标题： 高阶欧拉-庞加莱场方程 显示英文标题

标题： Higher-order Euler-Poincaré field equations

摘要： 我们为定义在主$G$-丛的高阶喷射丛上的$G$-不变拉格朗日场理论发展了一种约化理论，从而得到了高阶欧拉-泊松场方程。 为此，我们将哈密顿原理转移到约化配置丛上，该丛被识别为主$G$-丛的平坦连接（在一定阶数下）。 结果表明，重构条件总是满足，因此约化场方程的每一个解局部上都来源于原始（未约化）方程的解。 此外，证明了约化方程等价于诺特定理电流的守恒。 最后，通过研究李群上的多元高阶样条来说明该理论。 显示英文摘要

摘要： We develop a reduction theory for $G$-invariant Lagrangian field theories defined on the higher-order jet bundle of a principal $G$-bundle, thus obtaining the higher-order Euler-Poincar\'e field equations. To that end, we transfer the Hamilton's principle to the reduced configuration bundle, which is identified with the bundle of flat connections (up to a certain order) of the principal $G$-bundle. As a result, the reconstruction condition is always satisfied and, hence, every solution of the reduced field equations locally comes from a solution of the original (unreduced) equations. Furthermore, the reduced equations are shown to be equivalent to the conservation of the Noether current. Lastly, we illustrate the theory by investigating multivariate higher-order splines on Lie groups.