发表机构
Centre for Automation and Robotics (CSIC-UPM)（自动化与机器人中心（CSIC-UPM））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对中心半直积主丛上的协变场论，从拉格朗日与哈密顿视角建立约化方案，得到分解的Euler-Poincaré与Lie-Poisson方程、多辛公式及守恒律，并应用于矩阵微结构与jet群模型，验证了理论的有效性。
AI 中文摘要
设 $G$ 为李群，在向量空间 $V$ 上配备可交换的左、右线性表示，并设 $G\Join V$ 为相应的中心半直积。我们从拉格朗日与哈密顿两个视角，对结构群为 $G\Join V$ 的主丛上的协变场论发展了一套约化方案，得到了分解为 $\mathfrak g$ 分量与 $V$ 分量的约化Euler-Poincaré方程和Lie-Poisson场方程，其中 $\mathfrak g$ 是 $G$ 的李代数。我们还推导了约化多辛形式公式，以及Noether和能量-动量平衡律。我们的约化结果与结构群为标准半直积的主丛上的结果进行了比较，并研究了中心变量的作用：它们保留了原始的双侧几何数据，并通过心算子与菱形算子分离了左、右贡献。该框架随后应用于双侧矩阵微结构模型以及 $\mathbb R^r$ 上的二阶jet群 $G_r^2$。对于矩阵模型，共轭表示将 $\operatorname{Mat}(3)$ 分解为标量、向量和对称无迹扇区，澄清了其与指向矢和排列张量变量的精确关系，而不将其等同于标准向列动力学。对于 $(1+1)$ 维Minkowski底空间上的一维jet群，我们得到了一个半线性对称双曲系统和一个精确的行进转换前沿，其线性与二次jet扇区之间具有显式的能量-动量交换。其双曲靶标解释为线性化扰动提供了守恒的正能量。一个非线性数值实验展示了群值场的约化演化与重构，并通过网格细化与独立的波映射计算评估了一致性。
英文摘要
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.
