AI 中文总结
本文在k-余辛哈密顿场论中引入广义无穷小对称性，定义高阶嘉当与良好非几何对称两类，并建立诺特型定理以给出守恒量，推广至k-辛流形，提供实例验证。
AI 中文摘要
本文在$k$-余辛哈密顿场论的框架下研究高阶对称性和非几何对称性。我们引入了广义无穷小对称性的概念，该概念放宽了标准对称性条件，进而定义了两类新的对称性：高阶一般无穷小嘉当对称性和良好非几何对称性。对于每一类对称性，我们建立了诺特型定理，为精确的$k$-余辛哈密顿系统提供显式的守恒量。这些结果扩展并推广了先前在$k$-余辛背景下关于标准嘉当对称性的工作。此外，我们以并行方式将所有结果推广到$k$-辛流形上的哈密顿系统，展示了我们方法的统一性。我们提供了具体例子来说明所提出对称性的存在性和适用性。我们的框架为经典场论中广义对称性及其相关守恒律提供了一种系统的几何处理。
英文摘要
In this paper, we investigate higher-order symmetries and non-geometric symmetries within the framework of $k$-cosymplectic Hamiltonian field theory. We introduce the concept of generalized infinitesimal symmetry, which relaxes the standard symmetry condition, and then define two new classes of symmetries: higher-order general infinitesimal Cartan symmetries and good non-geometric symmetries. For each of these symmetry classes, we establish a Noether-type theorem that provides explicit conserved quantities for exact $k$-cosymplectic Hamiltonian systems. These results extend and generalize previous work on standard Cartan symmetries in the $k$-cosymplectic setting. Furthermore, we extend all of these results in a parallel fashion to Hamiltonian systems on $k$-symplectic manifolds, demonstrating the unifying nature of our approach. Concrete examples are provided to illustrate the existence and applicability of the proposed symmetries. Our framework offers a systematic geometric treatment of generalized symmetries and their associated conservation laws in classical field theories.
Comments23 pages
