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The Restricted Three-Body Problem
Published in G.A. Gurzadyan, Theory of Interplanetary Flights, 2020
Usually the problem of determination of the motion of any celestial body implies the traditional sequence starting from the statement of the problem up to achievemnt of the final aim, including the derivation of the system of differential equations, their integration for the given boundary or initial conditions and the estimation of the constants of integration. From the point of view of the general aim, i.e. obtaining rigorous solutions of the equations of motion, even the restricted version of the three-body problem appears to be unsolvable. The solution of this problem can be followed only up to some stage revealing certain properties of the infinitely small mass in the gravitational fields of the two finite bodies. However, even those qualitative results played a key role in the development of celestial mechanics, so that acquaintance with these results seems quite useful.
Celestial Mechanics and Astrodynamics
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
So far, we have only considered the gravitational interaction between two masses. Clearly, there are many planets and heavenly bodies in our solar system. The existence of a third body clearly will change the orbit of any mass. The so-called three-body problem in celestial mechanics was formulated by Newton, but he confessed that the problem was too difficult for him to solve it. In this section, we will present one of the few problems that can be solved analytically, called the restricted three-body problem obtained by Euler and Lagrange. The solutions are called Lagrangian Points. The restriction imposed in the formulation is that the motion of the three bodies is all within the same plane and the orbits are approximately circular. This celestial mechanics problem has been considered by the greatest mathematicians in history, including Newton, Euler, Lagrange, Laplace, Jacobi, Hill, Poincare, Gauss, Delaunay, Birkhoff, Kolmogorov, Painleve, Bendixson, Darwin, Moulton, Levi-Civita, Lyapunov, Sundman, Moser, and Arnold. Henri Poincaré’s stability theory of dynamic systems is, in fact, motivated by the three-body problem. One of the very first three-body problems is the lunar theory of the moon under the gravitation pull of the Earth and the Sun. In this section, we will examine the particular solution to the three-body problem called Lagrangian Points.
Introduction to Analytical Dynamics
Published in John G. Papastavridis, Tensor Calculus and Analytical Dynamics, 2018
If no functional relation(s) exist among the N {rP}, independently of any kinetic considerations, the system S is called free, or unconstrained. Then, the motion of its particles is found by solving its (generally nonlinear and coupled) differential equations of motion (Equations 4.2.6b and c): d2ys/dt2 = YS(t, y, dy/dt): known function of its arguments, plus appropriate initial (and/or boundary) conditions. The preeminent, or prototypical, areas of application of this unconstrained dynamics are in celestial mechanics, ballistics (e.g., the famous three-body problem). There, for an N-particle system, we have 3N equations of motion for the 3N unknown time functions {xP, yP, zP; P = 1, …, N) or {yS; S = 1, …, 3N}; i.e., a mathematically determinate problem.
Invariant Cantor sets in the parametrized Hénon-Devaney map
Published in Dynamical Systems, 2022
Hénon [2] introduced the nonlinear mapping defined by as an asymptotic form of the equations of motion of the restricted three body problem. Devaney [1] showed that this map is topologically conjugated to the baker transformation hence transitive with dense periodic orbits. More recently, Lenarduzzi [6] constructed a semi-conjugacy to a subshift of finite type and extended such a coding to a more general class of maps that can be seen as a map in a square with a fixed discontinuity. Leal and Muñoz [4] generalized the dynamics of the Hénon-Devaney map to a large class of homeomorphisms which are transitive in the whole plane. Leal and Muñoz [5] considered the same family as in this article and showed that placing 0<c<a<1 then appears non-compact global attractors of fractal type.