrix model, the conjugation representation decomposes $\\operatorname{Mat}(3)$ into scalar, vector, and symmetric-traceless sectors, clarifying its precise relation to director and alignment-tensor variables without identifying it with standard nematodynamics. For the one-dimensional jet group on a $(1+1)$-dimensional Minkowski base, we obtain a semilinear symmetric-hyperbolic system and an exact traveling conversion front with explicit energy-momentum exchange between the linear- and quadratic-jet sectors. Its hyperbolic-target interpretation yields a conserved positive energy for the linearized perturbations. A nonlinear numerical experiment illustrates the reduced evolution and reconstruction of the group-valued field, with consistency assessed through mesh refinement and an independent wave-map computation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04628",
  "author_total": 3,
  "authors": [
    {
      "name": "Miguel Ángel Berbel"
    },
    {
      "name": "Leonardo Colombo"
    },
    {
      "name": "Álvaro Rodríguez Abella"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cympwa7d5bg4d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04628",
  "fulltext_url": null,
  "id": "cympwa7d5bg4d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04628",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SG"
    ],
    [
      "openalex",
      "3103|Advanced Differential Geometry Research"
    ]
  ],
  "openalex": "W7220684032",
  "pdf_url": "https://arxiv.org/pdf/2610.04628",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cympwa7d5bg4d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2610.04628",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04628"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.SG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Covariant Euler-Poincaré Reduction for Field Theories on Centered Semidirect Products",
  "title_text": "Covariant Euler-Poincaré Reduction for Field Theories on Centered Semidirect Products",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/cympwa7d5bg4d",
  "venue": "arXiv",
  "versions": []
}