Non-integrability of the dumbbell and point mass problem

Abstract

This paper discusses a constrained gravitational three-body problem with two of the point masses separated by a massless inflexible rod to form a dumbbell. This problem is a simplification of a problem of a symmetric rigid body and a point mass, and has numerous applications in Celestial Mechanics and Astrodynamics. The non-integrability of this system is proven. This was achieved thanks to an analysis of variational equations along a certain particular solution and an investigation of their differential Galois group. Nowadays this approach is the most effective tool for study integrability of Hamiltonian and non-Hamiltonian systems.Keywords Three-body problem · Morales-Ramis theory · Differential Galois theory · Non-integrability Equations of motion, symmetries and reductionConsidered is the gravitational three-body problem with a single constraint. Three point masses, m 1 , m 2 and m 3 move in a plane under mutual gravitational interaction. Masses m 2 and m 3 are connected by a massless inflexible rod of length l > 0 to form a dumbbell. A pictorial description of the problem is given in Fig. 1. Various celestial objects are considered to possess such bimodal mass distribution because do not have enough gravitational force to form their shape into a spherical object, e.g. some asteroids such as (51) Nemausa and (216) Kleopatra; meteorites, especially large irons or nucleus of some comets e.g. Comet Borrelly (for detailed references see Povenmire 2002).The dumbbell satellite has attracted the attention of scientists since the middle of 20th century because it is suitable for an investigation of the general properties of the rigid body motion in a gravity field and provides important

Complex Dynamics of Some Hamiltonian Systems: Nonintegrability of Equations of Motion

“…where ( ) and ( ) are analytic at = 0 and (0) ̸ = 0. It follows that the differential Galois group can be only full triangular of (2, ) (for details see [36,37]). Hence, only Case 1 or 4 of the Kovacic's algorithm is possible.…”

“…Clearly, the normal variational equation (37) along Γ is a system of two uncoupled second order linear differential equations̈1…”

Meromorphic Non-Integrability of Several 3D Dynamical Systems

“…Secondly, there is no general approach to calculate the differential Galois group of a linear differential equation. In some particular cases, one can compute the differential Galois group by the property of the monodromy group or the solvability of second-order linear differential equations, see for instance [17,18]. For second-order linear differential equations with rational coefficients, the so-called Kovacic's algorithm [34] is a very effective tool to calculate the differential Galois group.…”

“…Roughly speaking, the Morales-Ramis theory shows that if the Hamiltonian system with n degrees of freedom admits n meromorphic first integrals which are in involution and independent, then the identity component of the differential Galois group of the normal variational equation should be commutative. Since then, Morales-Ramis theory has been considered as a powerful tool for the meromorphic non-integrability of the Hamiltonian system, and has been successfully applied by many scholars to large numbers of physical models, such as by the authors of [17][18][19][20].…”

Constrained n-Body Problems

“…Thus, formally, in order to apply the differential Galois theory approach to such a Hamiltonian system we have to extend it to the corresponding Poisson system introducing r as additional variable. However, in calculations one can work with the original Hamiltonian system, and the only trace of this extension is the fact that we study the integrability in the class of meromorphic functions of not only coordinates and momenta but also of r. This extension procedure as well as its application to a certain three-body problem was given in [7]. The similar trick is applied to all remaining Hamiltonian systems with algebraic potentials considered in this paper.…”