Angular Integral Research Articles

Generalized Integral Method for Solving the Neutron Transport Equation with the Hybridized Discontinuous Galerkin Method

Accurate modeling of the neutron transport equation (NTE) with anisotropic scattering is crucial to the understanding of neutron interactions within various mediums. The primary challenges in this domain are (1) the considerable computational resources demanded by anisotropic calculations and (2) the numerical instabilities that arise due to the transport correction approximation. This study introduces a novel, generalized integral method based on the hybridized discontinuous Galerkin framework for solving the second-order NTE with anisotropic scattering. This method employs a spherical harmonics expansion to define the partial current at the mesh interface and applies an angular integral approach to the flux treatment within the mesh. This dual approach facilitates an efficient computational process while preserving accuracy. The integral method has been validated through comparisons with the standard discrete ordinates method (SN) using two eigenvalue problems. The integral method showcases several significant improvements over the traditional SN method. First, it repositions the P0 scattering sources during the formulation process, effectively circumventing the convergence issues associated with transport correction. Second, this strategic repositioning substantially enhances the convergence rates of iterative calculations. Last, a standout feature of the integral method is its capability to perform angular integrals during assembling matrices, successfully reducing the floating-point operations for local flux retrieval and eliminating the ray effect.

Angular Integral Autocorrelation for Speed Estimation in Shear-Wave Elastography

The utilization of a reverberant shear-wave field in shear-wave elastography has emerged as a promising technique for achieving robust shear-wave speed (SWS) estimation. However, many types of estimators cannot accurately measure SWS within such a complicated 3D wave field. This study introduces an advanced autocorrelation estimator based on angular integration known as the angular integral autocorrelation (AIA) approach to address this issue. The AIA approach incorporates all the autocorrelation data from various angles during measurements, resulting in enhanced robustness to both noise and imperfect distributions in SWS estimation. The effectiveness of the AIA estimator for SWS estimation is first validated using a k-Wave simulation of a stiff branching tube in a uniform background. Furthermore, the AIA estimator is applied to ultrasound elastography experiments, magnetic resonance imaging (MRI) experiments, and optical coherence tomography (OCT) studies across a range of different excitation frequencies on tissues and phantoms, including in vivo scans. The results verify the capacity of the AIA approach to enhance the accuracy of SWS estimation and the signal-to-noise ratio (SNR), even within an imperfect reverberant shear-wave field. Compared to simple autocorrelation approaches, the AIA approach can also successfully visualize and define lesions while significantly improving the estimated SWS and SNR in homogeneous background materials and providing improved elastic contrast between structures within the scans. These findings demonstrate the robustness and effectiveness of the AIA approach across a wide range of applications, including ultrasound elastography, magnetic resonance elastography (MRE), and optical coherence elastography (OCE), for accurately identifying the elastic properties of biological tissues in diverse excitation scenarios.

Lagrangian and Hamiltonian Formalisms for Relativistic Mechanics with Lorentz-Invariant Evolution Parameters in 1 + 1 Dimensions

This article presents alternative Hamiltonian and Lagrangian formalisms for relativistic mechanics using proper time and proper Lagrangian coordinates in 1 + 1 dimensions as parameters of evolution. The Lagrangian and Hamiltonian formalisms for a hypothetical particle with and without charge are considered based on the relativistic equation for the dynamics and integrals of particle motion. A relativistic invariant law for the conservation of energy and momentum in the Lorentz representation is given. To select various generalized coordinates and momenta, it is possible to modify the Lagrange equations of the second kind due to the relativistic laws of conservation of energy and momentum. An action function is obtained with an explicit dependence on the velocity of the relativistic particles. The angular integral of the particle motion is derived from Hamiltonian mechanics, and the displacement Hamiltonian is obtained from the Hamilton–Jacobi equation. The angular integral of the particle motion θ is an invariant form of the conservation law. It appears only at relativistic intensities and is constant only in a specific case. The Hamilton–Jacobi–Lagrange equation is derived from the Hamilton–Jacobi equation and the Lagrange equation of the second kind. Using relativistic Hamiltonian mechanics, the Euler–Hamilton equation is obtained by expressing the energy balance through the angular integral of the particle motion θ. The given conservation laws show that the angular integral of the particle motion reflects the relativistic Doppler effect for particles in 1 + 1 dimensions. The connection between the integrals of the particle motion and the doubly special theory of relativity is shown. As an example of the applicability of the proposed invariant method, analyses of the motion of relativistic particles in circularly polarized, monochromatic, spatially modulated electromagnetic plane waves and plane laser pulses are given, and comparisons are made with calculations based on the Landau and Lifshitz method. To allow for the analysis of the oscillation of a particle in various fields, a phase-plane method is presented.

On the stability of ring relative equilibria in the N-body problem on with Hodge potential

AbstractIn this paper, we study the stability of the ring solution of the N-body problem in the entire sphere $\mathbb {S}^2$ by using the logarithmic potential proposed in Boatto et al. (2016, Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 472, 20160020) and Dritschel (2019, Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 377, 20180349), derived through a definition of central force and Hodge decomposition theorem for 1-forms in manifolds. First, we characterize the ring solution and study its spectral stability, obtaining regions (spherical caps) where the ring solution is spectrally stable for $2\leq N\leq 6$ , while, for $N\geq 7$ , the ring is spectrally unstable. The nonlinear stability is studied by reducing the system to the homographic regular polygonal solutions, obtaining a 2-d.o.f. Hamiltonian system, and therefore some classic results on stability for 2-d.o.f. Hamiltonian systems are applied to prove that the ring solution is unstable at any parallel where it is placed. Additionally, this system can be reduced to 1-d.o.f. by using the angular momentum integral, which enables us to describe the phase portraits and use them to find periodic ring solutions to the full system. Some of those solutions are numerically approximated.

A Theoretical Analysis of Favorable Propagation on Massive MIMO Channels with Generalized Angle Distributions

Massive MIMO obtains the multiuser performance gain based on the favorable propagation (FP) assumption, defined as the mutual orthogonality of different users’ channel vectors. Until now, most of the theoretical analyses of FP are based on uniform angular distributions and only consider the horizontal dimension. However, the real propagation channel contains full dimensions, and the spatial angle varies with the environment. Thus, it remains unknown whether the FP condition holds in real deployment scenarios and how it impacts the massive MIMO system performance. In this paper, we analyze the FP condition theoretically based on a cluster-based three-dimensional (3D) MIMO channel with generalized angle distributions. Firstly, the FP condition’s unified mathematical expectation and variance expressions with full-dimensional angular integral are given. Since the closed-form expressions are hard to derive, we decompose generalized angle distributions, i.e., wrapped Gaussian (WG), Von Mises (VM), and truncated Laplacian (TL) into the functions of Bessel and Cosine basis by introducing Jacobi-Anger expansions and Fourier series. Thus the closed-form expressions of the FP condition are derived. Based on the above, we theoretically analyze the asymptotically FP condition under generalized angle distributions and then compare the impact of angular spreads on the FP performance. Furtherly, the FP condition is also investigated by numerical simulations and practical measurements. It is observed that environments with larger angle spreads and larger antenna spacing are more likely to realize FP. This paper provides valuable insights for the theoretical analysis of the practical application of massive MIMO systems.

Motion control of the spherical robot rolling on a vibrating plane

This paper investigates the controlled motion of a spherical robot on a plane performing horizontal periodic oscillations. The spherical robot is modeled by a balanced dynamically asymmetric sphere (the Chaplygin sphere) with three noncoplanar gyrostats placed inside it. The motion of the sphere is controlled by the controlled rotation of the internal gyrostats. The paper addresses two control problems concerning the construction of controls which generate motion along a trajectory given either on a moving plane or in a fixed frame of reference. It is shown that, using a control torque constant in the fixed frame of reference, the general problem can be reduced to the problem of control on the zero level set of the angular momentum integral. It is proved that, on the zero level set of the angular momentum integral, the system under consideration is completely controllable according to the Rashevsky – Chow theorem. Control algorithms for the motion of the sphere along an arbitrary prescribed trajectory are constructed. Examples are given of controls for the sphere rolling in a straight line in an arbitrary direction and in a circle, and for the sphere turning so that the position of the center of mass, both relative to the moving plane and relative to the fixed frame of reference, remains unchanged.

On the enhancement of boundary layer skin friction by turbulence: an angular momentum approach

Turbulence enhances the wall shear stress in boundary layers, significantly increasing the drag on streamlined bodies. Other flow features such as free stream pressure gradients and streamwise boundary layer growth also strongly influence the local skin friction. In this paper, an angular momentum integral (AMI) equation is introduced to quantify these effects by representing them as torques that alter the shape of the mean velocity profile. This approach uniquely isolates the skin friction of a Blasius boundary layer in a single term that depends only on the Reynolds number most relevant to the flow's engineering context, so that other torques are interpreted as augmentations relative to the laminar case having the same Reynolds number. The AMI equation for external flows shares this key property with the so-called FIK relation for internal flows (Fukagata et al., Phys. Fluids, vol. 14, 2002, pp. L73–L76). Without a geometrically imposed boundary layer thickness, the length scale in the Reynolds number for the AMI equation may be chosen freely. After a brief demonstration using Falkner–Skan boundary layers, the AMI equation is applied as a diagnostic tool on four transitional and turbulent boundary layer direct numerical simulation datasets. Regions of negative wall-normal velocity are shown to play a key role in limiting the peak skin friction during the late stages of transition, and the relative strengths of terms in the AMI equation become independent of the transition mechanism a very short distance into the fully turbulent regime. The AMI equation establishes an intuitive, extensible framework for interpreting the impact of turbulence and flow control strategies on boundary layer skin friction.

Wave-function geometry of band crossing points in two dimensions

Geometry of the wave function is a central pillar of modern solid state physics. In this work, we unveil the wave-function geometry of two-dimensional semimetals with band crossing points (BCPs). We show that the Berry phase of BCPs are governed by the quantum metric describing the infinitesimal distance between quantum states. For generic linear BCPs, we show that the corresponding Berry phase is determined either by an angular integral of the quantum metric, or equivalently, by the maximum quantum distance of Bloch states. This naturally explains the origin of the $\ensuremath{\pi}$-Berry phase of a linear BCP. In the case of quadratic BCPs, the Berry phase can take an arbitrary value between 0 and $2\ensuremath{\pi}$. We find simple relations between the Berry phase, maximum quantum distance, and the quantum metric in two cases: (i) when one of the two crossing bands is flat; (ii) when the system has rotation and/or time-reversal symmetries. To demonstrate the implication of the continuum model analysis in lattice systems, we study tight-binding Hamiltonians describing quadratic BCPs. We show that, when the Berry curvature is absent, a quadratic BCP with an arbitrary Berry phase always accompanies another quadratic BCP so that the total Berry phase of the periodic system becomes zero. This work demonstrates that the quantum metric plays a critical role in understanding the geometric properties of topological semimetals.

A Real-Time Estimation Method of Roll Angle and Angular Rate Based on Geomagnetic Information

In the course of the guidance transformation of the rotating projectile, the accurate acquisition of the roll angle and roll angle rate is very important to the attitude determination and guidance control of the rotating projectile. However, due to the impact of high rotation and high overload of projectile, MEMS gyros have problems such as limited range, saturation, overload, and even performance degradation, which make the roll angle rate unable to be output normally. At the same time, because the MEMS gyro estimation of roll angle is in the form of angular rate integral, the roll angle cannot be estimated normally if the roll angle rate cannot be accurately obtained. In order to solve this problem, a real-time estimation of projectile roll angle and roll rate based on geomagnetic information under high dynamic and high overload conditions is presented. Firstly, according to the motion characteristics of the rotating projectile, the motion model of the projectile is established, and the roll angle and roll angle rate of the projectile are estimated by Kalman filtering algorithm under the conditions of high axial rotation and high overload. Considering the high dynamic characteristics of the rotating projectile, based on the Kalman filter, the algorithm of the forgetting filter with the forgetting factor is further adopted to estimate the roll angle and roll angle rate, so as to reduce the error caused by the estimation delay in the process of high-speed dynamic change. Simulation data and semiphysical test results show that the accuracy of roll angle estimated by this method reaches about 2° in semiphysical test, which is one time higher than that calculated by the system. In the semiphysical experiment, the accuracy of the estimated roll rotation rate reaches 5 °/s, which is more than 6 times higher than that obtained by direct derivation. In the high dynamic stage, compared with the pure Kalman filter, the accuracy of roll angle with forgetting factor estimation is improved by an order of magnitude, and the accuracy of roll angle rate is improved by 4 times, which meets the desired accuracy of rotating projectile.

Derivation and application of a Green function propagator suitable for nonparaxial propagation over a two-dimensional domain.

The goal of optical simulation is to determine the performance characteristics of an optical system from a knowledge of its physical construction and how it affects light sent through it. To produce meaningful results efficiently, two simulation approaches are available for passing light through a system, geometrical raytracing and wave optics. Within the wave optics realm, there are many techniques for determining the optical fields within a system, both numerical and analytical. A few of the numerical techniques are finite-difference, finite-element, and FFT-based; analytical techniques include modal expansions, coupled wave theory, series expansions, and Green function propagators. A propagator is a function that gives the light fields at any specified location if they are known at a source location; this is possible because the light fields, electric and magnetic, satisfy a differential equation, in the case of time harmonic fields, the Helmholtz equation. The propagator is a transfer function for the fields and often takes the form of an integral, in which case, the integrand is a product of the transfer function with the source field distribution, and the integration is performed over the source field coordinates. The integrand transfer function, also known as a Green function or propagation kernel, is a solution of the Helmholtz equation. An approximation is often used in finding a solution to the Helmholtz equation, called the paraxial approximation, in which the second derivative in the propagation direction is dropped. If no approximation is made, and all second derivatives are kept, the solution is nonparaxial. In the present paper, a Green function for the propagator of the Helmholtz equation over two-dimensional domains is derived, differing in functional form from previous work on two-dimensional propagation. An angular spectrum integral is evaluated and the resulting Green function, the propagator kernel, is a nonparaxial analytic solution of the Helmholtz equation. The propagator could be applied directly to the electric and magnetic field components; instead, it is applied to the Hertz vector components. The Hertz vector is a potential function, similar to the vector potential, defined such that the electric and magnetic fields are found by taking derivatives of it. An advantage of the Hertz vector is that only it needs be propagated, versus two, electric and magnetic, vectors. In this paper, the derived propagator is applied to Hertz vector components defined by Legendre polynomial expansions, and derivatives are taken of the propagated Hertz vector components to calculate the associated electric and magnetic fields. The Green function propagator and all field quantities produced by its application are closed form analytic expressions.

Isotropic-nematic phase transition of uniaxial variable softness prolate and oblate ellipsoids.

Onsager's theory of the isotropic-nematic phase separation of rod shaped particles is generalized to include particle softness and attractions in the anisotropic interparticle force field. The procedure separates a scaled radial component from the angular integral part, the latter being treated in essentially the same way as in the original Onsager formulation. Building on previous treatments of more idealised hard-core particle models, this is a step toward representing more realistic rod-like systems and also allowing temperature (and in principle specific chemical factors) to be included at a coarse grained level in the theory. The focus of the study is on the coexisting concentrations and associated coexistence properties. Prolate and oblate ellipsoids are considered in both the small and very large aspect ratio limits. Approximations to the terms in the angular integrals derived assuming the very large (prolate) and very small (oblate) aspect ratios limits are compared with the formally exact treatment. The approximation for the second virial coefficient matches the exact solution for aspect ratios above about 20 for the prolate ellipsoids and less than ca. 0.05 for the oblate ellipsoids from the numerical evaluation of the angular integrals. The temperature dependence of the coexistence density could be used to help determine the interaction potential of two molecules. The method works at temperatures above a certain threshold temperature where the second virial coefficient is positive.

First integrals of generalized Ermakov systems via the Hamiltonian formulation

We obtain first integrals of the generalized two-dimensional Ermakov systems, in plane polar form, via the Hamiltonian approaches. There are two methods used for the construction of the first integrals, viz. the standard Hamiltonian and the partial Hamiltonian approaches. In the first approach, [Formula: see text] and [Formula: see text] in the Ermakov system are related as [Formula: see text]. In this case, we deduce four first integrals (three of which are functionally independent) which correspond to the Lie algebra sl[Formula: see text] in a direct constructive manner. We recover the results of earlier work that uses the relationship between symmetries and integrals. This results in the complete integrability of the Ermakov system. By use of the partial Hamiltonian method, we discover four new cases: [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text]; [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text]; [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text] arbitrary and [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text] arbitrary, where the [Formula: see text]s are constants in all cases. In the last two cases, we find that there are three operators each which give rise to three first integrals each. In both these cases, we have complete integrability of the Ermakov system. The first two cases each result in two first integrals each. For every case, both for the standard and partial Hamiltonian, the angular momentum type first integral arises and this is a consequence of the operator which depends on a momentum coordinate which is a generalized symmetry in the Lagrangian context.

Towards an effective field theory on the light-shell

We discuss our work toward the construction of a light-shell effective theory (LSET), an effective field theory for describing the matter emerging from high-energy collisions and the accompanying radiation. We work in the highly simplified venue of 0-flavor scalar quantum electrodynamics, with a gauge invariant product of scalar fields at the origin of space-time as the source of high-energy charged particles. Working in this simple gauge theory allows us to focus on the essential features of LSET. We describe how the effective theory is constructed and argue that it can reproduce the full theory tree-level amplitude. We study the 1-loop radiative corrections in the LSET and suggest how the leading double-logs in the full theory at 1-loop order can be reproduced by a purely angular integral in the LSET.

Modeling Both Active and Passive Microwave Remote Sensing of Snow Using Dense Media Radiative Transfer (DMRT) Theory With Multiple Scattering and Backscattering Enhancement

In this paper, we incorporate the cyclical terms in dense media radiative transfer (DMRT) theory to model combined active and passive microwave remote sensing of snow over the same scene. The inclusion of cyclical terms is crucial if the DMRT is used to model both the active and passive contributions with the same model parameters. This is a necessity when setting out on a joint active/passive retrieval. Previously, the DMRT model has been applied to active and passive separately, and in each case with a separate set of model parameters. The traditional DMRT theory only includes the ladder terms of the Feynman diagrams. The cyclical terms are important in multiple volume scattering and volume–surface interactions. This leads to backscattering enhancement which represents itself as a narrow peak centered at backward direction. This effect is of less significance in passive remote sensing since emissivity is relating to the angular integral of bistatic scattering coefficients. The inclusion of cyclical terms in first-order radiative transfer (RT) accounts for the enhancement of the double bounce contribution and makes the results the same as that of distorted Born approximation in volume–surface interactions. In this paper, we develop the methodology of cyclical corrections within the framework of DMRT beyond first order to all orders of multiple scattering. The active DMRT equation is solved using a numerical iterative approach followed by cyclical corrections. Both quasi-crystalline approximation (QCA)–Mie theory with sticky spheres and bicontinuous media scattering model are used to illustrate the results. The cyclical correlation introduces around 1 dB increase in backscatter with a moderate snowpack optical thickness of ${\sim}\text{0.2}$ . The bicontinuous/DMRT model is next applied to compare with data acquired in the Nordic Snow Radar Experiment (NoSREx) campaign in the snow season of 2010–2011. The model results are validated against coincidental active and passive measurements using the same set of physical parameters of snow in all frequency and polarization channels. Results show good agreement in multiple active and passive channels.

Measuring the black hole spin direction in 3D Cartesian numerical relativity simulations

We show that the so-called flat-space rotational Killing vector method for measuring the Cartesian components of a black hole spin can be derived from the surface integral of Weinberg's pseudotensor over the apparent horizon surface when using Gaussian normal coordinates in the integration. Moreover, the integration of the pseudotensor in this gauge yields the Komar angular momentum integral in a foliation adapted to the axisymmetry of the spacetime. As a result, the method does not explicitly depend on the evolved lapse $\alpha$ and shift $\beta^i$ on the respective timeslice, as they are fixed to Gaussian normal coordinates, while leaving the coordinate labels of the spatial metric $\gamma_{ij}$ and the extrinsic curvature $K_{ij}$ unchanged. Such gauge fixing endows the method with coordinate invariance, which is not present in integral expressions using Weinberg's pseudotensor, as they normally rely on the explicit use of Cartesian coordinates.

Chaotic behavior of three interacting vortices in a confined Bose-Einstein condensate.

Motivated by recent experimental works, we investigate a system of vortex dynamics in an atomic Bose-Einstein condensate (BEC), consisting of three vortices, two of which have the same charge. These vortices are modeled as a system of point particles which possesses a Hamiltonian structure. This tripole system constitutes a prototypical model of vortices in BECs exhibiting chaos. By using the angular momentum integral of motion, we reduce the study of the system to the investigation of a two degree of freedom Hamiltonian model and acquire quantitative results about its chaotic behavior. Our investigation tool is the construction of scan maps by using the Smaller ALignment Index as a chaos indicator. Applying this approach to a large number of initial conditions, we manage to accurately and efficiently measure the extent of chaos in the model and its dependence on physically important parameters like the energy and the angular momentum of the system.

New formulation of the two body problem using a continued fractional potential

In the present work, the two body problem using a potential of a continued fractions procedure is reformulated. The equations of motion for two bodies moving under their mutual gravity is constructed. The integrals of motion, angular momentum integral, center of mass integral, total mechanical energy integral are obtained. New orbit equation is obtained. Some special cases are followed directly. Some graphical illustrations are shown. The only included constant of the continued fraction procedure is adjusted so as to represent the so called J 2 perturbation term of the Earth’s potential.

Experimental investigation of natural circulation in a symmetrical loop under large scale rolling motion conditions

In ocean environment, the ship motion significantly affects the natural circulation behavior in ship-based integrated-type reactor. This paper theoretically and experimentally investigated natural circulation characteristics in symmetrical loops under rolling condition. Experiments were carried out on a test loop with a symmetrical configuration by simulating the structure of an accrual reactor. The theoretical results revealed that only angular acceleration contributes to the resultant force under zero power rolling condition. In a closed circuit with a uniform cross-section area, the angular acceleration force integral is proportional to the angular acceleration and the area enclosed by the circuit. The integral value varies over time and causes flow oscillations. However, the angular acceleration force does not influence the flow status in the shared part of the two symmetrical neighbor circuits due to force interactions. Rolling experiments with a zero power load confirmed these results. Full power experiments under rolling condition exhibited observable flow rate and temperature oscillations in each branch of the flow channel. The oscillations in the side flow channels had the same values for both the period and the phase with the variation of rolling angle. The angular acceleration force was the main cause of this. The oscillations in the middle channel had a period half the value of the rolling period. The periodical variation of the vertical component of gravity caused this. The horizontal component of gravity was out-phasing with angular acceleration. Therefore, it alleviated oscillation in the side channels. The experimental results showed that for the same rolling period, as the rolling angle increased, the average flow rate decreased and oscillation amplitudes increased. Also, as the power load increased, the oscillations in the middle channel increased and the oscillation in the side channel decreased.

An Improved Quasi-Analytic Method for Direct Impulse Response Analysis of an Offset Reflector

For impulse-radiating-antenna applications, an improved quasi-analytic method of offset reflectors' direct impulse response analysis is presented in this article. The offset reflector is illuminated by a time-domain Gaussian beam. The proposed method can eliminate errors caused by numerical differentiation because the numerical differentiation of step response is avoided. Time-domain physical optic approximation is utilized to investigate the direct impulse response. With the appropriate coordinate transform, the direct impulse response integral problem can be reduced to one-dimensional angular integral, which is independent of the reflector's size. Adaptive interpolation is adopted when the impulse response changes rapidly.

Evolution of some quasicircular orbits in the planetary problem

We derive in the plane problem a new closed solution of the Lagrangian equations for resonant motion, concomitantly including zeroth order, and approximate first order and secular perturbations. A major aim is the determination of simple lower limits for the maximum eccentricities and variations of semimajor axes (intrinsic values). Applications of the general solution are made for each perturbation separately. (i) Zeroth and first order perturbations: a new closed solution for the principal zeroth order variation of semimajor axes is obtained. The maximum eccentricity and relative change of semimajor axis of any lunar orbit cannot be lower than 0.019 and 0.018, respectively. (ii) Secular perturbations: with the angular momentum integral our secular perturbations can be easily extended to the spatial problem. The planetary Lidov–Kozai problem is extended to retrograde orbits, showing that large variations of eccentricity and inclination occur for initially circular orbits, if initial mutual inclinations are between about 40° and 150°. (iii) Resonant perturbations: for first and second order resonances, and initially circular orbits our formulas generally approximate just the calculated orbital elements during the whole motion. As a new unexpected result, the numerical exploration of the asteroid belt reproduces most of its overall characteristics up to third order resonances within the restricted three-body problem and modest initial eccentricities ≤0.05.