Nonlinear resonant torus oscillations as a model of Keplerian disc warp dynamics

Observations of distorted discs have highlighted the ubiquity of warps in a variety of astrophysical contexts. This has been complemented by theoretical efforts to understand the dynamics of warp evolution. Despite significant efforts to understand the dynamics of warped discs, previous work fails to address arguably the most prevalent regime – nonlinear warps in Keplerian discs for which there is a resonance between the orbital, epicyclic and vertical oscillation frequencies. In this work, we implement a novel nonlinear ring model, developed recently by Fairbairn and Ogilvie, as a framework for understanding such resonant warp dynamics. Here we uncover two distinct nonlinear regimes as the warp amplitude is increased. Initially we find a smooth modulation theory which describes warp evolution in terms of the averaged Lagrangian of the oscillatory vertical motions of the disc. This hints towards the possibility of connecting previous warp theory under a generalised secular framework. Upon the warp amplitude exceeding a critical value, which scales as the square root of the aspect-ratio of our ring, the disc enters into a bouncing regime with extreme vertical compressions twice per orbit. We develop an impulsive theory which predicts special retrograde and prograde precessing warped solutions, which are identified numerically using our full equation set. Such solutions emphasise the essential activation of nonlinear vertical oscillations within the disc and may have important implications for energy and warp dissipation. Future work should search for this behaviour in detailed numerical studies of the internal flow structure of warped discs.

New insights into singularity analysis

In this work, we emphasize the use of singularity analysis in obtaining analytic solutions for equations for which standard Lie point symmetry analysis fails to make any lucid decision. We study the higher-dimensional Kadomtsev–Petviashvili, Boussinesq, and Kaup–Kupershmidt equations in a more general sense. With higher-order equations, there can be a commensurate number of resonances and when consistency for the full equation is examined at each resonance the constant of integration is supposed to vanish from the expression so that it remains arbitrary, but if there is an instance of this not happening, the consistency can be partially established by giving the offending constant the value from the defining equation. If consistency is otherwise not compromised, the equation can be said to be partially integrable, i.e., integrable on a surface of the complex space. Furthermore, we propose an approach that is meant to magnify the scope of singularity analysis for equations admitting higher values for resonances or positive leading-order exponent.

The Coupler Surface of the RSRS Mechanism

Two degree-of-freedom (2-DOF) closed spatial linkages can be useful in the design of robotic devices for spatial rigid-body guidance or manipulation. One of the simplest linkages of this type, without any passive DOF on its links, is the revolute-spherical-revolute-spherical (RSRS) four-bar spatial linkage. Although the RSRS topology has been used in some robotics applications, the kinematics study of this basic linkage has unexpectedly received little attention in the literature over the years. Counteracting this historical tendency, this work presents the derivation of the general implicit equation of the surface generated by a point on the coupler link of the general RSRS spatial mechanism. Since the derived surface equation expresses the Cartesian coordinates of the coupler point as a function only of known geometric parameters of the linkage, the equation can be useful, for instance, in the process of synthesizing new devices. The steps for generating the coupler surface, which is computed from a distance-based parametrization of the mechanism and is algebraic of order twelve, are detailed and a web link where the interested reader can download the full equation for further study is provided. It is also shown how the celebrated sextic curve of the planar four-bar linkage is obtained from this RSRS dodecic.

Predicting average velocity of a non-linear sliding microrobot

Among the locomotion concepts employed in the microrobotics, friction-based locomotion principles are of considerable importance. In this study, the dynamic modelling of friction drive microrobots subjected to the tangential and normal excitations is investigated, which have the same frequency, but are shifted in phase. The motion equation of the microrobot reveals a strongly non-linear differential equation with discontinuity for which the elastic force term is proportional to a signum function. Using the homotopy perturbation method, simple expressions are derived for predicting average velocity of the slider. The obtained results are in good agreement with those achieved from numerical integration of the full equation and experimental results, reported in the literature. Using the concept of step efficiency, some useful guidelines for the design and control of this type of microrobots are provided.

A COMPARISON OF THE PROPER-TIME EQUATION AND THE RENORMALIZATION GROUP β-FUNCTION IN STRING THEORY

It is known that there is a proportionality factor relating the β-function and the equations of motion viz. the Zamolodchikov metric. Usually this factor has to be obtained by other methods. The proper-time equation, on the other hand, is the full equation of motion. We explain the reasons for this and illustrate it by calculating corrections to Maxwell’s equation. The corrections are calculated to cubic order in the field strength, but are exact to all orders in derivatives. We also test the gauge covariance of the proper-time method by calculating higher (covariant) derivative corrections to the Yang-Mills equation.

THE BOLTZMANN EQUATION AND THE GROUP TRANSFORMATIONS

This paper is devoted to the investigation of group properties of the nonlinear Boltzmann equation. The complete Lie group of invariant transformations for the spatially inhomogeneous Boltzmann equation is constructed. The generalization to the Lie-Backlund groups is given for the spatially homogeneous case. It is shown that there are only two non-trivial group transformations for the Boltzmann equation in the wide class of Lie and Lie-Backlund transformations. Some consequences of these symmetry properties are discussed. The special role of Galileo group and the analogy between the spatially homogeneous Boltzmann equation and the full equation are also investigated.

A Quasi-Exact Inverteble Equation for Absorption Coefficient from Reflectance and Transmittance Measurements

Exact formulas describing reflectance and transmittance R and T of the system absorbing film%substrate%back ambient are examined. Basing on the consideration that, unlike R and T, the quantity T%(l-R) does not oscillate with wavelength, the entity of terms entering the full equation are evaluated, and some approximations are made. As a result, a non-oscillating analytical equation is obtained, that can be inverted and therefore utilized to compute the film absorption coefficient with an accuracy that is better than 3 %for α>103cm-1. The effect of further approximations, like fixing the refractive index of the film and the substrate, is also evaluated. A comparison with a normally used, simplified formula, as well as an example of application, are also reported.