标题： 商流形投影与分层动力学 显示英文标题

标题： Quotient Manifold Projections and Hierarchical Dynamics

摘要： 在本文中，我们探讨平滑动力系统中分层组织的数学结构。 我们首先明确什么是分层中的一个层次，以及较高层次如何尊重较低层次的动力学。 我们推导出一种数学构造，用于识别分层动力系统中的不同层次。 该构造通过相空间的商流形和满足与流动有关某些要求的李群来表达。 我们表明，到更高层次的投影可以与动力系统的对称性相关联。 我们还讨论了商流形投影如何与不变流形、运动不变量以及诺特定理相关联。 显示英文摘要

摘要： In this paper we explore the mathematical structure of hierarchical organization in smooth dynamical systems. We start by making precise what we mean by a level in a hierarchy, and how the higher le vels need to respect the dynamics on the lower levels. We derive a mathematical construction for identifying distinct levels in a hierarchical dynamics. The construction is expressed through a quotient manifold of the phase space and a Lie group that fulfills certain requirement with respect to the flow. We show that projections up to higher levels can be related to symmetries of the dynamical system. We also discuss how the quotient manifold projections relate to invariant manifolds, invariants of the motion, and Noether's theorem.