The aim of this paper is to study the global geometry of non-planar 3-body motions in the realms of equivariant Differential Geometry and Geometric Mechanics. This work was intended as an attempt at bringing together these two areas, in which geometric methods play the major role, in the study of the 3-body problem. It is shown that the Euler equations of a three-body system with non-planar motion introduce non-holonomic constraints into the Lagrangian formulation of mechanics. Applying the method of undetermined Lagrange multipliers to study the dynamics of three-body motions reduced to the level of moduli spaceM subject to the non-holonomic constraints yields the generalized EulerLagrange equations of non-planar three-body motions inM. As an application of the derived dynamical equations in the level ofM, we completely settle the question posed by A. Wintner in his book [The analytical foundations of Celestial Mechanics, Sections 394-396, 435 and 436. Princeton University Press (1941)] on classifying the constant inclination solutions of the three-body problem.
Identifying new stable dynamical systems, such as generic stable mechanical or electrical control systems, requires questing for the desired systems parameters that introduce such systems. In this paper, a systematic approach to construct generic stable dynamical systems is proposed. In fact, our approach is based on a simple identification method in which we intervene directly with the dynamics of our system by considering a continuous 1-parameter family of system parameters, being parametrized by a positive real variable , and then identify the desired parameters that introduce a generic stable dynamical system by analyzing the solutions of a special system of nonlinear functional-differential equations associated with the-varying parameters. We have also investigated the reliability and capability of our proposed approach. To illustrate the utility of our result and as some applications of the nonlinear differential approach proposed in this paper, we conclude with considering a class of coupled spring-mass-dashpot systems, as well as the compartmental systems-the latter of which provide a mathematical model for many complex biological and physical processes having several distinct but interacting phases.
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