Browsing by Author "Dmitrašinović, Veljko"

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    A guide to hunting periodic three-body orbits with non-vanishing angular momentum
    Janković, Marija; Dmitrašinović, Veljko; Šuvakov, Milovan
    A large number of periodic three-body orbits with vanishing angular momentum have been found in Newtonian gravity over the past 6 years due to a simple search method and to the contribution from practitioners outside the Celestial Mechanics community. Extension of such orbits to non-vanishing angular momentum has been lacking due to inter alia the absence of a sufficiently simple and widely known search method. We present a method, i.e., a general strategy plus detailed tactics (but not a specific algorithm, or a code), to numerically search for relative periodic orbits in the Newtonian three-body problem with three equal masses and non-vanishing angular momentum. We illustrate the method with an application to a specific, so-called Broucke–Hadjidemetriou–Hénon (BHH) family of periodic 3-body orbits: Our search yielded around 100 new “satellite” orbits, related to the original BHH orbits by a topological relation (defined in the text), with infinitely many orbits remaining to be discovered. We used the so-obtained orbits to test the period vs. topology relation that had previously been established, within a certain numerical accuracy, for orbits with vanishing angular momentum. Our method can be readily: (1) applied to families of periodic 3-body orbits other than the BHH one; (2) implemented using various standard algorithms for solving ordinary differential equations, such as the Bulirsch–Stoer and the Runge–Kutta–Fehlberg ones; (3) adapted to 3-body systems with distinct masses and/or coupling constants, including, but not limited to, Coulomb interaction. Our goal is to enable numerical searches for new orbits in as many families of orbits as possible, and thus to allow searches for other empirical relations, such as the aforementioned topology vs. period one.
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    Linear stability of periodic three-body orbits with zero angular momentum and topological dependence of Kepler’s third law: a numerical test
    Dmitrašinović, Veljko; Hudomal, Ana; Shibayama, Mitsuru; Sugita, Ayumu
    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.