AI 中文总结
本研究建立李群不变联络约化理论，提出联络依赖变分原理，导出含两类时间非局域性的积分-微分方程并给出数值格式，在海森堡群与旋转群上验证了理论有效性。
AI 中文摘要
我们对不变联络在几何力学中的作用开展系统研究。首先建立李群上左不变与右不变联络的约化完整理论，证明欧拉-庞加莱方程与李-泊松方程均独立于联络。随后引入一种新的联络依赖变分原理，其中拉格朗日量依赖于沿曲线平行运输回初始点的速度。对于嘉当-斯豪滕联络，该原理导出一个积分-微分欧拉-庞加莱方程，呈现出两类时间非局域性来源：平行运输编码的依赖路径项，以及曲率积分产生的依赖未来项。我们将该方程重新表述为两点边值问题，并提出一种两层数值格式。该一般理论在海森堡群上得到示例，方程因幂零性而简化；在旋转群上则保留了完整的积分-微分结构。
英文摘要
We present a systematic study on the role of invariant connections in geometric mechanics. We first develop a comprehensive theory of reduction under left- and right-invariant connections on Lie groups, showing that the Euler--Poincaré and Lie--Poisson equations are independent of the connection. We then introduce a novel connection-dependent variational principle, where the Lagrangian depends on the velocity parallel-transported back to the initial point of the curve. For Cartan--Schouten connections, this leads to an integro-differential Euler--Poincaré equation that exhibits two distinct sources of nonlocality in time: a path-dependent term encoded in the parallel transport and a future-dependent term arising from a curvature integral. We reformulate this equation as a two-point boundary value problem and present a two-level numerical scheme. The general theory is illustrated on the Heisenberg group, where the equations simplify due to nilpotency, and on the rotation group, where the full integro-differential structure is retained.
