Symplectic Reduction of Classical Mechanics on Shape Space

TL;DR

This paper develops a framework for classical mechanics on shape space by extending symplectic reduction to include scale transformations, emphasizing relationalism and deriving a unique metric on shape space.

Contribution

It introduces a method to perform symplectic reduction of Hamiltonian systems considering scale invariance, advancing relational formulations of classical mechanics.

Findings

Extended symplectic reduction to include scale transformations.

Derived a unique metric on shape space based on relational principles.

Formulated a reduced Hamiltonian and symplectic form on shape space.

Abstract

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.