Sahand Tokasi | Searcharxiv

arXiv2023

Reduction of Lagrangian Equations of Motion of Modified Newtonian Theory of Gravity with respect to the Similarity Group

The equivalence class of absolute configurations of a system under the group of similarity transformations $Sim(3)$ is called the shape of the system. The $Sim(3)$ invariant Lagrangian of the modified Newtonian theory ensures the existence of the its law of motion on shape space. To deduce the equations of motion for a system's shape degrees of freedom from its evolution equations for the $3N$ absolute configuration degrees of freedom, the Boltzman-Hamel equations of motion in an non-holonomic frame on the tangent space $T(Q)$ to the system's absolute configuration space $Q$ is adapted to the $Sim(3)$ fiber bundle structure of the configuration space. The derived equations of motion on shape space enable us, among other things, to predict the evolution of the shape of a classical system governed by this theory without any reference to its absolute position, orientation, or size in space. The paper will explain, that by treating the measuring instruments as part of the matter in the theory, how the mass metric $\textbf{M}$ on the configuration space $Q$ uniquely defines a metric on the reduced tangent bundle $\frac{T(Q)}{Sim(3)}$, and how the unique metric structure on shape space $S$ can be derived. After deriving the reduced equations of motion on shape space for a general $N$-body system, the shape equations of motion for a three-body system in suitable coordinates is given as an illustration.

math-ph↗

arXiv2022

Symplectic Reduction of Classical Mechanics on Shape Space

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.

physics.hist-ph↗