商流形投影与递阶动力学
Quotient Manifold Projections and Hierarchical Dynamics
摘要：
在本文中，我们探讨了光滑动力系统的层次组织的数学结构。我们首先要明确什么是层次结构中的一个层次，以及较高的层次需要如何尊重较低层次的动态。我们推导出一个数学结构，用于确定不同的层次动态。该构造通过相空间的商流形和满足关于流的某些要求的李群来表示。我们表明，预测到更高的水平可以与动力系统的对称性。我们还讨论了商流形投影与不变流形、运动不变量和Noether定理的关系。
英文摘要：
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.
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