Reduction of the Classical Mechanics with respect to the Similarity Group Sim(3)

Abstract

One of the most important goals of research in physics is to find the most basic and universal theories that describe our universe. Many theories assume the presence of absolute space and time in which physical objects are located, and physical processes occur. However, it is more fundamental to understand time as relative to the motion of another object, such as the number of swings of a pendulum and the position of an object primarily relative to other objects. The purpose of this thesis is to explain how classical mechanics can be formulated using the principle of relationalism (introduced below) on a most elementary space which is freed from absolute entities: shape space. In shape space, only the relative orientation and length of subsystems are considered. A sufficient requirement for the validity of the principle of relationalism is that when the scale variable of a system changes, all parameters of the theory that depend on the length change accordingly. In particular, the principle of relationalism requires an appropriate transformation of the coupling constants of the interaction potentials in classical physics. Consequently, this change leads to a transformation of Planck’s measuring units, which allows us to derive a metric on shape space in a unique way. In particular, we explain in two different ways how to find the unique metric of shape space, taking into account the crucial role of rulers in determining the geometry of a space. In order to find out the classical equations of motion on shape space, the method of ”symplectic reduction of Hamiltonian systems” is extended to include scale transformations. In particular, we will give the derivation of the reduced Hamiltonian and symplectic form on shape space, and in this way, the reduction of a classical system with respect to the entire similarity group is achieved. One can alternatively use the Lagrangian formalism of mechanics to derive the reduced equations of motion on shape space. It will be explained how the Principle of Relationalism makes the Lagrangian of the classical mechanics scale-invariant, which in turn ensures the existence of laws of motion on shape space. In order to find out these laws of motion, the Boltzman-Hammel equations of motion in an anholonomic frame on tangent space to system’s absolute configuration space T(Q), is adapted to the Sim(3)-fiber bundle structure of the configuration space Q. The derived equations of motion on shape space enable us, among others, to predict the evolution of the shape of a classical system without any reference to its absolute position, orientation, or size in absolute space. Under the action of the group of scale transformations Sc, the internal configuration space Q_int:=Q/E(3) becomes a fiber-bundle whose base space is shape space. It has been explicitly shown that the connection form of the Qint considered as the Sc-fiber-bundle is flat. After treating the general N-body system, shape equations of motion of a three body system are derived explicitly as an illustration of the general method, after which some cosmological implications of the scale-invariant classical mechanics are presented. In particular, we explain how the observed universe’s accelerated expansion follows from the conservation of the dilational momentum in the modified Newtonian theory. Finally, we compare the present work with two other approaches to relational physics and discuss their essential differences.