Showing papers on "Equations of motion published in 2015"
TL;DR: A new class of scalar-tensor theories of gravity that extend Horndeski, or "generalized Galileon," models show that the true propagating degrees of freedom obey well-behaved second-order equations and are thus free from Ostrogradski instabilities, in contrast to standard lore.
Abstract: We introduce a new class of scalar-tensor theories of gravity that extend Horndeski, or " generalized galileon " , models. Despite possessing equations of motion of higher order in derivatives, we show that the true propagating degrees of freedom obey well-behaved second-order equations and are thus free from Ostrogradski instabilities, in contrast to standard lore. Remarkably, the covariant versions of the original galileon Lagrangians—obtained by direct replacement of derivatives with covariant derivatives—belong to this class of theories. These extensions of Horndeski theories exhibit an uncommon, interesting phenomenology: The scalar degree of freedom affects the speed of sound of matter, even when the latter is minimally coupled to gravity. The discovery of the present cosmological acceleration has spurred the exploration of gravitational theories that could account for this effect. Many extensions of general relativity (GR) are based on the inclusion of a scalar degree of freedom (DOF) in addition to the two ten-sor propagating modes of GR (see e.g. [1] for a review). In this context, a recent important proposal is the so-called galileon models [2], with Lagrangians that involve second-order derivatives of the scalar field and lead, nevertheless , to equations of motions of second order. Such a property guarantees the avoidance of Ostrogradski in-stabilities, i.e. of the ghost-like DOF that are usually associated with higher time derivatives (see e.g. [3]). Initially introduced in Minkowski spacetime, galileons have then been generalized to curved spacetimes [4–6], where they turn out to be equivalent to a class of theories originally constructed by Horndeski forty years ago [7]. Today, Horndeski theories, which include quintessence, k-essence and f (R) models, constitute the main theoretical framework for scalar-tensor theories, in which cosmologi-cal observations are interpreted. The purpose of this Letter is to show that this framework is not as exhaustive as generally believed, and can in fact be extended to include new Lagrangians. Indeed, having equations of motion of second order in derivatives—while indeed sufficient—is not necessary to avoid Ostrogradski instabilities, as already pointed out in e.g. [8, 9]. The theories beyond Horndeski that we propose lead to distinct observational effects and are thus fully relevant for an extensive comparison of scalar-tensor theories with observations.
707 citations
TL;DR: In this paper, a review of the theory and theory of particle physics at low temperatures is presented, and the amplitude modes are decoupled from the phase oscillations only near particle-hole symmetry, where the equations of motion have an effective Lorentz symmetry and if there are no significant avenues for decay into other excitations.
Abstract: The order parameter and its variations in space and time in many different states in condensed matter physics at low temperatures are described by the complex function Ψ(r, t). These states include superfluids, superconductors, and a subclass of antiferromagnets and charge density waves. The collective fluctuations in the ordered state may then be categorized as oscillations of phase and amplitude of Ψ(r, t). The phase oscillations are the Goldstone modes of the broken continuous symmetry. The amplitude modes, even at long wavelengths, are well defined and are decoupled from the phase oscillations only near particle-hole symmetry, where the equations of motion have an effective Lorentz symmetry, as in particle physics and if there are no significant avenues for decay into other excitations. They bear close correspondence with the so-called Higgs modes in particle physics, whose prediction and discovery are very important for the standard model of particle physics. In this review, we discuss the theory and...
318 citations
TL;DR: In this article, a simple and refined trigonometric higher-order beam theory is developed for bending and vibration of functionally graded beams, and the neutral surface position for such beams in which the material properties vary in the thickness direction is determined.
Abstract: In the present work, a simple and refined trigonometric higher-order beam theory is developed for bending and vibration of functionally graded beams The beauty of this theory is that, in addition to modeling the displacement field with only 3 unknowns as in Timoshenko beam theory, the thickness stretching effect (ez ≠ 0) is also included in the present theory Thus, the present refined beam theory has fewer number of unknowns and equations of motion than the other shear and normal deformations theories, and it considers also the transverse shear deformation effects without requiring shear correction factors The neutral surface position for such beams in which the material properties vary in the thickness direction is determined Based on the present refined trigonometric higher-order beam theory and the neutral surface concept, the equations of motion are derived from Hamilton's principle Numerical results of the present theory are compared with other theories to show the effect of the inclusion of transverse normal strain on the deflections and stresses
307 citations
TL;DR: In this paper, the authors investigated the three-dimensional motion characteristics of perfect and imperfect Timoshenko microbeams under mechanical and thermal forces; the mechanical properties of the microbeam are considered temperature-dependent.
200 citations
TL;DR: In this article, free and forced vibration of a bi-directional functionally graded (BDFG) Timoshenko beam under the action of a moving load was investigated by means of Lagrange equations based on TBT and Euler-Bernoulli beam theory.
196 citations
TL;DR: In this article, a non-local hyperbolic refined plate model is presented for free vibration properties of functionally graded (FG) plates, where the material properties are assumed to vary only in the thickness direction and the effective properties for the FG nano-plate are computed using Mori-Tanaka homogenization scheme.
Abstract: In this paper, a new nonlocal hyperbolic refined plate model is presented for free vibration properties of functionally graded (FG) plates. This nonlocal nano-plate model incorporates the length scale parameter which can capture the small scale effect. The displacement field of the present theory is chosen based on a hyperbolic variation in the in-plane displacements through the thickness of the nano-plate. By dividing the transverse displacement into the bending and shear parts, the number of unknowns and equations of motion of the present theory is reduced, significantly facilitating structural analysis. The material properties are assumed to vary only in the thickness direction and the effective properties for the FG nano-plate are computed using Mori-Tanaka homogenization scheme. The governing equations of motion are derived based on the nonlocal differential constitutive relations of Eringen in conjunction with the refined four variable plate theory via Hamilton\'s principle. Analytical solution for the simply supported FG nano-plates is obtained to verify the theory by comparing its results with other available solutions in the open literature. The effects of nonlocal parameter, the plate thickness, the plate aspect ratio, and various material compositions on the dynamic response of the FG nano-plate are discussed.
195 citations
TL;DR: In this paper, size-dependent equations of motion for functionally graded cylindrical shell were developed using shear deformation model and rotation inertia, where material properties of the shell were assumed as continuously variable along thickness, consistent with the variation in the component's volume fraction based on power law distribution.
191 citations
TL;DR: In this paper, the authors derived stochastic partial differential equations (SPDEs) for fluid dynamics from a SVP by taking variations in the SVP to derive Stochastic Stratonovich fluid equations; writing their Ito representation; and then investigating the properties of these stochnastic fluid models in comparison with each other, and with corresponding deterministic fluid models.
Abstract: This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The paper proceeds by taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their Ito representation; and then investigating the properties of these stochastic fluid models in comparison with each other, and with the corresponding deterministic fluid models. The circulation properties of the stochastic Stratonovich fluid equations are found to closely mimic those of the deterministic ideal fluid models. As with deterministic ideal flows, motion along the stochastic Stratonovich paths also preserves the helicity of the vortex field lines in incompressible stochastic flows. However, these Stratonovich properties are not apparent in the equivalent Ito representation, because they are disguised by the quadratic covariation drift term arising in the Stratonovich to Ito transformation. This term is a geometric generalization of the quadratic covariation drift term already found for scalar densities in Stratonovich's famous 1966 paper. The paper also derives motion equations for two examples of stochastic geophysical fluid dynamics; namely, the Euler-Boussinesq and quasi-geostropic approximations.
186 citations
TL;DR: In this article, the von Karman plate theory is employed to model the system by retaining in-plane displacements and inertia, and the nonlinear dynamics are examined through plotting the frequency-response and force-response curves of the system.
172 citations
TL;DR: In this article, a geometrically imperfect microplate is modelled by means of the von Karman plate theory and Kirchhoff's hypotheses retaining all in-plane and out-of-plane displacements and inertia.
170 citations
TL;DR: In this paper, the in-plane and out-of-plane nonlinear size-dependent dynamics of a microplate resting on an elastic foundation, constrained by distributed rotational springs at boundaries, were examined via plotting the frequency response and force response curves.
Abstract: The aim of the current study is to examine the in-plane and out-of-plane nonlinear size-dependent dynamics of a microplate resting on an elastic foundation, constrained by distributed rotational springs at boundaries. Employing the von Karman plate theory as well as Kirchhoff’s hypotheses, the equations of motion for the in-plane and out-of-plane directions are derived by means of the Lagrange equations, based on the modified couple stress theory. The potential energies stored in a Winkler-type elastic foundation and the rotational springs at the edges of the microplate are taken into account. The set of second-order nonlinear ordinary differential equations, obtained via the Lagrange scheme, is recast into a double-dimensional set of first-order nonlinear ordinary differential equations with coupled terms by means of a change of variables. The linear natural frequencies of the system are obtained through use of an eigenvalue analysis upon the linear terms of the equations of motion. The nonlinear response, on the other hand, is obtained by means of the pseudo-arclength continuation method. The dynamical characteristics of the system are examined via plotting the frequency–response and force–response curves. The effect of the stiffness of the rotational and translational springs on the nonlinear size-dependent behaviour is also examined. Finally, the effect of employing the modified couple stress theory, rather than the classical theory, on the response is discussed.
TL;DR: A nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen is presented in this paper.
Abstract: This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler–Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton\'s principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.
TL;DR: The approach allows for general mobile robots to independently select new control inputs while avoiding collisions with each other and is capable of letting a non-homogeneous group of robots with non-linear equations of motion safely avoid collisions at real-time computation rates.
Abstract: Reciprocal collision avoidance has become a popular area of research over recent years Approaches have been developed for a variety of dynamic systems ranging from single integrators to car-like, differential-drive, and arbitrary, linear equations of motion In this paper, we present two contributions First, we provide a unification of these previous approaches under a single, generalized representation using control obstacles In particular, we show how velocity obstacles, acceleration velocity obstacles, continuous control obstacles, and LQR-obstacles are special instances of our generalized framework Secondly, we present an extension of control obstacles to general reciprocal collision avoidance for non-linear, non-homogeneous systems where the robots may have different state spaces and different non-linear equations of motion from one another Previous approaches to reciprocal collision avoidance could not be applied to such systems, as they use a relative formulation of the equations of motion and can, therefore, only apply to homogeneous, linear systems where all robots have the same linear equations of motion Our approach allows for general mobile robots to independently select new control inputs while avoiding collisions with each other We implemented our approach in simulation for a variety of mobile robots with non-linear equations of motion: differential-drive, differential-drive with a trailer, car-like, and hovercrafts We also performed physical experiments with a combination of differential-drive, differential-drive with a trailer, and car-like robots Our results show that our approach is capable of letting a non-homogeneous group of robots with non-linear equations of motion safely avoid collisions at real-time computation rates
TL;DR: In this paper, a spin-induced nonminimal couplings were derived for the binary inspiral problem, where all field modes below the orbital scale are integrated out, and the corresponding degrees of freedom of the spin can be directly obtained via a proper variation of the action, and Hamiltonians can be straightforwardly derived.
Abstract: We introduce a formulation for spinning gravitating objects in the effective field theory in the post-Newtonian scheme in the context of the binary inspiral problem. We aim at an effective action, where all field modes below the orbital scale are integrated out. We spell out the relevant degrees of freedom, in particular the rotational ones, and the associated symmetries. Building on these symmetries, we introduce the minimal coupling part of the point particle action in terms of gauge rotational variables, and construct the spin-induced nonminimal couplings, where we obtain the leading order couplings to all orders in spin. We specify the gauge for the rotational variables, where the unphysical degrees of freedom are eliminated already from the Feynman rules, and all the orbital field modes are integrated out. The equations of motion of the spin can be directly obtained via a proper variation of the action, and Hamiltonians may be straightforwardly derived. We implement this effective field theory for spin to derive all spin dependent potentials up to next-to-leading order to quadratic level in spin, namely up to the third post-Newtonian order for rapidly rotating compact objects. In particular, the proper next-to-leading order spin-squared potential and Hamiltonian for generic compact objects are also derived. For the implementations we use the nonrelativistic gravitational field decomposition, which is found here to eliminate higher-loop Feynman diagrams also in spin dependent sectors, and facilitates derivations. This formulation for spin is thus ideal for treatment of higher order spin dependent sectors.
TL;DR: In this paper, a general static spherically symmetric solution to the equations of motion is proposed in the framework of mimetic gravity, an extension of general relativity where the conformal degree of freedom of gravity is isolated in a covariant fashion.
Abstract: In this work, we analyse static spherically symmetric solutions in the framework of mimetic gravity, an extension of general relativity where the conformal degree of freedom of gravity is isolated in a covariant fashion. Here we extend previous works by considering in addition a potential for the mimetic field. An appropriate choice of such potential allows for the reconstruction of a number of interesting cosmological and astrophysical scenarios. We explicitly show how to reconstruct such a potential for a general static spherically symmetric space-time. A number of applications and scenarios are then explored, among which traversable wormholes. Finally, we analytically reconstruct potentials which leads to solutions to the equations of motion featuring polynomial corrections to the Schwarzschild spacetime. Accurate choices for such corrections could provide an explanation for the inferred flat rotation curves of spiral galaxies within the mimetic gravity framework, without the need for particle dark matter.
TL;DR: In this paper, it was shown that any Lagrangian for third order equations of motion, which evades the nondegeneracy condition, still leads to the Ostrogradsky instability.
Abstract: It is known that any nondegenerate Lagrangian containing time derivative terms higher than first order suffers from the Ostrogradsky instability, pathological excitation of positive and negative energy degrees of freedom. We show that, within the framework of analytical mechanics of point particles, any Lagrangian for third order equations of motion, which evades the nondegeneracy condition, still leads to the Ostrogradsky instability. Extension to the case of higher odd order equations of motion is also considered.
TL;DR: In this paper, a regular singularity-free modified gravity (MOG) solution was derived using a nonlinear field dynamics for the repulsive gravitational field component and a reasonable physical energy-momentum tensor.
Abstract: The field equations for scalar–tensor–vector gravity (STVG) or modified gravity (MOG) have a static, spherically symmetric black hole solution determined by the mass $$M$$
with two horizons. The strength of the gravitational constant is $$G=G_N(1+\alpha )$$
where $$\alpha $$
is a parameter. A regular singularity-free MOG solution is derived using a nonlinear field dynamics for the repulsive gravitational field component and a reasonable physical energy-momentum tensor. The Kruskal–Szekeres completion of the MOG black hole solution is obtained. The Kerr-MOG black hole solution is determined by the mass $$M$$
, the parameter $$\alpha $$
and the spin angular momentum $$J=Ma$$
. The equations of motion and the stability condition of a test particle orbiting the MOG black hole are derived, and the radius of the black hole photosphere and the shadows cast by the Schwarzschild-MOG and Kerr-MOG black holes are calculated. A traversable wormhole solution is constructed with a throat stabilized by the repulsive component of the gravitational field.
TL;DR: The solution of the equations that describe the dynamical evolution of black strings and branes to leading order in the expansion in the inverse of the number of dimensions D shows that thin enough black strings are unstable to developing inhomogeneities along their length, and at late times they asymptote to stable nonuniform black strings.
Abstract: We derive a simple set of nonlinear, (1+1)-dimensional partial differential equations that describe the dynamical evolution of black strings and branes to leading order in the expansion in the inverse of the number of dimensions D. These equations are easily solved numerically. Their solution shows that thin enough black strings are unstable to developing inhomogeneities along their length, and at late times they asymptote to stable nonuniform black strings. This proves an earlier conjecture about the end point of the instability of black strings in a large enough number of dimensions. If the initial black string is very thin, the final configuration is highly nonuniform and resembles a periodic array of localized black holes joined by short necks. We also present the equations that describe the nonlinear dynamics of anti-de Sitter black branes at large D.
TL;DR: In this paper, it was shown that general scalar-tensor theories of gravity are generically invariant under disformal transformations, regardless of whether the scalar field in the action of the assumed scalar tensor theory of gravity is the same or different than the scalars field involved in the transformation.
Abstract: We show that very general scalar-tensor theories of gravity (including, e.g., Horndeski models) are generically invariant under disformal transformations. However there is a special subset, when the transformation is not invertible, that yields new equations of motion which are a generalization of the so-called "mimetic" dark matter theory recently introduced by Chamsedinne and Mukhanov. These conclusions hold true irrespective of whether the scalar field in the action of the assumed scalar-tensor theory of gravity is the same or different than the scalar field involved in the transformation. The new equations of motion for our general mimetic theory can also be derived from an action containing an additional Lagrange multiplier field. The general mimetic scalar-tensor theory has the same number of derivatives in the equations of motion as the original scalar-tensor theory. As an application we show that the simplest mimetic scalar-tensor model is able to mimic the cosmological background of a flat FLRW model with a barotropic perfect fluid with any constant equation of state.
TL;DR: In this paper, the dynamics of spinning binaries of compact objects at the next-to-leading order in the quadratic-in-spin effects, which corresponds to the third post-Newtonian order (3PN), were investigated.
Abstract: We investigate the dynamics of spinning binaries of compact objects at the next-to-leading order in the quadratic-in-spin effects, which corresponds to the third post-Newtonian order (3PN). Using a Dixon-type multipolar formalism for spinning point particles endowed with spin-induced quadrupoles and computing iteratively in harmonic coordinates the relevant pieces of the PN metric within the near zone, we derive the post-Newtonian equations of motion as well as the equations of spin precession. We find full equivalence with available results. We then focus on the far-zone field produced by those systems and obtain the previously unknown 3PN spin contributions to the gravitational-wave energy flux by means of the multipolar post-Minkowskian (MPM) wave generation formalism. Our results are presented in the center-of-mass frame for generic orbits, before being further specialized to the case of spin-aligned, circular orbits. We derive the orbital phase of the binary based on the energy balance equation and briefly discuss the relevance of the new terms.
TL;DR: In this article, the exterior geometry of a spinning compact star is computed to second order in the angular momentum of the star and the deformability of the compact object induced by a perturbing tidal field.
Abstract: The deformability of a compact object induced by a perturbing tidal field is encoded in the tidal Love numbers, which depend sensibly on the object's internal structure. These numbers are known only for static, spherically-symmetric objects. As a first step to compute the tidal Love numbers of a spinning compact star, here we extend powerful perturbative techniques to compute the exterior geometry of a spinning object distorted by an axisymmetric tidal field to second order in the angular momentum. The spin of the object introduces couplings between electric and magnetic deformations and new classes of induced Love numbers emerge. For example, a spinning object immersed in a quadrupolar, electric tidal field can acquire some induced mass, spin, quadrupole, octupole and hexadecapole moments to second order in the spin. The deformations are encoded in a set of inhomogeneous differential equations which, remarkably, can be solved analytically in vacuum. We discuss certain subtleties in defining the tidal Love numbers in general relativity, which are due to the difficulty in separating the tidal field from the linear response of the object in the solution, even in the static case. By extending the standard procedure to identify the linear response in the static case, we prove analytically that the Love numbers of a Kerr black hole remain zero to second order in the spin. As a by-product, we provide the explicit form for a slowly-rotating, tidally-deformed Kerr black hole to quadratic order in the spin, and discuss its geodesic and geometrical properties.
TL;DR: In this article, the authors derived quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature.
Abstract: We consider a quantum mechanical system represented in phase space (referred to hereafter as “Wigner space”), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material.
TL;DR: In this paper, a unified treatment of dark energy and modified gravity is presented, where distinct conformal-disformal couplings of matter species to the gravitational sector are derived.
Abstract: We present a unifying treatment of dark energy and modified gravity that allows distinct conformal-disformal couplings of matter species to the gravitational sector. In this very general approach, we derive the conditions to avoid ghost and gradient instabilities. We compute the equations of motion for background quantities and linear perturbations. We illustrate our formalism with two simple scenarios, where either cold dark matter or a relativistic fluid is nonminimally coupled. This extends previous studies of coupled dark energy to a much broader spectrum of gravitational theories.
TL;DR: An extended hierarchy equation of motion (HEOM) is proposed and applied to study the dynamics of the spin-boson model by including the system reduced density matrix and auxiliary fields composed of these expansion functions, where the extended HEOM is derived for the time derivative of each element.
Abstract: An extended hierarchy equation of motion (HEOM) is proposed and applied to study the dynamics of the spin-boson model. In this approach, a complete set of orthonormal functions are used to expand an arbitrary bath correlation function. As a result, a complete dynamic basis set is constructed by including the system reduced density matrix and auxiliary fields composed of these expansion functions, where the extended HEOM is derived for the time derivative of each element. The reliability of the extended HEOM is demonstrated by comparison with the stochastic Hamiltonian approach under room-temperature classical ohmic and sub-ohmic noises and the multilayer multiconfiguration time-dependent Hartree theory under zero-temperature quantum ohmic noise. Upon increasing the order in the hierarchical expansion, the result obtained from the extended HOEM systematically converges to the numerically exact answer.
TL;DR: It is found that motion and melt convection enhance dendritic growth along the flow direction as well as the no-slip boundary condition and the shape preservation condition are satisfied in this method.
TL;DR: In this paper, a theoretical study is presented for peristaltic flow of a MHD fluid in an asymmetric channel, where the governing equations of motion and energy are simplified using long wave length and low Reynolds number approximation.
Abstract: In this article, a theoretical study is presented for peristaltic flow of a MHD fluid in an asymmetric channel. Effects of viscosity variation, velocity-slip as well as thermal-slip have been duly taken care of in the present study. The energy equation is formulated by including a heat source term which simulates either absorption or generation. The governing equations of motion and energy are simplified using long wave length and low Reynolds number approximation. The coupled non-linear differential equations are solved analytically by means of the perturbation method for small values of Reynolds model viscosity parameter. The salient features of pumping and trapping are discussed with particular focus on the effects of velocity-slip parameter, Grashof number and magnetic parameter. The study reveals that the velocity at the central region diminishes with increasing values of the velocity-slip parameter. The size of trapped bolus decreases and finally vanishes for large values of magnetic parameter.
TL;DR: A coordinate-free form of equations of motion for a complete model of a quadrotor UAV with a payload which is connected via a flexible cable according to Lagrangian mechanics on a manifold is derived.
Abstract: We derived a coordinate-free form of equations of motion for a complete model of a quadrotor UAV with a payload which is connected via a flexible cable according to Lagrangian mechanics on a manifold. The flexible cable is modeled as a system of serially-connected links and has been considered in the full dynamic model. A geometric nonlinear control system is presented to asymptotically stabilize the position of the quadrotor while aligning the links to the vertical direction below the quadrotor. Numerical simulation and experimental results are presented and a rigorous stability analysis is provided to confirm the accuracy of our derivations. These results will be particularly useful for aggressive load transportation that involves large deformation of the cable.
TL;DR: In this paper, a new type of holographic entanglement entropy, which appears on entangling surface without the rotational symmetry and reduces to usual Wald entropy on the Killing horizon, is investigated.
Abstract: The holographic entanglement entropy for the most general higher derivative gravity is investigated. We find a new type of Wald entropy, which appears on entangling surface without the rotational symmetry and reduces to usual Wald entropy on Killing horizon. Furthermore, we obtain a formal formula of HEE for the most general higher derivative gravity and work it out exactly for some squashed cones. As an important application, we derive HEE for gravitational action with one derivative of the curvature when the extrinsic curvature vanishes. We also study some toy models with non-zero extrinsic curvature. We prove that our formula yields the correct universal term of entanglement entropy for 4d CFTs. Furthermore, we solve the puzzle raised by Hung, Myers and Smolkin that the logarithmic term of entanglement entropy derived from Weyl anomaly of CFTs does not match the holographic result even if the extrinsic curvature vanishes. We find that such mismatch comes from the ‘anomaly of entropy’ of the derivative of curvature. After considering such contributions carefully, we resolve the puzzle successfully. In general, we need to fix the splitting problem for the conical metrics in order to derive the holographic entanglement entropy. We find that, at least for Einstein gravity, the splitting problem can be fixed by using equations of motion. How to derive the splittings for higher derivative gravity is a non-trivial and open question. For simplicity, we ignore the splitting problem in this paper and find that it does not affect our main results.
TL;DR: In this article, a new class of exact solutions of regular black holes in f(T) gravity with non-linear electrodynamics material content, with spherical symmetry in 4D, was obtained.
Abstract: We seek to obtain a new class of exact solutions of regular black holes in f(T) Gravity with non-linear electrodynamics material content, with spherical symmetry in 4D. The equations of motion provide the regaining of various solutions of General Relativity, as a particular case where the function f(T)=T. We developed a powerful method for finding exact solutions, where we get the first new class of regular black holes solutions in the f(T) Theory, where all the geometrics scalars disappear at the origin of the radial coordinate and are finite everywhere, as well as a new class of singular black holes.
TL;DR: In this paper, a spin-induced nonminimal couplings were derived for the binary inspiral problem, where all field modes below the orbital scale are integrated out, and the corresponding degrees of freedom of the spin can be directly obtained via a proper variation of the action, and Hamiltonians can be straightforwardly derived.
Abstract: We introduce a formulation for spinning gravitating objects in the effective field theory in the post-Newtonian scheme in the context of the binary inspiral problem. We aim at an effective action, where all field modes below the orbital scale are integrated out. We spell out the relevant degrees of freedom, in particular the rotational ones, and the associated symmetries. Building on these symmetries, we introduce the minimal coupling part of the point particle action in terms of gauge rotational variables, and construct the spin-induced nonminimal couplings, where we obtain the leading order couplings to all orders in spin. We specify the gauge for the rotational variables, where the unphysical degrees of freedom are eliminated already from the Feynman rules, and all the orbital field modes are integrated out. The equations of motion of the spin can be directly obtained via a proper variation of the action, and Hamiltonians may be straightforwardly derived. We implement this effective field theory for spin to derive all spin dependent potentials up to next-to-leading order to quadratic level in spin, namely up to the third post-Newtonian order for rapidly rotating compact objects. In particular, the proper next-to-leading order spin-squared potential and Hamiltonian for generic compact objects are also derived. For the implementations we use the nonrelativistic gravitational field decomposition, which is found here to eliminate higher-loop Feynman diagrams also in spin dependent sectors, and facilitates derivations. This formulation for spin is thus ideal for treatment of higher order spin dependent sectors.