形空间上经典力学的辛约化
Symplectic Reduction of Classical Mechanics on Shape Space
摘要：
物理学研究的首要目标之一是找到描述我们宇宙的最基本和最普遍的理论。许多理论假设存在一个绝对的空间和时间，物理对象位于其中，物理过程发生。然而，更基本的是将时间理解为相对于另一个物体的运动，例如钟摆的摆动次数，以及一个物体的位置主要是相对于其他物体。本文的目的是解释如何利用关系论原理（将在下面介绍），经典力学可以在一个最基本的空间上公式化，这个空间是从绝对实体中解放出来的：形状空间。在形状空间上，只考虑子系统的相对方向和长度。为了研究经典系统的形状随时间的演化，将“Hamilton系统的辛约化”方法推广到尺度变换，从而实现了经典系统关于全相似群的约化.关系主义原理有效性的一个必要条件是，改变系统的长度尺度，依赖于长度的理论的所有参数都相应地改变。特别地，关系论原理要求对经典物理中相互作用势的耦合常数进行适当的变换。这导致普朗克测量单位的变换，使我们能够以独特的方式导出形状空间的度量。在本文的后面，我们将解释形空间上的约化哈密顿量和辛形式的推导。
英文摘要：
One of the foremost goals of research in physics is to find the most basic and universal theories that describe our universe. Many theories assume the presence of an absolute space and time in which the physical objects are located and physical processes take place. However, it is more fundamental to understand time as relative to the motion of another object, e.g. the number of swings of a pendulum, and the position of an object primarily as relative to other objects. The goals of this paper is to explain, how using the principle of relationalism (to be introduced below), classical mechanics can be formulated on a most elementary space, which is freed from absolute entities: shape space. On shape space only the relative orientation and length of subsystems are taken into account. In order to find out how the shape of a classical system evolves in time, the method of "symplectic reduction of Hamiltonian systems" is extended to include scale transformations, and in this way the reduction of a classical system with respect to the full similarity group is achieved. A necessary requirement for the validity of the principle of relationalism is that changing the length scale of a system, all parameters of the theory that depend on the length, get changed accordingly. In particular, the principle of relationalism requires a proper transformation of the coupling constants of the interaction potentials in Classical Physics. This leads consequently to a transformation in Planck's measuring units, which enables us to derive a metric on shape space in a unique way. Later in this paper, we will explain the derivation of the reduced Hamiltonian and symplectic form on shape space.
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