标题： q-余辛几何，可积性与约化 显示英文标题

标题： q-Cosymplectic Geometry, Integrability and Reduction

摘要： 在本文中，我们定义了\( q \)-余辛流形的概念，在其上我们研究了哈密顿向量场、梯度向量场、局部梯度向量场和\( q \)-演化向量场。 建立了几个类似Liouville--Arnold的定理和一个\( q \)-余辛Marsden--Weinstein约化定理。 我们还提供了物理实例，说明该结构在多时间动力学（快慢动力系统）中的应用。 为了使我们的工作更加自洽，我们包含了某些结果的详细证明，这些结果可能与余辛流形已知的结果相似。 显示英文摘要

摘要： In the present paper, we define the concept of a \( q \)-cosymplectic manifold, on which we study the Hamiltonian, gradient, local gradient, and \( q \)-evolution vector fields. Several Liouville--Arnold-type theorems and a \( q \)-cosymplectic Marsden--Weinstein reduction theorem are established. We also provide physical examples illustrating the application of the structure to multitime dynamics (Fast-slow dynamical system). To make our work more self-contained, we include detailed proofs for some results that may resemble those known for cosymplectic manifolds.