PERIODIC SOLUTIONS OF THE RESTRICTED THREE BODY PROBLEM REPRESENTING ANALYTIC CONTINUATIONS OF KEPLERIAN ELLIPTIC MOTIONS

Periodic solution of restricted three body problem as analytic continuation of keplerian elliptical motion

Periodic orbits in the restricted three-body problem and Arnold's J+J^+-invariant

We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold's $J^+$-invariant, we introduce invariants of periodic orbits in planar Stark-Zeeman systems and study their behaviour.Comment: 36 Pages, 16 Figure

On the convex central configurations of the symmetric (ℓ + 2)-body problem

For the 4-body problem there is the following conjecture: Given arbitrary positive masses, the planar 4-body problem has a unique convex central configuration for each ordering of the masses on its convex hull. Until now this conjecture has remained open. Our aim is to prove that this conjecture cannot be extended to the (ℓ + 2)-body problem with ℓ ⩾ 3. In particular, we prove that the symmetric (2n + 1)-body problem with masses m1 = … = m2n−1 = 1 and m2n = m2n+1 = m sufficiently small has at least two classes of convex central configuration when n = 2, five when n = 3, and four when n = 4. We conjecture that the (2n + 1)-body problem has at least n classes of convex central configurations for n > 4 and we give some numerical evidence that the conjecture can be true. We also prove that the symmetric (2n + 2)-body problem with masses m1 = … = m2n = 1 and m2n+1 = m2n+2 = m sufficiently small has at least three classes of convex central configuration when n = 3, two when n = 4, and three when n = 5. We also conjecture that the (2n + 2)-body problem has at least [(n +1)/2] classes of convex central configurations for n > 5 and we give some numerical evidences that the conjecture can be true

Linear stability of periodic three-body orbits with zero angular momentum and topological dependence of Kepler's third law: a numerical test

We test numerically the recently proposed linear relationship between the scale-invariant period Ts.i. = T|E| 3/2, and the topology of an orbit, on several hundred planar Newtonian periodic three-body orbits. Here T is the period of an orbit, E is its energy, so that Ts.i. is the scale-invariant period, or, equivalently, the period at unit energy |E| = 1. All of these orbits have vanishing angular momentum and pass through a linear, equidistant configuration at least once. Such orbits are classified in ten algebraically well-defined sequences. Orbits in each sequence follow an approximate linear dependence of Ts.i., albeit with slightly different slopes and intercepts. The orbit with the shortest period in its sequence is called the ‘progenitor’: six distinct orbits are the progenitors of these ten sequences. We have studied linear stability of these orbits, with the result that 21 orbits are linearly stable, which includes all of the progenitors. This is consistent with the Birkhoff–Lewis theorem, which implies existence of infinitely many periodic orbits for each stable progenitor, and in this way explains the existence and ensures infinite extension of each sequence