Qualitative Methods In The Many Body Problem  

Original Title  Qualitative Methods In The Many Body Problem 
Author  G. F. Khilmi 
Publication date 
1961 
Topics  physics, dynamics, nbody problem, gravitation, invariant measure, evolutionary systems, celestial mechanics, method of continuous induction, dimensional analysis, measure theory, dynamical systems 
Collection  mirtitles, additional_collections 
Language  English 
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The nbody problem is the name usually given to the problem of the motion of a system of many particles attracting each other according to Newton’s law of gravitation. This is the classical problem of mathematical natural science, the significance of which goes far beyond the limits of its astronomical applications.
The nbody problem has been the main topic of celestial mechanics from the time of its inception as a science. Now that many of the problems of celestial mechanics have become part of geophysics, the central position of the wbody problem has been further strengthened.
The fundamental dynamical problem for a system of n gravitating bodies is the investigation and predetermination of the changes in position and velocity that the particles undergo as the time varies. However, this is a very complex nonlinear problem whose solution has not been possible under the presentday status of mathematical analysis.
In the first chapter, we give the equations and general integrals of the nbody problem, and we study the simplest theorems on the final motion due to Jacobi.
In the second chapter, we consider means of applying the method of dimensional analysis to the nbody problem. As far as we know, dimensional analysis has not been used before in the investigation of the final motion, though it does not yield definitive results in this area, it is very use ful in carrying out a preliminary analysis of the problem.
In the third chapter, we present our “method of continuous induction” and we consider some applications of it in which the final motion in the nbody problem is analyzed. This method allows us to obtain effective qualitative results, namely, it allows us to formulate sufficient conditions for the occurrence of certain types of final motions in the form of conditions on the initial state of the dynamical system.
The fourth chapter is devoted to the method of invariant measure. The application of this method to the manybody problem has necessitated working out a number of theorems on the measure theory of dynamical systems. These are presented at the beginning of the chapter. At the end of the chapter, we prove some very general theorems on the motion of a system of gravitating bodies using the method of invariant measure.
In the fifth chapter, an attempt is made to analyze some cases of the evolution of a system of n gravitating bodies on the basis of celestial mechanics. Here, we shall be concerned with the processes which are of cosmogonical interest, and which are accompanied by the conversion of mechanical energy into nonmechanical forms.

Qualitative Methods In The Many Body Problem
 G. F. Khilmi
Qualitative Methods In The Many Body Problem Original Title Qualitative Methods In The Man
Qualitative Methods In The Many Body Problem
 G. F. Khilmi
Qualitative Methods In The Many Body Problem Original Title Qualitative Methods In The Man