1,697 research outputs found
Integrability and chaos: the classical uncertainty
In recent years there has been a considerable increase in the publishing of
textbooks and monographs covering what was formerly known as random or
irregular deterministic motion, now named by the more fashionable term of
deterministic chaos. There is still substantial interest in a matter that is
included in many graduate and even undergraduate courses on classical
mechanics. Based on the Hamiltonian formalism, the main objective of this
article is to provide, from the physicist's point of view, an overall and
intuitive review of this broad subject (with some emphasis on the KAM theorem
and the stability of planetary motions) which may be useful to both students
and instructors.Comment: 24 pages, 10 figure
Bottlenecks to vibrational energy flow in OCS: Structures and mechanisms
Finding the causes for the nonstatistical vibrational energy relaxation in
the planar carbonyl sulfide (OCS) molecule is a longstanding problem in
chemical physics: Not only is the relaxation incomplete long past the predicted
statistical relaxation time, but it also consists of a sequence of abrupt
transitions between long-lived regions of localized energy modes. We report on
the phase space bottlenecks responsible for this slow and uneven vibrational
energy flow in this Hamiltonian system with three degrees of freedom. They
belong to a particular class of two-dimensional invariant tori which are
organized around elliptic periodic orbits. We relate the trapping and
transition mechanisms with the linear stability of these structures.Comment: 13 pages, 13 figure
Dynamical Evolution Induced by Planet Nine
The observational census of trans-Neptunian objects with semi-major axes
greater than ~250 AU exhibits unexpected orbital structure that is most readily
attributed to gravitational perturbations induced by a yet-undetected, massive
planet. Although the capacity of this planet to (i) reproduce the observed
clustering of distant orbits in physical space, (ii) facilitate dynamical
detachment of their perihelia from Neptune, and (iii) excite a population of
long-period centaurs to extreme inclinations is well established through
numerical experiments, a coherent theoretical description of the dynamical
mechanisms responsible for these effects remains elusive. In this work, we
characterize the dynamical processes at play, from semi-analytic grounds. We
begin by considering a purely secular model of orbital evolution induced by
Planet Nine, and show that it is at odds with the ensuing stability of distant
objects. Instead, the long-term survival of the clustered population of
long-period KBOs is enabled by a web of mean-motion resonances driven by Planet
Nine. Then, by taking a compact-form approach to perturbation theory, we show
that it is the secular dynamics embedded within these resonances that regulates
the orbital confinement and perihelion detachment of distant Kuiper belt
objects. Finally, we demonstrate that the onset of large-amplitude oscillations
of orbital inclinations is accomplished through capture of low-inclination
objects into a high-order secular resonance and identify the specific harmonic
that drives the evolution. In light of the developed qualitative understanding
of the governing dynamics, we offer an updated interpretation of the current
observational dataset within the broader theoretical framework of the Planet
Nine hypothesis.Comment: 22 pages, 13 figures, accepted for publication in the Astronomical
Journa
Poincare recurrences and transient chaos in systems with leaks
In order to simulate observational and experimental situations, we consider a
leak in the phase space of a chaotic dynamical system. We obtain an expression
for the escape rate of the survival probability applying the theory of
transient chaos. This expression improves previous estimates based on the
properties of the closed system and explains dependencies on the position and
size of the leak and on the initial ensemble. With a subtle choice of the
initial ensemble, we obtain an equivalence to the classical problem of Poincare
recurrences in closed systems, which is treated in the same framework. Finally,
we show how our results apply to weakly chaotic systems and justify a split of
the invariant saddle in hyperbolic and nonhyperbolic components, related,
respectively, to the intermediate exponential and asymptotic power-law decays
of the survival probability.Comment: Corrected version, as published. 12 pages, 9 figure
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