Andrzej J. Maciejewski

Reduction, relative equilibria and potential in the two rigid bodies problem

In this paper the problem of two, and thus, after a generalization, of an arbitrary finite number, of rigid bodies is considered. We show that the Newton-Euler equations of motion are Hamiltonian with respect to a certain non-canonical structure. The system possesses natural symmetries. Using them we shown how to perform reduction of the number of degrees of freedom. We prove that on every stage of this process equations of motion are Hamiltonian and we give explicite form corresponding of non-canonical Poisson bracket. We also discuss practical consequences of the reduction. We prove the existence of 36 non-Lagrangean relative equilibria for two generic rigid bodies. Finally, we demonstrate that our approach allows to simplify the general form of the mutual potential of two rigid bodies.

Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential

In this paper we study the integrability of natural Hamiltonian systems with a homogeneous polynomial potential. The strongest necessary conditions for their integrability in the Liouville sense have been obtained by a study of the differential Galois group of variational equations along straight line solutions. These particular solutions can be viewed as points of a projective space of dimension smaller by one than the number of degrees of freedom. We call them Darboux points. We analyze in detail the case of two degrees of freedom. We show that, except for a radial potential, the number of Darboux points is finite and it is not greater than the degree of the potential. Moreover, we analyze cases when the number of Darboux points is smaller than maximal. For two degrees of freedom the above-mentioned necessary condition for integrability can be expressed in terms of one nontrivial eigenvalue of the Hessian of potential calculated at a Darboux point. We prove that for a given potential these nontrivial ei...

On the HU Aquarii planetary system hypothesis

In this paper, we investigate the eclipse timing of the polar binary HU Aquarii that has been observed for almost two decades. Recently, Qian et al. attributed large (O–C) deviations between the eclipse ephemeris and observations to a compact system of two massive Jovian companions. We improve the Keplerian, kinematic model of the light travel time effect and re-analyse the whole currently available data set. We add almost 60 new, yet unpublished, mostly precision light curves obtained using the time high-resolution photopolarimeter Optical Timing Analyzer (OPTIMA), as well as photometric observations performed at the Monitoring Network of Telescopes/North, Physics Innovations Robotic Astronomical Telescope Explorer and Carlos Sanchez Telescope. We determine new mid-egress times with a mean uncertainty at the level of 1 s or better. We claim that because the observations that currently exist in the literature are non-homogeneous with respect to spectral windows (ultraviolet, X-ray, visual and polarimetric mode) and the reported mid-egress measurements errors, they may introduce systematics that affect orbital fits. Indeed, we find that the published data, when taken literally, cannot be explained by any unique solution. Many qualitatively different and best-fit two-planet configurations, including self-consistent, Newtonian N-body solutions may be able to explain the data. However, using high-resolution, precision OPTIMA light curves, we find that the (O–C) deviations are best explained by the presence of a single circumbinary companion orbiting at a distance of ∼4.5 au with a small eccentricity and having ∼7 Jupiter masses. This object could be the next circumbinary planet detected from the ground, similar to the announced companions around close binaries HW Vir, NN Ser, UZ For, DP Leo, FS Aur or SZ Her, and planets of this type around Kepler-16, Kepler-34 and Kepler-35.

On the Extrasolar Multiplanet System around HD 160691

We reanalyze the precision radial velocity (RV) observations of HD 160691 (μ Ara) by the Anglo-Australian Planet Search Team. The star is supposed to host two Jovian companions (HD 160691b, HD 160691c) in long-period orbits (~630 days and ~2500 days, respectively) and a hot Neptune (HD 160691d) in ~9 day orbit. We perform a global search for the best fits in the orbital parameter space with a hybrid code employing the genetic algorithm and simplex method. The stability of Keplerian fits is verified with the N-body model of the RV signal that takes into account the dynamical constraints (so called GAMP method). Our analysis reveals a signature of the fourth, as yet unconfirmed, Jupiter-like planet HD 160691e in ~307 day orbit. Overall, the global architecture of four-planet configuration recalls the solar system. All companions of μ Ara move in quasi-circular orbits. The orbits of two inner Jovian planets are close to the 2 : 1 mean motion resonance. The alternative three-planet system involves two Jovian planets in eccentric orbits (e ~ 0.3), close to the 4 : 1 MMR, but it yields a significantly worse fit to the data. We also verify a hypothesis of the 1 : 1 MMR in the subsystem of two inner Jovian planets in the four-planet model.

Orbital Configurations and Dynamical Stability of Multiplanet Systems around Sun-like Stars HD 202206, 14 Herculis, HD 37124, and HD 108874

We perform a dynamical analysis of the recently published radial velocity (RV) measurements of a few solar-type stars that host multiple Jupiter-like planets. In particular, we reanalyze the data for HD 202206, 14 Her, HD 37124, and HD 108874. We derive dynamically stable configurations that reproduce the observed RV signals, using GAMP (the genetic algorithm with MEGNO penalty). GAMP relies on N-body dynamics and makes use of genetic algorithms merged with a stability criterion. For this purpose, we use the maximal Lyapunov exponent computed with the dynamical fast indicator MEGNO. Through a dynamical analysis of the phase space in a neighborhood of the obtained best-fit solutions, we derive meaningful limits on the parameters of the planets. We demonstrate that GAMP is especially well suited to the analysis of the RV data that only partially cover the longest orbital period and/or are related to multiplanet configurations involved in low-order mean motion resonances (MMRs). Our analysis reveals a presence of a second Jupiter-like planet in the 14 Her system (14 Her c) that is involved in a 3 : 1 or 6 : 1 MMR with the known companion 14 Her b. We also show that the dynamics of the HD 202206 system may be qualitatively different when coplanar and mutually inclined orbits of the companions are considered. We demonstrate that the two outer planets in the HD 37124 system may reside in a close neighborhood of the 5 : 2 MMR. Our results confirm that the HD 108874 system may be very close to a, or locked in an exact, 4 : 1 MMR.

Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces

We formulate the necessary conditions for the maximal super-integrability of a certain family of classical potentials defined in the constant curvature two-dimensional spaces. We give examples of homogeneous potentials of degree -2 on

Where is the Second Planet in the HD 160691 Planetary System

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Full spectrum of the Rabi model

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Non-integrability of ABC flow

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