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Lato Lato is a toy that consists of two identical solid balls that are secured together with a rope and can be used as a learning tool in physics. In this study, we examined why the Lato Lato toy uses two balls with the same mass rather than balls with different masses. The collision of the two Lato Lato balls can be considered an elastic collision. The two balls move symmetrically at the same velocity and form a semi-circular route with opposing motion directions when we use Lato Lato with the same ball mass ( m 2 = m 1 ). The angular momentum of the two balls is also the same but in opposite directions, so they cancel each other out. Hence, the two balls travel steadily when Lato Lato is performed. This condition leads to a balanced movement of the Lato Lato system and prevents it from collapsing. On the other hand, because of their disparate masses ({m_1}$?> m 2 > m 1 ), the two balls of Lato Lato move in an imbalanced manner. A lighter ball travels farther because it moves more quickly than a heavier ball. The angular momentum of the two balls is in opposite directions but not of the same magnitude, so they do not cancel each other out. This results in unstable movement of the two balls when Lato Lato is played. Therefore, the two balls no longer collide centrally, like in Lato Lato, with the same mass, but the collision is glancing. Then, the motion of the two balls facilitates collapse easily. This work can enrich students' understanding of collisions beyond the collisions of two objects commonly discussed in textbooks. [ABSTRACT FROM AUTHOR]

In the constrained Cosserat theory of a rod with a rigid cross-section, the balance of angular momentum is satisfied by a restriction on the constitutive equations that requires a second order tensor to be symmetric. The kinematics of the rod are determined by satisfying the balances of linear and director momentum and kinematic constraints. In contrast, the Antman model for a special Cosserat theory of rods proposes constitutive equations directly for the force and mechanical moment applied to the rod and the kinematics are determined by the balances of linear and angular momentum. These two models differ by their treatment of angular momentum. This note poses and answers the question: Are the solutions of these two models identical for the same strain energy? [ABSTRACT FROM AUTHOR]

We obtain a nonlinear second-order differential equation that describes forced vibrations in a milking machine. Approximate analytical solutions of the obtained differential equations are given. [ABSTRACT FROM AUTHOR]

In this study, the particle motion around the naked singularity and black hole of Kerr–Newman spacetime is investigated with a special attention on the closed timelike orbits. It is found that both in the naked singularity (NS) and in black hole (BH), the singularity is concealed by causality violating regions, and the Cauchy surface consistently resides inside the inner horizon in nonextremal black holes. For neutral particles and particles with an identical charge to the source, only particles with positive angular momentum are permitted to traverse the closed timelike curves. Conversely, for particles with the opposite charge to the source, the strong Coulomb attraction draws all particles inside the Cauchy surface, allowing them to be present in the closed timelike curves irrespective of their angular momentum. However, in both the NS and BH (both extremal and nonextremal), test particles are confined at a considerable distance from the singular point such that there always exists an empty region surrounding the singularity which prevents particles from interacting with it. The radius of the empty surface that depends on the source parameters and the particle characteristics is investigated with an accurate expression. [ABSTRACT FROM AUTHOR]

To win a taekwondo competition, more points must be scored, and the key to scoring points is to improve the motor performance of the kick. However, the relationship between core stability and the effectiveness of athletic performance, particularly in executing the wing kick, remains ambiguous, impeding coaches and athletes in their efforts to improve athletic performance. A total of 26 male participants (height: 167.21 ± 7.29 cm; weight: 62.07 ± 8.03 kg; age: 24.69 ± 2.53 years) participated, and they have been practicing Taekwondo for more than 10 years. Kinematic and kinetic data were collected using 13 infrared cameras at 120 Hz, muscle activity data were collected using epidermal electromyography, and participants’ core stability level were measured using the Sahrmann Core Stability Test (SCST) for correlation with other variables. Core stability levels showed a strong negative correlation with execution time (r = −0.7144, p < 0.001) of the aerial phase of the wing kick as did angular momentum in most lower limb segments (thigh X axis, r = −0.6435, p < 0.001, shank X axis, r = −0.62008, p < 0.001, foot X axis, r = −0.7033, p < 0.001). Bilateral rectus abdominis (RA) showed strong correlations with muscle activation (left rectus abdominis (LRA) of left foot take off-left knee max extension (LTO-LKMF) phase, r = 0.5351, p < 0.001, right rectus abdominis (RRA) of LTO-LKMF phase, r = −0.5628, p < 0.001), and high levels of core stability were associated with better core-muscle coupling. We conclude that incorporating core stability training into taekwondo training has the potential to improve wing kick performance.

A novel physical effect has been found to be illuminated of being physically self-consistent to take into account in further calculating the dynamics of magnetically-induced asteroid rotation, surface of which has mostly a conducting structure (whereas such mostly metallic-formed asteroid is assumed to be moving in an external magnetic field of planet but in the meantime to be orbiting preferably outside its sphere of effective attraction). The last condition of asteroid surface's having conducting structure means the existence of the applied external torques stemming from the physical interactions between external magnetic field and iron asteroid itself (including long-term effect due to torques stemming from arising eddy-currents on metallic surface of asteroid). Such eddy-currents should heat the conducting surface of asteroid with further non-uniformly re-directing the heat flow from surface of asteroid into outer space via fluxes of thermal photons which carry momentum (in orthogonal direction to the surface which is in most cases far from the ideal surface of sphere). This leads by taking into account the overall outcome of heat flow (from the non-ideal surface of asteroid) to a kind of long-term magnetic second-grade YORP effect. System of Euler equations for aforementioned dynamics of asteroid magnetic rotation has been investigated in regard to existence of semi-analytical solution. Various perturbations (such as collisions, YORP effect) may destabilize the rotation of asteroid deviating it from the current spin state, whereas the electric(eddy)current-induced dissipation of energy reduces kinetic one of asteroid spin. So, dynamics of the asteroid rotation should result in a spinning about maximal-inertia axis with the proper spin state corresponding to minimal energy with a fixed angular momentum. • New physical effect is found for magnetically-induced metallic asteroid rotation. • Metallic-formed asteroid is assumed to be moving in an external magnetic field. • Conducting structure means torques stemming from an eddy-currents on its surface. • Overall outcome of non-iniform heat flow forms magnetic second-grade YORP effect. • Evolution of spin towards rotation about maximal-inertia axis is approximated. [ABSTRACT FROM AUTHOR]

We consider semiclassically scaled, weakly nonlinear Schrödinger equations with external confining potentials and additional angular-momentum rotation term. This type of model arises in the Gross–Pitaevskii theory of trapped, rotating quantum gases. We construct asymptotic solutions in the form of semiclassical wave packets, which are concentrated in both space and in frequency around an classical Hamiltonian phase-space flow. The rotation term is thereby seen to alter this flow, but not the corresponding classical action. [ABSTRACT FROM AUTHOR]

The purpose of this study was to determine the effects of thrower-discus system rotation on performance in discus throwing. The resultant angular momentum of the thrower-discus system and direction cosines of the angular momentum vector were calculated from kinematic data at five critical instants of discus throwing for 28 female and 15 male elite discus throwers, and compared between their long and short trials in actual competition. Compared to short trials, elite female throwers' long trials had significantly less angular momentum at right foot takeoff and left foot takeoff (P ≤ 0.023), and significantly less top-to-left rotation at right foot takeoff, left foot takeoff, left foot touchdown, and release (P ≤ 0.007), and significantly greater right-to-left rotation at left foot touchdown and release (P ≤ 0.009) in long trials. Compared to their short trials, elite male throwers had significantly less top-to-left and top-to-front rotations at left foot touchdown (P ≤ 0.042), and significantly less total angular momentum at release (P = 0.043) in long trials. Elite discus throwers should control the amount of rotations of the thrower-discus system to reduce the top-to-left roll of the discus during the flight to extend the aerodynamic distance. [ABSTRACT FROM AUTHOR]

This study developed a teaching material that makes the conservation of angular momentum, which students find very difficult, more concrete. The fact that angular momentum is a vector concept, the direction of the vector and the importance of this vector direction in angular momentum conservation are visualized. The teaching material consists of a table mounted on a freely rotating shaft, two electric motors mutually connected to the table, and a speed adjuster circuit used to change the rotation speeds of the motors. When electric motors begin to rotate, the angular momentum of the motors changes. How the angular momentum of the system is preserved thanks to the changing angular momentum of the parts interacting with each other, the directions of the vectors formed, the inertia torque, angular velocity, angular momentum relationship, and the relationship of the parts forming a system with each other have been tried to be explained in detail through the teaching material. When the motors were operated in different directions and speeds, the change in the rotation direction of the table was observed. This observation concretizes that angular momentum and angular momentum conservation are vector concepts. The right-hand rule, which describes the angular momentum vector direction with the material, can be demonstrated to students in practice. Video recordings showing how the teaching material works have been shared with readers. [ABSTRACT FROM AUTHOR]

The flywheel is a widespread mechanical component used for the storage of kinetic energy and angular momentum. It typically consists of cylindrical inertia rotating about its axis on rolling bearings, which involves undesired friction, lubrication, and wear. This paper presents an alternative mechanism that is functionally equivalent to a classical flywheel while relying exclusively on limited-stroke flexure joints. This novel one-degree-of-freedom zero-force mechanism has no wear and requires no lubrication: it is thus compatible with extreme environments, such as vacuum, cryogenics, or ionizing radiation. The mechanism is composed of two coupled pivoting rigid bodies whose individual angular momenta vary during motion but whose sum is constant at all times when the pivoting rate is constant. The quantitative comparison of the flexure-based flywheel to classical ones based on a hollow cylinder as inertia shows that the former typically stores 6 times less angular momentum and kinetic energy for the same mass while typically occupying 10 times more volume. The freedom of design of the shape of the rigid bodies offers the possibility of modifying the ratio of the stored kinetic energy versus angular momentum, which is not possible with classical flywheels. For example, a flexure-based flywheel with rigid pivoting bodies in the shape of thin discs stores 100 times more kinetic energy than a classical flywheel with the same angular momentum. A proof-of-concept prototype was successfully built and characterized in terms of reaction moment generation, which validates the presented analytical model. [ABSTRACT FROM AUTHOR]

The article considers the mechanisms of the formation of transverse energy flows in polychromatic fields with different structures. It is shown that the magnitude of the Poynting vector transverse component and the angular momentum for symmetric polychromatic beams are the same as in coherent monochromatic fields. Changes in the characteristics of energy flows due to the transformation and destruction of beam symmetry are demonstrated. The data of computer ulation are presented. [ABSTRACT FROM AUTHOR]

While walking, ground reaction forces point from the centre of pressure to the neighbourhood of a focal point, namely the virtual pivot point (VPP), that adjusts angular momentum around the centre of mass (CoM). This study explores how age and speed affect the VPP quality and position during walking. Analysing an experimental dataset reveals high quality of the VPP in the sagittal plane for both young and elderly groups, regardless of speed. However, in the frontal plane, the VPP quality decreases with increasing speed, with elderly participants exhibiting significantly lower quality. Although not a direct measure of balance, VPP quality reflects changes in whole-body angular momentum owing to ageing and speed. Additionally, a template model is used to reproduce the VPP quality and position trends observed in the experiment. Simulation results highlight the sensitivity of VPP quality to leg force feedback and show that changing VPP height has minimal effect on gait speed. Furthermore, energy redistribution occurs through increased hip extension and leg damping, associated with a greater horizontal VPP distance from the CoM, observed in elderly walking. This study shows promise for analysing gait based on VPP, potentially aiding clinical interventions and supporting locomotion in the elderly.

Sorting orbital angular momentum (OAM) and spin angular momentum (SAM) of photons is crucial for optical communications, optical storage, and quantum information processing, wherein complicated optical setups or bulky devices are conventionally utilized to determine the angular momentum via repetitive interference or diffractive projection measurements. Recently, advancements in compact and single‐shot OAM detection have been made via coordinate transformation metasurfaces. Nevertheless, most of these strategies require precise separation and alignment between multiple components and cannot simultaneously distinguish full angular momentum states. Besides, the single‐shot maximum identifiable OAM mode numbers are generally limited to 60 due to the tradeoff between the mode space and reduced‐cross‐talk mode interval. Here, this work puts forward and experimentally demonstrates a monolithic spiral metalens (SML) for ultrahigh‐capacity sorting full angular momentum states. The SML focus the incident vortex beam into multiple‐turn spiral ring rather than a circle ring on the focal plane according to its topological charge, with the mode order ranging from −250 to 250. Consequently, the maximum identifiable OAM mode number is up to 168 under a single‐shot measurement, which largely surpass the best results reported hitherto. The experimental results obtained in the telecom band underpin the design approach, which may find great potential in high‐capacity communication and storage applications. [ABSTRACT FROM AUTHOR]

Spin angular momentum (SAM) and Orbital angular momentum (OAM) are two distinct forms of electromagnetic angular momentum (AM). Generally, circularly-polarized (CP) wave is deemed as the typical form of spin wave, which carries SAM only. In this work, the OAM characteristic of spin wave is found after stating the CP wave into the cylindrical coordinate: the spin wave contains two components along r and ϕ directions, which carry both SAM and OAM. Based on the OAM character of spin wave, an electromagnetic waveform with SAM in the center and SAM-OAM beside is demonstrated. In addition, since spin wave carries OAM naturally, a spin–orbit coupling mechanism is performed, which demands no spin Pancharatnam–Berry phase nor any phase modulation structure. All the results have been experimentally verified. Our research can be utilized in the theoretical studies of rotational electromagnetic and may be potentially applied in metamaterial, information and quantum research areas.

Background In classical electrodynamics, light-matter interactions are modelled using Maxwell equations. The solution of Maxwell equations, which is typically given by means of the electric and magnetic field, is vectorial in nature. Yet it is well known that light-matter interactions can be approximately described in a scalar (polarization independent) way for many optical applications. While the accuracy of the scalar approximation can be theoretically computed, to the best of our knowledge, it has never been determined experimentally. Here, we introduce Vortex Circular Dichroism (VCD), an optical measurement that has the required features to assess the vectoriality of diffraction. Methods VCD is measured as the differential transmission (or absorption) of left and right circularly polarized vortex beams. We test the VCD measurement with two different systems: i) an experimental set of single circular nano-apertures drilled in a gold film with diameters ranging from 150 to 1950 nm; and ii) a theoretical set of golden spheres with the same diameters as the nano-apertures. Results We observe that in both systems, VCD > 0 for smaller diameters, VCD ≲ 0 for intermediate values and VCD ≈ 0 for larger values of the diameter. Furthermore, the simulations show that a diffraction process characterized by a VCD ≈ 0 (VCD ≠ 0) is polarization-independent (polarization-dependent). As a result, we relate VCD ≠ 0 to a vectorial diffraction, and VCD ≈ 0 to a scalar one. Conclusions Overall, our results show compelling evidence that it is possible to experimentally assess the scalar/vectorial regime of a diffraction process, and that the VCD technique possesses the required features to measure the vectoriality of diffraction processes involving plasmonic cylindrically symmetric structures.

Recently, Ahn et al. created dumbbell-shaped silica nanoparticles, then optically levitated in vacuum, capable of spinning at 300 billion revolutions per minute, driven only by the force and torque of laser. This is by far the fastest spinning object in the world. We show that in this system, there is an additional change in momentum (or extra force) that is proportional to the change in the angular momentum of the particle (or torque). We suggest that this effect also applies to microscopic particles. In the experiment of Ahn et al., although the angular momentum of the particle itself L is large, the rate of change of the angular momentum L ̇ may not be large. Therefore, experimentally, the physical effects associated with changes in angular momentum can be verified either by applying very strong force and torque to the particle, or by allowing the axis of rotation of the particle to find a way to change very quickly. [ABSTRACT FROM AUTHOR]

The Jordan–Schwinger map allows us to go from a matrix representation of any arbitrary Lie algebra to an oscillator (bosonic) representation. We show that any Lie algebra can be considered for this map by expressing the algebra generators in terms of the oscillator creation and annihilation operators acting in the Hilbert space of quantum oscillator states. Then, to describe quantum states in the probability representation of quantum oscillator states, we express their density operators in terms of conditional probability distributions (symplectic tomograms) or Husimi-like probability distributions. We illustrate this general scheme by examples of qubit states (spin-1/2 s u (2) -group states) and even and odd Schrödinger cat states related to the other representation of s u (2) -algebra (spin-j representation). The two-mode coherent-state superpositions associated with cyclic groups are studied, using the Jordan–Schwinger map. This map allows us to visualize and compare different properties of the mentioned states. For this, the s u (2) coherent states for different angular momenta j are used to define a Husimi-like Q representation. Some properties of these states are explicitly presented for the cyclic groups C 2 and C 3 . Also, their use in quantum information and computing is mentioned. [ABSTRACT FROM AUTHOR]

Since the 1970s, astronomers have struggled with the issue of how matter can be accreted to promote black-hole growth. While low-angular-momentum stars may be devoured by a black hole, they are not a sustainable source of fuel. Gas, which could potentially provide an abundant fuel source, presents another challenge due to its enormous angular momentum. While viscous torques are not significant, gas is subject to gravity torques from non-axisymmetric potentials such as bars and spirals. Primary bars can exchange angular momentum with the gas within corotation, causing it to spiral inwards until reaching the inner Lindblad resonance. An embedded nuclear bar can then take over. As the gas reaches the black hole's sphere of influence, the torque becomes negative, fueling the center. Dynamical friction also accelerates the infall of gas clouds closer to the nucleus. However, because of the Eddington limit, growing a black hole from a stellar-mass seed is a slow process. The existence of very massive black holes in the early universe remains a puzzle that could potentially be solved through direct collapse of massive clouds into black holes or super-Eddington accretion. [ABSTRACT FROM AUTHOR]

The combination of multi-component Bose–Einstein condensates (BECs) and phase imprinting techniques provides an ideal platform for exploring nonlinear dynamics and investigating the quantum transport properties of superfluids. In this paper, we study abundant density structures and corresponding dynamics of phase-separated binary Bose–Einstein condensates with phase-imprinted single vortex or vortex dipole. By adjusting the ratio between the inter-species and intra-species interactions, and the locations of the phase singularities, the typical density profiles such as ball-shell structures, crescent-gibbous structures, Matryoshka-like structures, sector-sector structures and sandwich-type structures appear, and the phase diagrams are obtained. The dynamics of these structures exhibit diverse properties, including the penetration of vortex dipoles, emergence of halfvortex dipoles, co-rotation of sectors, and oscillation between sectors. The pinning effects induced by a potential defect are also discussed, which is useful for controlling and manipulating individual quantum states. [ABSTRACT FROM AUTHOR]

Limiters that are used to stabilise the discontinuous Galerkin method fail to preserve its angular momentum conservation property. We propose a simple technique that restores the conservation of angular momentum while preserving the second-order accuracy of the scheme when paired with an explicit time integrator. Numerical experiments confirm the conservation property of the technique paired with two limiters [Giuliani and Krivodonova. 2018. "Analysis of Slope Limiters on Unstructured Triangular Meshes." Journal of Computational Physics 374: 1–26; Giuliani and Krivodonova. 2019. "A Moment Limiter for the Discontinuous Galerkin Method on Unstructured Triangular Meshes." SIAM Journal on Scientific Computing 41 (1): A508–A537]. We show that the technique eliminates numerical artifacts that arise from limiting solutions to the Euler equations. [ABSTRACT FROM AUTHOR]

The global perspective presented here is built on earlier papers discussing the dynamics of the upper branch of the Hadley cell in the two solsticial seasons. The role of the tropics is made explicit in a conceptual model that is presented and evaluated. The fluctuation of deep tropical convection in longitude and time is seen as crucial. The filamentary outflows from such convective events move westward and across the equator deep into the winter hemisphere. The horizontal tilt of the cross-equatorial flow implies a significant upper-tropospheric flux of westerly momentum from the winter tropics to the summer hemisphere. These properties are related to the cross-equatorial propagation of wave activity triggered by deep tropical convection in the summer hemisphere. The filaments carry with them near-equatorial values of absolute vorticity and potential vorticity. After turning anticyclonically, some filaments move eastward and poleward to the equatorial edge of the winter subtropical jet. There is strong evidence they can interact with the eddies on this jet and enhance their poleward westerly momentum flux. In the global perspective, tropical and extratropical systems and the interaction between them are all important for the dynamics of the upper branch of the Hadley cell. significance statement: The Hadley cell is the large-scale overturning in the atmosphere with air in the upper troposphere moving from the equatorial region to near 30° in the winter hemisphere. In the standard view it is midlatitude weather systems that are responsible for removing angular momentum from this upper branch of the Hadley cell. Here it is proposed that tropical systems and their interaction with the midlatitude systems are also important. Insight into the role of the tropics in the dynamics of the Hadley cell can be obtained by considering it as the sum of many events of active deep convection occurring in different longitudes and at different times. [ABSTRACT FROM AUTHOR]

Atmospheric angular momentum (AAM) is used to study the variability of Earth's atmospheric circulation during the past 45 years, a time of considerable climate change. Using global AAM, two interdecadal states are defined covering the periods 1977–98 (hereinafter P1) and 1999–2022 (P2). Global AAM decreased from P1 to P2 and was accompanied by weakened subtropical jet streams in both hemispheres, strong convection around the northern Maritime Continent, and a strengthened sea surface temperature (SST) gradient across the tropical Pacific Ocean. The period differences project onto 1) internal interdecadal Pacific variability (IPV), 2) a postulated transient ocean thermostat response to greenhouse gas and aerosol emissions, and 3) circulation anomalies related to the ozone hole. During 1977–2023, the first two processes are forcing the climate toward larger Pacific Ocean SST gradients and a poleward expansion of the Indo-Pacific warm pool (IPWP), especially into the Northern Hemisphere. The ozone hole produces its own distinct pattern of anomalies in the Southern Hemisphere that tend to become persistent in the early 1990s. The zonal and vertical mean AAM variations during P1 have frequent westerly wind anomalies between 40°N and 40°S with poleward propagation on interannual time scales. During P2, the circulation is dominated by subtropical easterly wind anomalies, poleward-shifted jets, and weaker propagation. Locally, the zonal mean anomalies manifest as midlatitude ridges that lead to continental droughts. Case studies illustrate the weakened subtropical jet streams of P2 and examine the factors behind a transition to La Niña in early 2020 that maintains the P2 pattern. [ABSTRACT FROM AUTHOR]

Featured Application: This study is a first step for real-time monitoring the risk of falls during gait. The goal of this study was to analyze instability during stepping at different speeds in young and older adults. To this aim, the anteroposterior and the mediolateral distances between the body center of mass (COM) and the minimum moment axis (MMA) were computed. A total of 15 young adults (25 y.o. [19–29]) and 15 older adults (68.7 y.o. [63–77]) volunteered for this study. For the computation of the distances, a complete biomechanical protocol combining two force platforms and a 3D motion capture analysis system was setup. The subjects were equipped with 47 reflective markers and were modeled as a frictionless multibody system with 19 segments, 18 joints and 42 degrees of freedom. They were asked to perform a series of stepping tasks at fast and spontaneous speeds. The stepping was divided into five phases, with successive swing and double-stance phases. Greater instability was observed during the swing phases. The distances reveal a significant higher instability at fast speed for both groups (p < 0.001) for all the phases compared with spontaneous speeds. The anteroposterior distance was significantly greater for older adults, highlighting greater instability compared to young adults, while no differences were observed for the mediolateral distance all along the five phases, suggesting higher risks of backward and forward falls during stepping. [ABSTRACT FROM AUTHOR]

Complex numbers were created by introducing the imaginary unit i to represent the square root of -1, allowing solutions to equations that involved square roots of negative numbers. Complex numbers were further extended to quaternionic numbers by introducing additional imaginary units, namely “j” and “k”. Quaternions are non-commutative and involve four components, with each component being a real number or a multiple of an imaginary unit. Usually complex numbers are used to represent wave functions in quantum mechanics. Solutions using quaternions to square well, spin and angular momentum, Dirac equations have been obtained by many researchers. In this article, we have made use of quaternions to study the generalization of adding angular momenta, the digital signal processing of a quaternionic function and circular potential of a particle in real Hilbert space and have obtained quaternionic solutions in terms of Bessel functions.

Based on the Richards-Wolf formalism, we obtain two different exact expressions for the angular momentum (AM) density in the focus of a vortex beam with the topological charge n and with right circular polarization. One expression for the AM density is derived as the cross product of the position vector and the Poynting vector and has a nonzero value at the focus for an arbitrary integer number n. The other expression for the AM density is deduced as a sum of the orbital angular momentum (OAM) and the spin angular momentum (SAM). We reveal that at the focus of the light field under analysis, the latter turns zero at n = –1. While both these expressions are not equal to each other at each point of space, 3D integrals thereof are equal. Thus, exact expressions are obtained for densities of AM, SAM and OAM at the focus of a vortex beam with right-hand circular polarization and the identity for the densities AM = SAM + OAM is shown to be violated. Besides, it is shown that the expressions for the strength vectors of the electric and magnetic fields near the sharp focus, obtained by adopting the Richards-Wolf formalism, are exact solutions of the Maxwell's equations. Thus, Richards–Wolf theory exactly describes the behavior of light near the sharp focus in free space.

The study of spin–orbit coupling (SOC) of light is crucial to explore the light–matter interactions in sub-wavelength structures. By designing a plasmonic lattice with chiral configuration that provides parallel angular momentum and spin components, one can trigger the strength of the SOC phenomena in photonic or plasmonic crystals. Herein, we explore the SOC in a plasmonic crystal, both theoretically and experimentally. Cathodoluminescence (CL) spectroscopy combined with the numerically calculated photonic band structure reveals an energy band splitting that is ascribed to the peculiar spin–orbit interaction of light in the proposed plasmonic crystal. Moreover, we exploit angle-resolved CL and dark-field polarimetry to demonstrate circular-polarization-dependent scattering of surface plasmon waves interacting with the plasmonic crystal. This further confirms that the scattering direction of a given polarization is determined by the transverse spin angular momentum inherently carried by the SP wave, which is in turn locked to the direction of SP propagation. We further propose an interaction Hamiltonian based on axion electrodynamics that underpins the degeneracy breaking of the surface plasmons due to the spin–orbit interaction of light. Our study gives insight into the design of novel plasmonic devices with polarization-dependent directionality of the Bloch plasmons. We expect spin–orbit interactions in plasmonics will find much more scientific interests and potential applications with the continuous development of nanofabrication methodologies and uncovering new aspects of spin–orbit interactions.

Formation of a compound nucleus and production of its evaporation residue are hindered by quasi-fission process. It reduces the evaporation residue cross section since it takes place before the target and projectile combines to form a composite nucleus. We study the quasifission barriers (Bqf) for 14 fusion reactions in the formation of superheavy element 298Og within the framework of the di-nuclear system model. In particular, the influence of entrance channel parameters on quasifission barriers has been investigated. We notice that the quasifission barriers decreases with an increase in angular momentum and fusion barrier height. We constructed semi-empirical formulae for Bqf as a function of mean fissility. The fusion reaction 48Ca+250Cf poses larger Bqf. The larger evaporation cross section is observed for this fusion reaction for 3n evaporation channel with the cross-section of 0.81pb. Similarly, a smaller evaporation cross-section of about 0.033pb is observed for 51Cr+247Pu fusion reaction for 4n evaporation channel with both projectile and target being deformed, unlike to the previous reaction with spherical projectile and deformed target.

Transfemoral prosthesis users (TFPUs) typically have a high risk of balance loss and falling. Whole-body angular momentum ( ${\overrightarrow {H}}_{{\text {WB}}}{)}$ is a common measure for assessing dynamic balance during human walking. However, little is known about how unilateral TFPUs maintain this dynamic balance through segment-to-segment cancellation strategies. Better understanding of the underlying mechanisms of dynamic balance control in TFPUs is required to improve gait safety. Thus, this study aimed to evaluate dynamic balance in unilateral TFPUs during walking at a self-selected constant speed. Fourteen unilateral TFPUs and fourteen matched controls performed level-ground walking at a comfortable speed on a straight, 10-m-long walkway. In the sagittal plane, the TFPUs had a greater and smaller range of $\overrightarrow {H}_{{\text {WB}}}$ compared to controls during intact and prosthetic steps, respectively. Further, the TFPUs generated greater average positive and negative $\overrightarrow {H}_{{\text {WB}}}$ than did the controls during intact and prosthetic steps, respectively, which may necessitate larger step-to-step postural changes in the forward and backward rotation about the body center of mass (COM). In the transverse plane, no significant difference was observed in the range of $\overrightarrow {H}_{{\text {WB}}}$ between groups. However, the TFPUs displayed smaller negative average $\overrightarrow {H}_{{\text {WB}}}$ in the transverse plane than did the controls. In the frontal plane, the TFPUs and controls demonstrated similar range of $\overrightarrow {H}_{{\text {WB}}}$ and step-to-step whole-body dynamic balance owing to the employment of different segment-to-segment cancellation strategies. Our findings should be interpreted and generalized with caution for the demographic features in our participants.

The flywheel is a widespread mechanical component used for the storage of kinetic energy and angular momentum. It typically consists of cylindrical inertia rotating about its axis on rolling bearings, which involves undesired friction, lubrication, and wear. This paper presents an alternative mechanism that is functionally equivalent to a classical flywheel while relying exclusively on limited-stroke flexure joints. This novel one-degree-of-freedom zero-force mechanism has no wear and requires no lubrication: it is thus compatible with extreme environments, such as vacuum, cryogenics, or ionizing radiation. The mechanism is composed of two coupled pivoting rigid bodies whose individual angular momenta vary during motion but whose sum is constant at all times when the pivoting rate is constant. The quantitative comparison of the flexure-based flywheel to classical ones based on a hollow cylinder as inertia shows that the former typically stores 6 times less angular momentum and kinetic energy for the same mass while typically occupying 10 times more volume. The freedom of design of the shape of the rigid bodies offers the possibility of modifying the ratio of the stored kinetic energy versus angular momentum, which is not possible with classical flywheels. For example, a flexure-based flywheel with rigid pivoting bodies in the shape of thin discs stores 100 times more kinetic energy than a classical flywheel with the same angular momentum. A proof-of-concept prototype was successfully built and characterized in terms of reaction moment generation, which validates the presented analytical model.

The statistical mechanics and thermodynamics of a system of symmetric rigid quantum rotors in the field of centrifugal forces are considered. Using the theory of canonical ensembles, we find one-particle distribution function, and the partition function and thermodynamic potential are expressed as a function of temperature and angular velocity. The equation of state in the angular velocity–angular momentum–temperature coordinates is explicitly found in the high-temperature and ultra-cold cases. The macroscopic parameters of the system show a dependence on the angular velocity, so the chiral effects are present in the system. The magnitude of chiral effects is estimated. The entropy and angular momentum graphs are plotted in the ultra-cold case. [ABSTRACT FROM AUTHOR]

We consider the focusing mass supercritical nonlinear Schrödinger equation with rotation i u t = - 1 2 Δ u + 1 2 V (x) u - | u | p - 1 u + L Ω u , (x , t) ∈ R N × R , where N = 2 or 3 and V(x) is an anisotropic harmonic potential. Here L Ω is the quantum mechanical angular momentum operator. We establish conditions for global existence and blow-up in the energy space. Moreover, we prove strong instability of standing waves under certain conditions on the rotation and the frequency of the wave. Finally, we construct orbitally stable standing waves solutions by considering a suitable local minimization problem. Those results are obtained for nonlinearities which are L 2 -supercritical. [ABSTRACT FROM AUTHOR]

Legged robots excel in navigating challenging natural environments, such as steep obstructions or wide gaps in the ground. In addition to rough terrain, they may confront unexpected impact forces during their leaping gaits. While facing external disturbances, legged robots should maintain and (if necessary) restore their stability while completing their gaits. To this end, external disturbances and body orientation errors should be identified, and appropriate actions have to be taken to restore the balance of the robot and to provide advantageous landing circumstances. This paper briefly surveys the developments for balance and posture control of legged robots. The primary focus of these studies is on balancing legged robots under external disturbances or performing dynamic gaits. This paper also includes a brief focus on the literature that present research on balance and posture control strategies using the angular momentum approach. [ABSTRACT FROM AUTHOR]

Photonic spin‐orbit interactions describe the interactions between spin angular momentum and orbital angular momentum of photons, which play essential roles in subwavelength optics. However, the influence of frequency dispersion on photonic angular‐momentum coupling is rarely studied. Here, by elaborately designing the contribution of the geometric phase and waveguide propagation phase, the dispersion‐enabled symmetry switching of photonic angular‐momentum coupling is experimentally demonstrated. This notion may induce many exotic phenomena and be found in enormous applications, such as the spin‐Hall effect, optical calculation, and wavelength division multiplexing systems. As a proof‐of‐concept demonstration, two metadevices, a multi‐channel vectorial vortex beam generator and a phase‐only hologram, are applied to experimentally display optical double convolution, which may offer additional degrees of freedom to accelerate computing and a miniaturization configuration for optical convolution without collimation operation. These results may provide a new opportunity for complex vector optical field manipulation and calculation, optical information coding, light‐matter interaction manipulation, and optical communication. [ABSTRACT FROM AUTHOR]

In this paper, a novel magnetohydrodynamic attitude adjustment method is proposed, which uses a spherical magnetohydrodynamic attitude control device for the three-axis attitude adjustment. This paper mainly studies the distribution characteristics of the magnetic field and liquid flow field, and it does not involve attitude control performance and other contents. The magnetohydrodynamic method is used to solve the liquid flow field, which is coupled with the magnetic field. Then, the angular momentum and output torque of the attitude controller are calculated. The calculations aim to investigate the effects of current intensity and frequency on the magnetic induction intensity, flow velocity, angular momentum, and output torque in the spherical magnetohydrodynamic attitude control device. It was found that the velocity, angular momentum, and output torque increased when the current frequency and intensity increased. In addition, to achieve a larger flow rate and more efficient performance of the output angular momentum, priority should be given to increasing the current intensity rather than the current frequency. Thus, an optimal combination of the current intensity and frequency is proposed based on the numerical results. Finally, some implications of the proposed method are further discussed. [ABSTRACT FROM AUTHOR]

In this study, persistent rainfall (PR) over South China (SC) is divided into two types. One type occurs multiple times in succession [defined as multiple PR (MPR)]; another type represents isolated PR (IPR), for which no new PR occurs for 10 days after the previous PR. The spatiotemporal structures of the 10-30-day intraseasonal oscillations (ISOs) associated with the two types of PR are compared and analyzed. The results reveal that the low-level moisture and air temperature perturbations always have a leading phase relative to the anomalous precipitation. In addition, the positive low-level moisture tendency appears in the MPR ending phase, whereas that in the IPR is close to zero. This difference results in convective development after the MPR ending phase, though not after the IPR. The moisture budget shows that the difference in moisture tendency between MPR and IPR is mainly due to meridional advection, including advections by the mean meridional flow across the perturbation moisture gradient and by the perturbation meridional flow across the mean moisture gradient. For the former, the difference is attributed to the perturbation moisture gradients, while the mean moisture gradients are responsible for the difference of the latter. Furthermore, an essential cause of the difference is the influence of higher-latitude disturbances that affect the IPR more significantly than the MPR. Two associated mechanisms are proposed. One is the perturbation stacking effect, and the other is the effect of angular momentum conservation. By contrast, the low-level temperature anomalies are not the key factor causing the difference between MPR and IPR. [ABSTRACT FROM AUTHOR]

For modern Earth, the annual-mean equatorial winds in the upper troposphere are flowing from east to west (i.e., easterly winds). This is mainly due to the deceleration effect of the seasonal cross-equatorial Hadley cells, against the relatively weaker acceleration effect of coupled Rossby and Kelvin waves excited from tropical convection and latent heat release. In this work, we examine the evolution of equatorial winds during the past 250 million years using one global Earth system model, the Community Earth System Model version 1.2.2 (CESM1.2.2). Three climatic factors different from the modern Earth—solar constant, atmospheric CO2 concentration, and land–sea configuration—are considered in the simulations. We find that the upper-tropospheric equatorial winds change sign to westerly flows (called equatorial superrotation) in certain eras, such as 250–230 and 150–50 Ma. The strength of the superrotation is below 4 m s−1, comparable to the magnitude of the present-day easterly winds. In general, this phenomenon occurs in a warmer climate within which the tropical atmospheric circulation shifts upward in altitude, stationary and/or transient eddies are relatively stronger, and/or the Hadley cells are relatively weaker, which in turn are due to the changes of the three factors, especially CO2 concentration and land–sea configuration. [ABSTRACT FROM AUTHOR]

The tight focusing of an optical vortex with an integer topological charge (TC) and linear polarization was considered. We showed that the longitudinal components of the spin angular momentum (SAM) (it was equal to zero) and orbital angular momentum (OAM) (it was equal to the product of the beam power and the TC) vectors averaged over the beam cross-section were separately preserved during the beam propagation. This conservation led to the spin and orbital Hall effects. The spin Hall effect was expressed in the fact that the areas with different signs of the SAM longitudinal component were separated from each other. The orbital Hall effect was marked by the separation of the regions with different rotation directions of the transverse energy flow (clockwise and counterclockwise). There were only four such local regions near the optical axis for any TC. We showed that the total energy flux crossing the focus plane was less than the total beam power since part of the power propagated along the focus surface, while the other part crossed the focus plane in the opposite direction. We also showed that the longitudinal component of the angular momentum (AM) vector was not equal to the sum of the SAM and the OAM. Moreover, there was no summand SAM in the expression for the density of the AM. These quantities were independent of each other. The distributions of the AM and the SAM longitudinal components characterized the orbital and spin Hall effects at the focus, respectively. [ABSTRACT FROM AUTHOR]

A mechanical counterpart of cyclotron motion is found. The configuration of the corresponding device is determined; it may be called a stabilized rotor. The natural rotational frequency (speed) of the rotor is fixed. (It does not depend on the moment of inertia or the angular momentum.) Remarkably, it is the same as the natural oscillation frequency of a pendulum with identical parameters. [ABSTRACT FROM AUTHOR]

The paper focuses on applying the octonions to explore the influence of the external torque on the angular momentum of fluid elements, revealing the interconnection of the external torque and the vortices of vortex streets. J. C. Maxwell was the first to introduce the quaternions to study the physical properties of electromagnetic fields. The contemporary scholars utilize the quaternions and octonions to investigate the electromagnetic theory, gravitational theory, quantum mechanics, special relativity, general relativity, curved spaces, and so forth. The paper adopts the octonions to describe the electromagnetic and gravitational theories, including the octonionic field potential, field strength, linear momentum, angular momentum, torque, and force. In case the octonion force is equal to zero, it is able to deduce eight independent equations, including the fluid continuity equation, current continuity equation, and force equilibrium equation. Especially, one of the eight independent equations will uncover the interrelation of the external torque and angular momentums of fluid elements. One of its inferences is that the direction, magnitude, and frequency of the external torque must impact the direction and curl of the angular momentum of fluid elements, altering the frequencies of Karman vortex streets within the fluids. It means that the external torque is interrelated with the velocity circulation, by means of the liquid viscosity. The external torque is able to exert an influence on the direction of downwash flows, improving the lift and drag characteristics generated by the fluids. [ABSTRACT FROM AUTHOR]

We have considered here a novel particular model for dynamics of a non-rigid asteroid rotation, assuming the added mass model instead of the concept of Viscoelastic Oblate Rotators to describe the physically reasonable response of a 'rubble pile' volumetric material of asteroid with respect to the action of a projectile impacting its surface. In such a model, the response is approximated as an inelastic collision in which the projectile pushes the 'rubble pile' parts of the asteroid together to form a mostly solidified plug in the crater during the sudden impact on the asteroid's surface. Afterwards, the aforementioned 'solidified plug' (having no sufficient adhesion inside the after-impact crater) will be pushed outside the asteroid's surface by centrifugal forces, forming a secondary rotating companion around the asteroid. Thus, according to the fundamental law of angular momentum conservation, the regime of the asteroid's rotation should be changed properly. Namely, changes in rotational dynamics stem from decreasing the asteroid's mass (due to the fundamental law of angular momentum conservation). As the main finding, we have presented a new solving procedure for a semi-analytical estimation of the total mass of the aforementioned 'solidified plug', considering the final spin state of rotation for the asteroid with minimal kinetic energy reduced during a long time period by the inelastic (mainly, tidal) dissipation. The asteroid is assumed to be rotating mainly along the maximal inertia axis with a proper spin state corresponding to minimal energy with a fixed angular momentum. [ABSTRACT FROM AUTHOR]

Biped robots swing their legs alternately to achieve highly dynamic walking, which is the basic ability required for them to perform tasks. However, swinging of the swinging leg in the air will disturb the interaction between the supporting leg and the ground and affect the upper body's balance during dynamic walking. To allow the robot to use its own intrinsic motion characteristics to maintain stable movement like a human when its lower limbs are affected by unknown disturbances during dynamic walking, the ability to use its arms to resist disturbances is essential. This article presents a hybrid momentum compensation control method for torque-controlled biped robots to adapt to unknown disturbances during dynamic walking. First, a hybrid angular momentum and linear momentum regulator is designed to compensate for the disturbance caused by the swinging leg. Second, based on real-time dynamic state changes of the legs, a mixed-momentum quadratic programming controller is designed to realize stable dynamic walking. The proposed method allows the force-controlled robot to maintain its balance while walking down an unknown platform, and it maintains good straightness in the forward direction of dynamic motion. The proposed method's effectiveness is verified experimentally on the BHR-B2 force-controlled biped robot platform. [ABSTRACT FROM AUTHOR]

Using noncommutative infinitesimal rotations, we work out a formulation for angular momentum which leads to an expression for nuclear magnetic resonance (NMR). We also indicate a cosmic footprint of these considerations, much like the 21 cm radio signal of cosmic Hydrogen. Here, the motivation is that there are a number of "second-order effects" which provide results over and above the usual well-known results. [ABSTRACT FROM AUTHOR]

Forward and reverse springboard somersaulting dives use similar approaches with a hurdle step prior to the final board contact phase during which forward rotation is produced in forward takeoffs and backward rotation in reverse takeoffs. This study compared forward and reverse takeoffs for joint strength, activation complexity, technique kinematics, and rotation potential. A planar 8-segment torque-driven computer simulation model of springboard diving takeoff was used to determine isometric joint strength by matching performances of a forward 2½ somersault dive and a reverse 1½ somersault dive. Activation complexity for the reverse takeoff was increased to achieve a similar closeness of match as for the forward takeoff. Takeoff technique was optimised to maximise rotation potential of forward and reverse somersaulting dives. Kinematics at touchdown, lowest point and takeoff were compared for the optimised forward and reverse takeoff simulations. It was found that the optimised reverse somersaulting dive exhibited greater isometric strength for ankle plantarflexion and shoulder flexion, greater joint torque activation complexity for ankle plantarflexion and for knee flexion. There was also less forward motion during board depression, more hip extension and knee flexion during the later stages of board recoil, less capacity for rotation potential, and greater vertical velocity at takeoff. [ABSTRACT FROM AUTHOR]

The interactions between electromagnetic (EM) waves and Metamaterials have always been considered as research focus of electronic information science and technology. In this work, we focus on the dynamical properties of EM waves with vortex structure propagating in left-handed materials (LHMs) exhibiting dispersive characteristics. The analytical expressions for the electric and magnetic field components of vortex EM waves without the limitation of the paraxial condition are derived using the spectral domain method. A canonical approach is utilized to explore the dynamical properties, including the energy, momentum, spin and orbital angular momentum, of vortex EM waves in dispersive LHMs. Numerical results show that the orientation of transverse energy flow in the LHM is reversed compared to that in the free space. The waves with higher angular frequency are beneficial to improve the dynamical characteristics. Furthermore, the topological charge and polarization state have a significant effect on the dynamical quantities. This study is expected to provide useful insights into the interactions of vortex EM waves with metamaterials, and their further applications. [ABSTRACT FROM AUTHOR]

Based on the Richards-Wolf (RW) formalism, we obtain two different exact expressions for the angular momentum (AM) density of light in the focus of an optical vortex with a topological charge n and right circular polarization. One expression for the AM density is derived as the cross product of the position vector and the Poynting vector and has a nonzero value in the focus for an arbitrary integer n. Another expression for the AM density is equal to a sum of the orbital angular momentum (OAM) and the spin angular momentum (SAM) and, in the focus of a considered light field, is equal to zero at n = −1. These expressions are not equal at each point in space, but their 3D integrals are equal. Thus, we derive exact expressions for the AM, SAM and OAM densities in the focus of an optical vortex with right circular polarization and demonstrate that the identity for the densities AM = SAM + OAM is not valid. In addition, we show that the expressions for the strength vectors of the electric and magnetic field near the tight focus, obtained on the basis of the RW formalism, are exact solutions of Maxwell's equations. Thus, the RW theory exactly describes the behavior of light near the tight focus in free space. [ABSTRACT FROM AUTHOR]

A quaternion solution of the dynamic problem on the optimal turn of a solid (for example, a spacecraft) from a known initial to a given final angular position is presented. Optimization of the control program is carried out by using a combined indicator that combines a quadratic performance criterion and turn time; the minimized functional combines energy costs and maneuver duration in a given proportion. Based on the maximum principle, quaternion models, and methods for studying the controlled motion of a solid (spacecraft), a solution of the problem has been obtained. The construction of the optimal rotation is based on a differential equation relating the angular momentum and the orientation quaternion of a solid. The conditions of optimality are written in analytical form and the properties of the optimal motion are studied. Analytical equations and calculation formulas for finding the optimal control are presented. The control law is formulated as an explicit dependence of the control variables on the phase coordinates. Key relations that determine the optimal values of the parameters of the angular momentum control algorithm are given. In the case of a dynamically symmetric body, a complete solution of the turn problem in a closed form is obtained: analytical dependences as explicit functions of time for control variables and relations for calculating the parameters of the control law are given. A numerical example and the results of numeric simulation of the rotation of a spacecraft as a solid under optimal control that demonstrate the practical feasibility of the proposed control method are given. [ABSTRACT FROM AUTHOR]

The Jordan–Schwinger map allows us to go from a matrix representation of any arbitrary Lie algebra to an oscillator (bosonic) representation. We show that any Lie algebra can be considered for this map by expressing the algebra generators in terms of the oscillator creation and annihilation operators acting in the Hilbert space of quantum oscillator states. Then, to describe quantum states in the probability representation of quantum oscillator states, we express their density operators in terms of conditional probability distributions (symplectic tomograms) or Husimi-like probability distributions. We illustrate this general scheme by examples of qubit states (spin-1/2 su(2)-group states) and even and odd Schrödinger cat states related to the other representation of su(2)-algebra (spin-j representation). The two-mode coherent-state superpositions associated with cyclic groups are studied, using the Jordan–Schwinger map. This map allows us to visualize and compare different properties of the mentioned states. For this, the su(2) coherent states for different angular momenta j are used to define a Husimi-like Q representation. Some properties of these states are explicitly presented for the cyclic groups C2 and C3. Also, their use in quantum information and computing is mentioned.

Since the 1970s, astronomers have struggled with the issue of how matter can be accreted to promote black-hole growth. While low-angular-momentum stars may be devoured by a black hole, they are not a sustainable source of fuel. Gas, which could potentially provide an abundant fuel source, presents another challenge due to its enormous angular momentum. While viscous torques are not significant, gas is subject to gravity torques from non-axisymmetric potentials such as bars and spirals. Primary bars can exchange angular momentum with the gas within corotation, causing it to spiral inwards until reaching the inner Lindblad resonance. An embedded nuclear bar can then take over. As the gas reaches the black hole’s sphere of influence, the torque becomes negative, fueling the center. Dynamical friction also accelerates the infall of gas clouds closer to the nucleus. However, because of the Eddington limit, growing a black hole from a stellar-mass seed is a slow process. The existence of very massive black holes in the early universe remains a puzzle that could potentially be solved through direct collapse of massive clouds into black holes or super-Eddington accretion.

Legged robots can walk stably down slopes without any actuation or control. The locomotion is called passive dynamic walking. Its dynamics at foot strike are governed by impact forces. The impulsive dynamics have been modelled in several approaches, where all the approaches assume that the stance leg instantaneously loses contact with the ground at foot strike. The loss-of-contact assumption has been introduced so as to guarantee conservation of angular momentum at foot strike; however, it is shown in this paper that the loss-of-contact assumption is neither necessary nor sufficient for that purpose. This paper provides a necessary and sufficient condition for conservation of angular momentum at foot strike in a modelling framework for multibody dynamics with impact. A compass-like biped robot is considered in this paper. Its impulsive dynamics are modelled by the relationship between impact forces and change of momentum with constraints. A necessary and sufficient condition for conservation of angular momentum is derived by examining the validity of foot velocities and impact forces in the impulsive model.

Based on the novel prescription for the power function, (x+iy)m, the new expression for Ψ(x,y|m), the eigenfunction of the operator of the third component of the angular momentum, M^z, is presented. These functions are normalizable, single valued and, distinct to the traditional presentation, (x+iy)m=ρmeimϕ, are invariant under the rotations at 2π for any, not necessarily integer, m—the eigenvalue of M^z. For any real m the functions Ψ(x,y|m) form an orthonormal set, therefore they may serve as a quantum mechanical eigenfunction of M^z. The eigenfunctions and eigenvalues of the angular momentum operator squared, M^2, derived for the two different prescriptions for the square root, (m2)1/2, (m2)1/2=|m| and (m2)1/2=±m, are reported. The normalizable eigenfunctions of M^2 are presented in terms of hypergeometric functions, admitting integer as well as non-integer eigenvalues. It is shown that the purely integer spectrum is not the most general solution but is just the artifact of a particular choice of the Legendre functions as the pair of linearly independent solutions of the eigenvalue problem for the M^2.