Variational Construction of Orbits Realizing Symbolic Sequences in the Planar Sitnikov Problem

Using the variational method, Chenciner and Montgomery (2000 Ann. Math. 52 881–901) proved the existence of an eight-shaped orbit of the planar three-body problem with equal masses. Since then a number of solutions to the N-body problem have been discovered. On the other hand, symbolic dynamics is one of the most useful methods for understanding chaotic dynamics. The Sitnikov problem is a special case of the three-body problem. The system is known to be chaotic and was studied by using symbolic dynamics (J. Moser, Stable and random motions in dynamical systems, Princeton University Press, 1973). In this paper, we study the limiting case of the Sitnikov problem. By using the variational method, we show the existence of various kinds of solutions in the planar Sitnikov problem. For a given symbolic sequence, we show the existence of orbits realizing it. We also prove the existence of periodic orbits

Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem

We study the isosceles three-body problem and show that there exist infinitely many families of relative periodic orbits converging to heteroclinic cycles between equilibria on the collision manifold in Devaney's blown-up coordinates. Towards this end, we prove that two types of heteroclinic orbits exist in much wider parameter ranges than previously detected, using self-validating interval arithmetic calculations, and we appeal to the previous results on heteroclinic orbits. Moreover, we give numerical computations for heteroclinic and relative periodic orbits to demonstrate our theoretical results. The numerical results also indicate that the two types of heteroclinic orbits and families of relative periodic orbits exist in wider parameter regions than detected in the theory and that some of them are related to Euler orbits

Variational proof of the existence of periodic orbits in the spatial Hill problem and its constrained problems

The Hill problem models the motion of a particle near a planet. In this paper, we show the existence of symmetric periodic orbits in the spatial Hill problem by using the variational method. We also study the problem under a constraint on a prescribed plane and show the existence of periodic orbits in the problem. The obtained orbits are applicable to artificial satellites around the Earth and other planets

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

Minimizing Periodic Orbits with Regularizable Collisions in the n-Body Problem

In the Newtonian n-body problem, there are various subsystems with two degrees of freedom, such as the collinear three-body problem and the isosceles three-body problem. After we determine a normal form of the Lagrangians of these subsystems, we prove the existence of periodic solutions with regularizable collisions for these systems. Our result includes several examples, such as Schubart’s orbit with or without equal masses, among others