Classical Physics

   

Chaos in N-Body Problem.

Authors: Michail Zak

via conservative forces. These forces can be of gravitational origin (celestial mechanics), inter-molecular origin (molecular dynamics), or representing (structural biology). In The n-body problem as a classic astronomical and physical problem that naturally follows from the two- body problem first solved by Newton in his Principia in 1687. The efforts of many famous mathematicians have been devoted to this difficult problem, including Euler and Lagrange (1772), Jacobi (1836), Hill (1878), Poincaré (1899), Levi-Civita (1905), and Birkhoff (1915). However, despite centuries of exploration, there is no clear structure of the solution of the general n- or even three-body problem as there are no coordinate transformations that can simplify the problem, and there are more and more evidences that, in general, the solutions of n-body problems are chaotic. Failure to find a general analytical structure of the solution shifted the effort towards numerical methods. Many ODE solvers offer a variety of advance numerical methods for the solution. 2. Chaos in classical dynamics We start this section with revisiting mathematical formalism of chaos in a non-traditional way that is based upon the concept of orbital instability. The concept of randomness entered Newtonian dynamics almost a century ago: in 1926, Synge, J. introduced a new type of instability - orbital instability- in classical mechanics, [4], that can be considered as a precursor of chaos formulated a couple of decades later, [5]. The theory of chaos was inspired by the fact that in recent years, in many different domains of science (physics, chemistry, biology, engineering), systems with a similar strange behavior were frequently encountered displaying irregular and unpredictable behavior called chaotic. Currently the theory of chaos that describes such systems is well established. However there are still two unsolved problem remain: prediction of chaos (without numerical runs), and analytical description of chaos in term of the probability density that would formally follow from the original ODE. This paper proposes a contribution to the solution of these problems illustrated by chaos in inertial systems a. Orbital instability as a precursor of chaos. Chaos is a special type of instability when the system does not have an alternative stable state and displays an irregular aperiodic motion. Obviously this kind of instability can be associated only with ignorable variables, i.e. with such variables that do not contribute into energy of the system. In order to demonstrate this kind of instability, consider an inertial motion of a particle M of unit mass on a smooth pseudosphere S having a constant negative curvature G0, Fig. 1. The n-body problem is the problem of predicting the individual motions of a group of objects 1 interacting with each other the most common version, the trajectories of the objects are determined by numerically solving the Newton's equations of motion for a system of interacting particles. Non-conservative version of the interaction forces became important in case of the n-body problem that incorporates the effects of the Coulomb potential radiation pressure, Poynting-Robertson (P-R) drag, and solar wind drag. The general method of numerical solution of the corresponding system of ODE was originally conceived within theoretical physics in the late 1950s,,[1,2], but is applied today mostly in chemical physics, materials science and the modeling of biomolecules. The most significant “side effect “of the existing numerical methods for n-body problems becomes chaos when different numerical runs with the same initial conditions result in different trajectories. Although numerical errors can contribute to chaos, nevertheless the primary origin of chaos is physical instability, [3]. In this work, a general approach to probabilistic description of chaos in n-body problem with conservative

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[v1] 2017-01-17 20:26:55

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