SE（d）-约化在Stokes和Navier-Stokes流体中的作用
The role of SE(d)-reduction for swimming in Stokes and Navier-Stokes fluids
摘要：
稳定的游泳似乎是周期性的和稳定的。这些特征正是极限环的定义，所以我们问“我们能把游泳看作是一个极限环吗？“在本文中，我们无法全面回答这个问题。然而，我们将发现，通过SE（d）对称性的约化使我们更接近。在进行对称化约后，我们将找到一个稳定的不动点，它对应于死水中的静止物体。然后，我们将推测存在的周期轨道，这是“近似”的极限环在约化系统。当我们提升这些周期轨道从减少的相空间，我们得到动态鲁棒的相对周期轨道，其中每个周期与前一个由SE（d）相位。显然，由非零平移和恒等旋转组成的SE（d）相位意味着定向游泳，而非平凡旋转对应于具有恒定转弯半径的转弯。
英文摘要：
Steady swimming appears both periodic and stable. These characteristics are the very definition of limit cycles, and so we ask "Can we view swimming as a limit cycle?" In this paper we will not be able to answer this question in full. However, we shall find that reduction by SE(d)-symmetry brings us closer. Upon performing reduction by symmetry, we will find a stable fixed point which corresponds to a motionless body in stagnant water. We will then speculate on the existence of periodic orbits which are "approximately" limit cycles in the reduced system. When we lift these periodic orbits from the reduced phase space, we obtain dynamically robust relatively periodic orbits wherein each period is related to the previous by an SE(d)-phase. Clearly, an SE(d) phase consisting of nonzero translation and identity rotation means directional swimming, while non-trivial rotations correspond to turning with a constant turning radius.
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