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Register a new userChaotic dynamics in the planar gravitational many-body problem with rigid body rotations
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The discovery of Plutos small moons in the last decade brought attention to the dynamics of the dwarf planets satellites. With such systems in mind, we study a planar $N$-body system in which all the bodies are point masses, except for a single rigid body. We then present a reduced model consisting of a planar $N$-body problem with the rigid body treated as a 1D continuum (i.e. the body is treated as a rod with an arbitrary mass distribution). Such a model provides a good approximation to highly asymmetric geometries, such as the recently observed interstellar asteroid Oumuamua, but is also amenable to analysis. We analytically demonstrate the existence of homoclinic chaos in the case where one of the orbits is nearly circular by way of the Melnikov method, and give numerical evidence for chaos when the orbits are more complicated. We show that the extent of chaos in parameter space is strongly tied to the deviations from a purely circular orbit. These results suggest that chaos is ubiquitous in many-body problems when one or more of the rigid bodies exhibits non-spherical and highly asymmetric geometries. The excitation of chaotic rotations does not appear to require tidal dissipation, obliquity variation, or orbital resonance. Such dynamics give a possible explanation for routes to chaotic dynamics observed in $N$-body systems such as the Pluto system where some of the bodies are highly non-spherical.
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We study chaos and Levy flights in the general gravitational three-body problem. We introduce new metrics to characterize the time evolution and final lifetime distributions, namely Scramble Density $mathcal{S}$ and the LF index $mathcal{L}$, that are derived from the Agekyan-Anosova maps and homology radius $R_{mathcal{H}}$. Based on these metrics, we develop detailed procedures to isolate the ergodic interactions and Levy flight interactions. This enables us to study the three-body lifetime distribution in more detail by decomposing it into the individual distributions from the different kinds of interactions. We observe that ergodic interactions follow an exponential decay distribution similar to that of radioactive decay. Meanwhile, Levy flight interactions follow a power-law distribution. Levy flights in fact dominate the tail of the general three-body lifetime distribution, providing conclusive evidence for the speculated connection between power-law tails and Levy flight interactions. We propose a new physically-motivated model for the lifetime distribution of three-body systems and discuss how it can be used to extract information about the underlying ergodic and Levy flight interactions. We discuss mass ejection probabilities in three-body systems in the ergodic limit and compare it to previous ergodic formalisms. We introduce a novel mechanism for a three-body relaxation process and discuss its relevance in general three-body systems.
We outline a new method suggested by Conway (2016) for solving the two-body problem for solid bodies of spheroidal or ellipsoidal shape. The method is based on integrating the gravitational potential of one body over the surface of the other body. When the gravitational potential can be analytically expressed (as for spheroids or ellipsoids), the gravitational force and mutual gravitational potential can be formulated as a surface integral instead of a volume integral, and solved numerically. If the two bodies are infinitely thin disks, the surface integral has an analytical solution. The method is exact as the force and mutual potential appear in closed-form expressions, and does not involve series expansions with subsequent truncation errors. In order to test the method, we solve the equations of motion in an inertial frame, and run simulations with two spheroids and two infinitely thin disks, restricted to torque-free planar motion. The resulting trajectories display precession patterns typical for non-Keplerian potentials. We follow the conservation of energy and orbital angular momentum, and also investigate how the spheroid model approaches the two cases where the surface integral can be solved analytically, i.e. for point masses and infinitely thin disks.
We investigate a small vortex-lattice system of four co-rotating vortices in an atomic Bose--Einstein condensate and find that the vortex dynamics display chaotic behaviour after a system quench introduced by reversing the direction of circulation of a single vortex through a phase-imprinting process. By tracking the vortex trajectories and Lyapunov exponent, we show the onset of chaotic dynamics is not immediate, but occurs at later times and is accelerated by the close-approach and separation of all vortices in a scattering event. The techniques we develop could potentially be applied to create locally induced chaotic dynamics in larger lattice systems as a stepping stone to study the role of chaotic events in turbulent vortex dynamics.
The motion of a point mass in the J2 problem has been generalized to that of a rigid body in a J2 gravity field for new high-precision applications in the celestial mechanics and astrodynamics. Unlike the original J2 problem, the gravitational orbit-rotation coupling of the rigid body is considered in the generalized problem. The existence and properties of both the classical and non-classical relative equilibria of the rigid body are investigated in more details in the present paper based on our previous results. We nondimensionalize the system by the characteristic time and length to make the study more general. Through the study, it is found that the classical relative equilibria can always exist in the real physical situation. Numerical results suggest that the non-classical relative equilibria only can exist in the case of a negative J2, i.e., the central body is elongated; they cannot exist in the case of a positive J2 when the central body is oblate. In the case of a negative J2, the effect of the orbit-rotation coupling of the rigid body on the existence of the non-classical relative equilibria can be positive or negative, which depends on the values of J2 and the angular velocity. The bifurcation from the classical relative equilibria, at which the non-classical relative equilibria appear, has been shown with different parameters of the system. Our results here have given more details of the relative equilibria than our previous paper, in which the existence conditions of the relative equilibria are derived and primarily studied. Our results have also extended the previous results on the relative equilibria of a rigid body in a central gravity field by taking into account the oblateness of the central body.
Linking thermodynamic variables like temperature $T$ and the measure of chaos, the Lyapunov exponents $lambda$, is a question of fundamental importance in many-body systems. By using nonlinear fluid equations in one and three dimensions, we prove that in thermalised flows $lambda propto sqrt{T}$, in agreement with results from frustrated spin systems. This reveals an underlying universality and provides evidence for recent conjectures on the thermal scaling of $lambda$. We also reconcile seemingly disparate effects -- equilibration on one hand and pushing systems out-of-equilibrium on the other -- of many-body chaos by relating $lambda$ to $T$ through the dynamical structures of the flow.
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