Topic
Hyperbolic trajectory
About: Hyperbolic trajectory is a research topic. Over the lifetime, 137 publications have been published within this topic receiving 1681 citations. The topic is also known as: hyperbolic orbit.
Papers published on a yearly basis
Papers
TL;DR: ‘Oumuamua—the first known interstellar object to have entered the Solar System—is probably a comet, albeit with unusual dust and chemical properties owing to its origin in a distant solar system.
Abstract: ‘Oumuamua (1I/2017 U1) is the first known object of interstellar origin to have entered the Solar System on an unbound and hyperbolic trajectory with respect to the Sun1. Various physical observations collected during its visit to the Solar System showed that it has an unusually elongated shape and a tumbling rotation state1–4 and that the physical properties of its surface resemble those of cometary nuclei5,6, even though it showed no evidence of cometary activity1,5,7. The motion of all celestial bodies is governed mostly by gravity, but the trajectories of comets can also be affected by non-gravitational forces due to cometary outgassing8. Because non-gravitational accelerations are at least three to four orders of magnitude weaker than gravitational acceleration, the detection of any deviation from a purely gravity-driven trajectory requires high-quality astrometry over a long arc. As a result, non-gravitational effects have been measured on only a limited subset of the small-body population9. Here we report the detection, at 30σ significance, of non-gravitational acceleration in the motion of ‘Oumuamua. We analyse imaging data from extensive observations by ground-based and orbiting facilities. This analysis rules out systematic biases and shows that all astrometric data can be described once a non-gravitational component representing a heliocentric radial acceleration proportional to r−2 or r−1 (where r is the heliocentric distance) is included in the model. After ruling out solar-radiation pressure, drag- and friction-like forces, interaction with solar wind for a highly magnetized object, and geometric effects originating from ‘Oumuamua potentially being composed of several spatially separated bodies or having a pronounced offset between its photocentre and centre of mass, we find comet-like outgassing to be a physically viable explanation, provided that ‘Oumuamua has thermal properties similar to comets.
224 citations
TL;DR: This paper develops two accurate and fast algorithms for the computation of the stable and unstable manifolds of hyperbolic trajectories of two-dimensional, aperiodically time-dependent vector fields and describes the computational requirements of all of these methods.
110 citations
TL;DR: In this paper, the authors present the results of a theoretical investigation of droplets walking on a rotating vibrating fluid bath, and describe the droplet's trajectory in terms of an integro-differential equation that incorporates the influence of its propulsive wave force.
Abstract: We present the results of a theoretical investigation of droplets walking on a rotating vibrating fluid bath. The droplet’s trajectory is described in terms of an integro-differential equation that incorporates the influence of its propulsive wave force. Predictions for the dependence of the orbital radius on the bath’s rotation rate compare favourably with experimental data and capture the progression from continuous to quantized orbits as the vibrational acceleration is increased. The orbital quantization is rationalized by assessing the stability of the orbital solutions, and may be understood as resulting directly from the dynamic constraint imposed on the drop by its monochromatic guiding wave. The stability analysis also predicts the existence of wobbling orbital states reported in recent experiments, and the absence of stable orbits in the limit of large vibrational forcing.
76 citations
10 Apr 2007
TL;DR: An analysis of a point mass, point foot, planar inverted pendulum model for bipedal walking, which derives expressions for a conserved quantity, the "orbital energy", given a smooth center of mass trajectory, and how the parameters of this polynomial and step length can be changed in order to achieve a desired next step velocity.
Abstract: We present an analysis of a point mass, point foot, planar inverted pendulum model for bipedal walking. Using this model, we derive expressions for a conserved quantity, the "orbital energy", given a smooth center of mass trajectory. Given a closed form center of mass trajectory, the equation for the orbital energy is a closed form expression except for an integral term, which we show to be the first moment of area under the center of mass path. Hence, given a center of mass trajectory, it is straightforward and computationally simple to compute phase portraits for the system. In fact, for many classes of trajectories, such as those in which height is a polynomial function of center of mass horizontal displacement, the orbital energy can be solved in closed form. Given expressions for the orbital energy, we can compute where the foot should be placed or how the center of mass trajectory should be modified in order to achieve a desired velocity on the next step. We demonstrate our results using a planar biped simulation with light legs and point mass body. We parameterize the center of mass trajectory with a fifth order polynomial function. We demonstrate how the parameters of this polynomial and step length can be changed in order to achieve a desired next step velocity.
66 citations
TL;DR: In this paper, an extension of the standard time-dependent WKB theory, which can be applied to propagate coherent states and other strongly localized states for long times, is presented.
Abstract: We derive an extension of the standard time-dependent WKB theory, which can be applied to propagate coherent states and other strongly localized states for long times. It in particular allows us to give a uniform description of the transformation from a localized coherent state to a delocalized Lagrangian state, which takes place at the Ehrenfest time. The main new ingredient is a metaplectic operator that is used to modify the initial state in a way that the standard time-dependent WKB theory can then be applied for the propagation. We give a detailed analysis of the phase space geometry underlying this construction and use this to determine the range of validity of the new method. Several examples are used to illustrate and test the scheme and two applications are discussed. (i) For scattering of a wave packet on a barrier near the critical energy, we can derive uniform approximations for the transition from reflection to transmission. (ii) A wave packet propagated along a hyperbolic trajectory becomes a Lagrangian state associated with the unstable manifold at the Ehrenfest time; this is illustrated with the kicked harmonic oscillator.
54 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2020 | 3 |
| 2019 | 1 |
| 2018 | 7 |
| 2017 | 5 |
| 2016 | 9 |