Dynamics of multibody chains in circular orbit: non-integrability of equations of motion

Abstract

This paper discusses the dynamics of systems of point masses joined by massless rigid rods in the field of a potential force. The general form of equations of motion for such systems is obtained. The dynamics of a linear chain of mass points moving around a central body in an orbit is analysed. The non-integrability of the chain of three masses moving in a circular Kepler orbit around a central body is proven. This was achieved thanks to an analysis of variational equations along two particular solutions and an investigation of their differential Galois groups.

Dynamics of constrained many body problems in constant curvature two-dimensional manifolds

“…We consider dynamics of a rigid body with point mass moving in a circle lying in a plane perpendicular to third principal axis b 3 (figure 3). We assume that the centre of the circle lies on the principal axis.…”

“…Further description in this section is just a shorter version of that presented in [3]. Let m i and q i denote the mass and the radius vector of point P i , respectively.…”

“…1} and e (2) = { 1 4 , 3 4 , 1 2 , 3 2 }. According to the algorithm, we have to check if there exists a polynomial of degree d(e) which is a solution of appropriate differential equation.…”

“…In this form, the kinetic energy as well as constraint J(r 0 , r 1 ) = l > 1 are invariant with respect to natural action of group SO(1, 2) on R 3 . Hence, as in the case of the spherical dumbbell, we define a frame {b 1 , b 2 , b 3 } in which the dumbbell rests. As we assumed that l > 1, vectors r 0 and r 1 are not parallel, so z = C(r 0 , r 1 ) = 0.…”

“…where matrix A 1 (x) is defined in (4.8). Explicit form of vector Ω is the following: 3 and Ω 3 =q 1 sinh q 2 sin q 3 +q 2 cos q 3 . (6.18) Components of this vector can be taken as quasi-velocities and we can write equations of motion in the Chetaev form (6.19) which have two first integrals…”