The Manev problem (a two-body problem given by a potential of the form A/r+B/r2, where r is the distance between particles and A,B are positive constants) comprises several important physical models, having its roots in research done by Isaac Newton. We provide its analytic solution, then completely describe its global flow using McGehee coordinates and topological methods, and offer the physical interpretation of all solutions. We prove that if the energy constant is negative, the orbits are, generically, precessional ellipses, except for a zero-measure set of initial data, for which they are ellipses. For zero energy, the orbits are precessional parabolas, and for positive energy they are precessional hyperbolas. In all these cases, the set of initial data leading to collisions has positive measure.
The restricted (equilateral) four-body problem consists of three bodies of masses m 1 , m 2 and m 3 (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh's critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them.
Abstract. We discuss a recently proposed variational principle for deriving the variational equations associated to any Lagrangian system. The principle gives simultaneously the Lagrange and the variational equations of the system. We define a new Lagrangian in an extended configuration space -which we call D'Alambert's-comprising both the original coordinates and the compatible "virtual displacements" joining two solutions of the original system. The variational principle is Hamilton's with the new Lagrangian. We use this formulation to obtain constants of motion in the Jacobi equations of any Lagrangian system with symmetries. These constants are related to constants in the original system and so with symmetries of the original Lagrangian. We cast our approach in an intrinsic coordinate free formulation. Our results can be of interest for reducing the dimensions of the equations that characterize perturbations in a Lagrangian control system.
Given an m-parameterized family of n-dimensional vector fields, such that: (i) for some value of the parameters, the family has an equilibrium point, (ii) its linearization has a double zero eigenvalue and no other eigenvalue on the imaginary axis, sufficient conditions on the vector field are given such that the dynamics on the two-dimensional center manifold is locally topologically equivalent to the versal deformation of the planar Takens–Bogdanov bifurcation.
The basic theory of Differential Galois and in particular MoralesRamis theory is reviewed with focus in analyzing the non-integrability of various problems of few bodies in Celestial Mechanics. The main theoretical tools are: Morales-Ramis theorem, the algebrization method of Acosta-Blázquez and Kovacic's algorithm. Morales-Ramis states that if Hamiltonian system has an additional meromorphic integral in involution in a neighborhood of a specific solution, then the differential Galois group of the normal variational equations is abelian. The algebrization method permits under general conditions to recast the variational equation in a form suitable for its analysis by means of Kovacic's algorithm. We apply these tools to various examples of few body problems in Celestial Mechanics: (a) the elliptic restricted three body in the plane with collision of the primaries; (b) a general Hamiltonian system of two degrees of freedom with homogeneous potential of degree −1; here we perform McGehee's blow up and obtain the normal variational equation in the form of an hypergeometric equation. We recover Yoshida's criterion for non-integrability. Then we contrast two methods to compute the Galois group: the well known, based in the Schwartz-Kimura table, and the lesser based in Kovacic's algorithm. We apply these methodology to three problems: the rectangular four body problem, the anisotropic Kepler problem and two uncoupled Kepler problems in the line; the last two depend on a mass
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