Computational Dynamics for N-Body Systems with Conserving Algorithms

Abstract

This paper provides new and di fferent perspectives dealing with computational dynamics for N-body systems and the associated conserving algorithms by design. In contrast to the traditonal setting of Newtonian mechanics with vector formalism where most developments are routinely conducted, and also in contrast to other frameworks with scalar formalism such as Lagrangian and Hamiltonian, the paper introduces new avenues and computationally convenient and attractive alternatives via the so-called Total Energy framework in configuration space. A measurable built in scalar descriptive function is employed via the Total Enery framework which is very natural for numerical discretizations to be routinely conducted, and it provides good and improved physical insight and computationally attractive features. In particular, although the equivalences of the various frameworks can be shown for holonomic-sceleronomic systems, the focus of this expositon and equivalences and the developments underlying conserving algorithms by design are strictly limited to the additonal restrictions, namely, the mass matrix is constant and the kinteic energy is not dependent upon the generalized coordinates. Past e fforts via various original methods of development can be also be readily explained, and various conserving algorithms by design are presented with simple illustrative examples. I. Introduction Of interest in this paper are computational dynamics applications for N-body systems via the use of conserving algorithms by design. In general, classical mechanics is classified into three branches, namely, Newtonian, Lagrangian, and Hamiltonian mechanics. We note that the distinction between Newtonian, Lagrangian, and Hamiltonian mechanics stems from the notion of space. The relevant distinctions are highlighted next, followed by detailed descriptions of the various classical mechanics frameworks. Subsequently, in contrast to past practices, we introduce the notion of a Total Energy framework as an alternative. And, we strongly advocate the Total Energy framework as a new and viable alternative for conducting numerical discretizations. We particularly defer such aspects and draw contrasts to the classical frameworks throughout the various sections of this paper. Newtonian mechanics requires the existence of an inertial frame of reference in three-dimensional Euclidean space. For a Newtonian dynamical system of N-particles, a three-dimensional motion of the dynamical system is described by vector quantities in Euclidean space. Conservative or non-conservative forces are exerted on the dynamical system. The Newtonian dynamical systems are often subject to constraints which are given in terms of Cartesian coordinate variables. Alternately, Lagrangian mechanics does not need vector quantities requiring the inertial frame of reference. Instead of vector quantities having both magnitude and direction, a scalar function having only magnitude, called the Lagrangian, is required to be defined to describe the motion of the dynamical system. In addition, the peculiarity of Lagrangian mechanics is the notion and introduction of the concept of generalized coordinates, thereby, making it possible to remove the dynamical system constraints. The space where the constraints associated with the inertial frame of reference disappear is called configuration space. The Lagrangian dynamical system (Lagrange’s equation of motion) is given in the configuration space in terms of generalized coordinates. It should be noted that the Newtonian dynamical system (Newton’s second law of motion) is usually defined in the inertial frame of reference, whereas the Lagrangian is defined on the velocity phase space (tangent bundle). Note that the configuration space belongs to Euclidean N-space. On the other hand, through introducing the notion and concepts of canonical coordinates and Legendre transformation, the scalar function, called the Hamiltonian, is defined in Hamiltonian mechanics. These canonical coordinates,