Covariant Euler-Poincaré Reduction for Field Theories on Centered Semidirect Products
Abstract
Let G be a Lie group equipped with commuting left and right linear representations on a vector space V, and let GJoin V be the associated centered semidirect product. We develop a reduction scheme for covariant field theories on principal bundles with structure group GJoin V from both Lagrangian and Hamiltonian perspectives, obtaining the reduced Euler-Poincaré and Lie-Poisson field equations decomposed into their g- and V-components, where 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 ℝr, Gr2. For the matrix model, the conjugation representation decomposes 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.
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BibTeX
@misc{berbel2026covariant,
title = {{Covariant Euler-Poincaré Reduction for Field Theories on Centered Semidirect Products}},
author = {Miguel Ángel Berbel and Leonardo Colombo and Álvaro Rodríguez Abella},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04628},
eprint = {2610.04628},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04628},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Covariant Euler-Poincaré Reduction for Field Theories on Centered Semidirect Products
AU - Miguel Ángel Berbel
AU - Leonardo Colombo
AU - Álvaro Rodríguez Abella
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04628
UR - https://arxiv.org/abs/2610.04628
AB - Let G be a Lie group equipped with commuting left and right linear representations on a vector space V, and let GJoin V be the associated centered semidirect product. We develop a reduction scheme for covariant field theories on principal bundles with structure group GJoin V from both Lagrangian and Hamiltonian perspectives, obtaining the reduced Euler-Poincaré and Lie-Poisson field equations decomposed into their g- and V-components, where 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 ℝ^r, G_r^2. For the matrix model, the conjugation representation decomposes 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.
ER -