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The non-deterministic behavior of the Duffing oscillator is classically attributed to the coexistence of two steady states in a double-well potential. However, this interpretation fails in the quantum-mechanical perspective which predicts a single unique steady state. Here, we measure the non-equilibrium dynamics of a superconducting Duffing oscillator and experimentally reconcile the classical and quantum descriptions as indicated by the Liouvillian spectral theory. We demonstrate that the two classically regarded steady states are in fact quantum metastable states. They have a remarkably long lifetime but must eventually relax into the single unique steady state allowed by quantum mechanics. By engineering their lifetime, we observe a first-order dissipative phase transition and reveal the two distinct phases by quantum state tomography. Our results reveal a smooth quantum state evolution behind a sudden dissipative phase transition and form an essential step towards understanding the intriguing phenomena in driven-dissipative systems. Classical mechanics predicts a bistability in the dynamics of the Duffing oscillator, a key model of nonlinear dynamics. By performing quantum simulations of the model, Chen et al. explain the bistability by quantum metastable states with long lifetimes and reveal a first-order dissipative phase transition. [ABSTRACT FROM AUTHOR]

The proper and precise reproduction of the molecular electrostatic potential (MEP) is crucial to describe correctly electrostatic interactions in molecular modeling. Most of the classical molecular mechanics force fields for biomolecules and drug‐like molecules use the atom‐centered point charges to describe MEP. However, it has been systematically pointed out in literature that such an approximation is not always enough, and some groups, like amino group or heavy halogens, require the use of anisotropic model for better description of their MEP. At the same time, the formally charged groups have not been as extensively and systematically studied as their neutral counterparts. In this report, we demonstrate that the anisotropic models for formally charged groups do bring improvements in the reference MEP reproduction, that are comparable in magnitude to those for neutral groups. [ABSTRACT FROM AUTHOR]

Vibrated monolayers of granular particles confined into horizontal cavities form a variety of fluid patterns with orientational order that resemble equilibrium liquid-crystal phases. In some cases, one can identify nematic and smectic patterns that can be understood in terms of classical statistical mechanics of hard bodies. Low aspect ratio cylinders project as rectangles and form uniaxial, or 2-atic, and tetratic, or 4-atic, nematic phases. Other polygonal particles may exhibit different liquid-crystal phases, in general $$p$$ p -atic phases, of higher symmetries. We give a brief summary of theoretical work on rectangles and triangles, and provide some experimental results on vibrated monolayers. In the case of equilateral triangles, the theory predicts an exotic triatic phase, or 6-atic phase, with six-fold symmetry and three equivalent directors. The right-angled triangles exhibit a 4-atic phase with strong octatic (8-atic) correlations. Experiments on cylinders show 4-atic textures and, even more remarkable, geometric frustration caused by confinement excites topological defects, which seem to follow the same topological rules as standard liquid crystals. Some of our findings can be understood with the help of simulations of hard particles subject to thermal equilibrium, although standard Density-Functional Theories fail to account for the correct equilibrium phases in some cases. [ABSTRACT FROM AUTHOR]

We study three-dimensional potentials of the form V = U (x p + y p + z p) , where U is an arbitrary function of C 2 -class, and p ∈ Z , which produces a preassigned two-parametric family of spatial regular orbits given in the solved form f (x , y , z) = c 1 , g (x , y , z) = c 2 ( c 1 , c 2 = const). These potentials have to satisfy two linear PDEs, which are the basic equations of the 3D inverse problem of Newtonian dynamics. The functions f and g can be represented uniquely by the "slope functions" α (x , y , z) and β (x , y , z) . The orbital functions α (x , y , z) and β (x , y , z) have to satisfy three differential conditions according to the theory of the inverse problem. If these conditions are satisfied, then we can find such a potential analytically. We offer pertinent examples of potentials that are mainly used in physical problems. The values obtained for p lead to cases of well-known potentials, such as the Newtonian, cored, logarithmic, polynomial and quadratic ones. New families of orbits produced by the 3D harmonic oscillator are found. Pertinent examples are given and cover all cases. Two-dimensional potentials belong to a special category of potentials and are studied separately. The families of straight lines in 3D space are also examined. [ABSTRACT FROM AUTHOR]

We study the motion of a test particle on the x y − plane. The particle trajectories are given by a one-parameter family of orbits f (x , y) = c, where c = const. By using the tools of the 2D inverse problem of Newtonian dynamics, we find two-dimensional potentials that produce a pre-assigned monoparametric family of regular orbits f (x , y) = c that can be represented by the "slope function" γ = f y f x uniquely. We apply a new methodology in order to find potentials depending on specific arguments, i.e., potentials of the form V (x , y) = P (u) where u = x 2 + y 2 , x y , x 3 − y 3 , x y ( x , y ≠ 0). Then, we establish one differential condition for the family of orbits f (x , y) = c. If it is satisfied, it guarantees the existence of such a potential, generating the above family of planar orbits. Then, the potential function V = V (x , y) is found by quadratures. For known families of curves, e.g., ellipse, the logarithmic spiral, the lemniscate of Bernoulli, and circles, we find homogeneous and polynomial potentials that are compatible with this family of orbits. We offer pertinent examples that cover all of the cases, and we examine which of these potentials are integrable. We also study one-dimensional potentials. The families of straight lines in 2D space are also examined. [ABSTRACT FROM AUTHOR]

We consider the motion of a weakly relativistic charged particle with an arbitrary spin in central potential e / r in terms of classical mechanics. We show that the spin–orbital interaction causes the precession of the plane of orbit around the vector of total angular momentum. The angular velocity of precession depends on the distance of the particle from the centre. The effective potential for in-plane motion is central, with the corrections to Coulomb terms coming from spin–orbital interaction. The possible orbits of a quantum particle are determined by the Bohr–Sommerfeld quantization rule. We give examples of orbits corresponding to small quantum numbers, which were obtained by numerical integration of equations of motion. The energies of stationary states are determined by spin–orbital interaction. [ABSTRACT FROM AUTHOR]

The generalized Reissner-type operator of three-dimensional micropolar mechanics of solids is presented, on the basis of which the generalized Reissner-type operator of three-dimensional micropolar mechanics of thin solids with one small size is obtained under the new parameterization of the domains of these bodies. From the last Reissner-type operator, in turn, the generalized Reissner-type variational principle of three-dimensional micropolar mechanics of thin solids with one small size is derived under the new parametrization of the domains of these bodies. It should be noted that the advantage of the new parameterization is that it is experimentally more accessible than other parameterizations (Nikabadze in Development of the method of orthogonal polynomials in the classical and micropolar mechanics of elastic thin bodies, MSU Publishing House, 2014; Contemp Math. Fundam Dir 55:3–194, 2015; J Math Sci 225:1, 2017). Further, applying the method of orthogonal polynomials (expansion of unknown quantities in series in terms of a system of orthogonal polynomials), from the generalized Reissner-type variational principle of three-dimensional micropolar mechanics of thin solids with one small size under the new parameterization of the domains of these bodies, the Reissner variational principle of micropolar mechanics of thin solids with one small size in the moments with respect to the system of Legendre polynomials is derived. In addition, the method is described for obtaining the variational principles of Lagrange and Castigliano of micropolar mechanics of thin solid with one small size under the new parametrization of the domains of these bodies in moments with respect to systems of the first and second kind Chebyshev polynomials. The paper is a continuation of the work "Nikabadze, Ulukhanyan, On some variational principles in the three-dimensional micropolar theories of solid"; therefore, before reading this paper, the authors invite the interested reader to familiarize themselves with the work (Nikabadze and Ulukhanyan in On some variational principles in the three-dimensional micropolar theories of solids, submitted). [ABSTRACT FROM AUTHOR]

In Paper I of this series, the elastic collision was described via simultaneous processes, where the energy is conserved at any moment. In this paper, we critically review the kinetic theory of gases, which was developed based on Newtonian mechanics, and show that it violates the principle of the conservation of energy. By placing the energy conservation at the beginning of the deductive formalism, we derive the ideal gas law via equally strong simultaneous counter-processes at the walls, namely, momentum work and volume work. Several new insights into the state variables of an ideal gas are obtained: (i) pressure cannot be expressed via the kinetic energy of an ideal gas, and (ii) temperature can be interpreted as a particle-related (microscopic) state variable. The historical choice to set a zero point of the potential energy for a confined ideal gas needs to be corrected, and the internal energy of an ideal gas turns out to include more forms of energy than specified in the kinetic theory of gases. Finally, and importantly, we show that the process approach to an ideal gas and thus to collisions is experimentally confirmed. [ABSTRACT FROM AUTHOR]

Deep space missions, and particularly cislunar endeavors, are becoming a major field of interest for the space industry, including for the astrodynamics research community. While near-Earth missions may be completely covered by perturbed Keplerian dynamics, deep space missions require a different modeling approach, where multi-body gravitational interactions play a major role. To this end, the Restricted Three-Body Problem stands out as an insightful first modeling strategy for early mission design purposes, retaining major dynamical transport structures while still being relatively simple. Dynamical Systems Theory and classical Hamiltonian Mechanics have proven themselves as remarkable tools to analyze deep-space missions within this context, with applications ranging from ballistic capture trajectory design to stationkeeping. In this work, based on this premise, a Hamiltonian derivation of the Restricted Three-Body Problem co-orbital dynamics between two spacecraft is introduced in detail. Thanks to the analytical and numerical models derived, connections between the relative and classical Keplerian and CR3BP problems are shown to exist, including first-order linear solutions and an inherited Hamiltonian normal form. The analytical linear and higher-order models derived allow the theoretical finding and unveiling of natural co-orbital phase space structures, including relative periodic and quasi-periodic orbital families, which are further exploited for general proximity operation applications. In particular, a novel reduced-order, optimal low-thrust stationkeeping controller is derived in the relative Floquet phase space, hybridizing the classical State Dependent Ricatti Equation (SDRE) with Koopman control techniques for efficient unstable manifold regulation. The proposed algorithm is demonstrated and validated within several end-to-end low-cost stationkeeping missions, and comparison against classical continuous stationkeeping algorithms presented in the literature is also addressed to reveal its enhanced performance. Finally, conclusions and open lines of research are discussed. [ABSTRACT FROM AUTHOR]

Collagen fibers and their orientation greatly influence an artery's mechanical characteristics, determining its transversely isotropic behavior. It is generally assumed that these fibers are deposited along a preferred direction to maximize the load bearing capacity of the vessel wall. This implies a large spatial variation in collagen orientation which can be reconstructed in numerical models using so‐called reorientation algorithms. Until now, these algorithms have used the classical continuum mechanics modeling framework which requires knowledge of tissue‐level parameters and the artery's stress‐free reference state, which is inaccessible in a clinical context. We present an algorithm to compute the preferred fiber distribution compatible with the constrained mixture theory, which orients two collagen fiber families according to the loading experienced by the isotropic non‐collagenous extracellular matrix, without requiring prior knowledge of the stress‐free state. Because consensus is lacking whether stress or stretch is the determining factor behind the preferred fiber distribution, we implemented both approaches and compared the results with experimental microstructural data of an abdominal aorta. The stress‐based algorithm was able to describe several experimentally observed transitions of the fiber distribution across the intima, media and adventitia. [ABSTRACT FROM AUTHOR]

This paper explores the possibilities of applying physics-informed neural networks (PINNs) in topology optimization (TO) by introducing a fully self-supervised TO framework based on PINNs. This framework solves the forward elasticity problem by the deep energy method (DEM). Instead of training a separate neural network to update the density distribution, we leverage the fact that the compliance minimization problem is self-adjoint to express the element sensitivity directly in terms of the displacement field from the DEM model. Thus, no additional neural network is needed for the inverse problem. The method of moving asymptotes is used as the optimizer for updating density distribution. The implementation of Neumann, Dirichlet, and periodic boundary conditions is described in the context of the DEM model. Three numerical examples are presented to demonstrate framework capabilities: (i) compliance minimization in 2D under different geometries and loading, (ii) compliance minimization in 3D, and (iii) maximization of homogenized shear modulus to design 2D metamaterial unit cells. The results show that the optimized designs from the DEM-based framework are very comparable to those generated by the finite element method and shed light on a new way of integrating PINN-based simulation methods into classical computational mechanics problems. [ABSTRACT FROM AUTHOR]

The healing process of chronic wounds continues to be a current clinical challenge, worsened by the risk of microbial infections and bacterial resistance to the most frequent antibiotics. In this work, non-antibiotic nanohybrids based on chlorhexidine dihydrochloride and clay minerals have been developed in order to design advanced therapeutic systems aimed to enhance wound healing in chronic lesions. To prepare the nanohybrids, two methodologies have been compared: the intercalation solution procedure and the spray-drying technique, the latter as a one-step process able to reduce preparation times. Nanohybrids were then fully studied by solid state characterization techniques. Computational calculations were also performed to assess the interactions between the drug and the clays at the molecular level. In vitro human fibroblast biocompatibility and antimicrobial activity against Staphylococcus aureus and Pseudomonas aeruginosa were assessed to check biocompatibility and potential microbicidal effects of the obtained nanomaterials. The results demonstrated the effective organic/inorganic character of the nanohybrids with homogeneous drug distribution into the clayey structures, which had been confirmed by classical mechanics calculations. Good biocompatibility and microbicidal effects were also observed, especially for the spray-dried nanohybrids. It was suggested that it could be due to a greater contact area with target cells and bacterial suspensions. [ABSTRACT FROM AUTHOR]

The vigorous development of additive manufacturing technology promotes the tendency of lattice infill design to be miniaturized. The representative scale of the deformation field becomes equivalent to the length scale of the microstructures when the characteristics size of the lattice infill design is down to micron or nanoscale, and strain gradient effects or size effects arise in the structure. Due to the lack of microscopic material properties, the typical topology optimization framework based on classical mechanics for lattice infill structure cannot represent the performance difference induced by the size effect. Therefore, single-scale lattice infill structure design and couple stress theory are combined to explore the size effect on topology optimization. Numerical examples show that when the size ratio of the macroscopic feature size to the material characteristic length is reduced to a comparable range, the topological configurations and the target compliance values exhibit a significant size effect. [ABSTRACT FROM AUTHOR]

The conceptual analysis of quantum mechanics brings to light that a theory inherently consistent with observations should be able to describe both quantum and classical systems, i.e., quantum–classical hybrids. For example, the orthodox interpretation of measurements requires the transient creation of quantum–classical hybrids. Despite its limitations in defining the classical limit, Ehrenfest's theorem makes the simplest contact between quantum and classical mechanics. Here, we generalized the Ehrenfest theorem to bipartite quantum systems. To study quantum–classical hybrids, we employed a formalism based on operator-valued Wigner functions and quantum–classical brackets. We used this approach to derive the form of the Ehrenfest theorem for quantum–classical hybrids. We found that the time variation of the average energy of each component of the bipartite system is equal to the average of the symmetrized quantum dissipated power in both the quantum and the quantum–classical case. We expect that these theoretical results will be useful both to analyze quantum–classical hybrids and to develop self-consistent numerical algorithms for Ehrenfest-type simulations. [ABSTRACT FROM AUTHOR]

The relation between the energy-dependent particle and wave descriptions of electron–matter interactions on the nanoscale was analyzed by measuring the delocalization of an evanescent field from energy-filtered amplitude images of sample/vacuum interfaces with a special aberration-corrected electron microscope. The spatial field extension coincided with the energy-dependent self-coherence length of propagating wave packets that obeyed the time-dependent Schrödinger equation, and underwent a Goos–Hänchen shift. The findings support the view that wave packets are created by self-interferences during coherent–inelastic Coulomb interactions with a decoherence phase close to Δφ = 0.5 rad. Due to a strictly reciprocal dependence on energy, the wave packets shrink below atomic dimensions for electron energy losses beyond 1000 eV, and thus appear particle-like. Consequently, our observations inevitably include pulse-like wave propagations that stimulate structural dynamics in nanomaterials at any electron energy loss, which can be exploited to unravel time-dependent structure–function relationships on the nanoscale. [ABSTRACT FROM AUTHOR]

The Luria–Delbrück mutation model is a cornerstone of evolution theory and has been mathematically formulated in a number of ways. In this paper, we illustrate how this model of mutation rates can be derived by means of classical statistical mechanics tools—in particular, by modeling the phenomenon resorting to methodologies borrowed from classical kinetic theory of rarefied gases. The aim is to construct a linear kinetic model that can reproduce the Luria–Delbrück distribution starting from the elementary interactions that qualitatively and quantitatively describe the variations in mutated cells. The kinetic description is easily adaptable to different situations and makes it possible to clearly identify the differences between the elementary variations, leading to the Luria–Delbrück, Lea–Coulson, and Kendall formulations, respectively. The kinetic approach additionally emphasizes basic principles which not only help to unify existing results but also allow for useful extensions. [ABSTRACT FROM AUTHOR]

Infrared spectra have been used in many chemical applications, and theoretical calculations have been useful for analyzing these experimental results. While quantum mechanics is used for calculating the spectra for small molecules, classical mechanics is used for larger systems. However, a systematic understanding of the similarities and differences between the two approaches is not clear. Previous studies focused on peak position and relative intensities of the spectra obtained by various quantum and classical methods, but here, we included "absolute" intensities in the evaluation. The infrared spectrum of a one‐dimensional (1D) harmonic oscillator (HO) and Morse oscillator were examined using four treatments: quantum, Wigner, truncated Wigner, and classical microcanonical treatments. For a 1D HO with a linear dipole moment function (DMF), the quantum and Wigner treatments give nearly the same spectra. On the other hand, the truncated Wigner underestimates the fundamental transition's intensity by half. In the case of cubic DMF, the truncated Wigner and classical methods fail to reproduce the relative intensity between the fundamental and second overtone transitions. Unfortunately, all the Wigner and classical methods fail to agree with the quantum results for a Morse oscillator with just 1% anharmonicity. [ABSTRACT FROM AUTHOR]

Peridynamics (PD) differs from classical continuum mechanics and other nonlocal theories that do not involve spatial derivatives of the displacement field. PD is based on the integral equation instead of differential equations to handle discontinuities and other singularities. In this paper, the standard peridynamic Reissner–Mindlin plate model (standard PD) accounting for the shear deformation is chosen to describe the thick plate kinematics. Unfortunately, when applied to very thin plate structures, the standard PD encounters shear locking phenomenon, leading to incorrect solutions. This shear locking can be successfully alleviated using the developed reduced or selective integration method. In particular, this technique has been implemented in the standard PD, which allows an accurate result for a wide range from very thin ( L / t > 10 3 ) to thick ( L / t ∼ 10 ) plate structures. It can also accelerate the computational time for particular dynamic problems using fewer neighboring integration particles. Several numerical examples are solved to demonstrate the effectiveness of the proposed method for modeling plate structures. [ABSTRACT FROM AUTHOR]

We combine classical continuum mechanics with the recently developed calculus for mixed-dimensional problems to obtain governing equations for flow in, and deformation of, fractured materials. We present models in both the context of finite and infinitesimal strain, and discuss nonlinear (and non-differentiable) constitutive laws such as friction models and contact mechanics in the fracture. Using the theory of well-posedness for evolutionary equations with maximal monotone operators, we show well-posedness of the model in the case of infinitesimal strain and under certain assumptions on the model parameters. [ABSTRACT FROM AUTHOR]

Consideration has been given to the well-known argument that in a conservative mechanical system existing in an equilibrium state, entropy reaches a maximum. First, using the example of a system of material points modeling a rarefied gas, it has been shown that a maximum of entropy defined through the number of microstates is achieved on a solution that cannot be approximated by an exponential distribution corresponding to the Maxwell–Boltzmann distribution. There are many such solutions and each of them satisfies both the condition of invariability of the number of points and the condition of constancy of the system′s total energy. Next, consideration has been given to an alternative definition of entropy through the density of distribution of the probability of a mechanical system being in an assigned volume of phase space. It has been shown that a conservative mechanical system of general type assigned by its Hamiltonian, in selecting various sets in the phase space for seeking an extremum, has an entropy maximum on various distributions in the case of one and the same energy. A conclusion has been drawn on the occurrence of a Maxwell–Boltzmann distribution in conservative systems of classical mechanics for reasons unrelated to entropy maximization. [ABSTRACT FROM AUTHOR]

This investigation was carried out with the purpose of presenting the flow of micropolar fluid flowing in the microchannel placed parallel to the ground. The prime aim of the work was to study the behavior of micropolar fluid and the response of the microrotation component when the two significant mechanisms namely Brownian movement and thermophoresis are accounted for, as these effects are mainly concerned with the motion of the particles of nano‐dimensions. For the flow of micropolar, we account for the extra kinematics variables combined with the classical continuum mechanics namely microinertia moment tensor and gyration tensor. Magnetic effect and suction/injection of the fluid through the channel walls are also facilitated. The influence on the fluid concentration due to the presence of activation energy was accounted in the present examination. On considering all of these effects, equations are carefully modeled and the solution was attained with the aid of Runge–Kutta Fehlberg 4–5th order method using a shooting scheme. The results have deciphered that the presence of material parameter elevates the microrotation component on the upper half of the channel and depletes it at the lower half. The microinertia parameter shows the opposite behavior of the material parameter. Brownian motion parameter is found to enhance the thermal profile and concentration profile. Lesser entropy was generated when the material parameter was high. [ABSTRACT FROM AUTHOR]

From ancient times to the Newtonian mechanics system, people defaulted to the speed of light as infinity. People usually measure the speed of a moving object directly with the help of optical or electronic equipment and take the measured speed as the actual speed of the object. In fact, the speed of light is limited. When an object moves at high speed, we must correct the measured speed of the object. However, at present, people only use Einstein's special theory of relativity to deal with the high-speed motion of objects, and there is no theory to correct the measured speed of objects under the condition of high-speed motion. Here, we report that the optical measurement speed of an object is affected by the observer's position effect and obtain the relationship between the optical measurement speed and the actual speed. On this basis, this paper explains the reason why Newton's classical mechanics is suitable for the case of low-speed motion, and there is a large deviation between the optical measurement speed and the actual speed under the condition of high-speed motion. This paper also explains the velocity limit of high-energy particles and the phenomena of superluminal and negative velocity and analyzes the observation of microparticle motion. This paper provides a theory to explain and deal with the high-speed motion of objects, which is of great scientific significance. [ABSTRACT FROM AUTHOR]

This work is based on the following three premises: (1) Equality c = u with u = (-2U)1/2 as the total escape speed (from all the masses in the universe) and U the total gravitational potential; given a value of the universe mass, as shared by many physicists, u tends to c, hence constant on Earth. Moreover, the above equality implies the massiveness of light, which regards the second premise: (2) Structure of light: Longitudinal/sinusoidal particles, photons, moving along rays, having parameters k (their length) and frequency v (their number, of the same ray, flowing in a time unit); now, if photons and electron should have, at their impact, opposite direction, a circling electron could fall into its nucleus, hence the third premise. (3) A structure of the electron where its charge is not uniformly distributed, but it can be considered as a point particle fixed on the electron surface, facing the atom nucleus during the electron orbits; the electron charge corresponds to the photons-electron impact point, where photons are absorbed and released. Due to these structures, the photons-electron impacts will force the involved electron to move, with a radial velocity w, toward wider orbits. On these bases, and according to the classical mechanics laws, we found, but not limited to, these results: The measured speed of light, constant on Earth because of the equality #1, turns out to be also invariant whatever the relative motion observersource (of light) is: In short, during the interaction photons-electron, due to both the electron radial velocity and its Doppler effect, if the photons frequency decreases, then their length will increase (and vice versa) always inducing the invariant c. [The equality c = u has a cosmological reason: If c > u, all the visible masses, following the photons mass moving toward the infinity, would be dispersed from each other; if c < u all masses would collapse, while, for c = u, the photons mass (as well as neutrinos mass, as shown) will ensure an endless balance between dispersion and collapse.] The gravitational redshift and the claimed time dilation depend, like c, on the variation of the total potential. On H atom, the number of the electron circular orbits turns out to be n = 1, 2...137; the electron orbital speed along its ground-state (g-s) orbit becomes v1 = c/137, while the electron charge g-s orbital speed is vo = ac with a being the fine-structure constant. As for the claimed fall of circling electrons into their nucleus due to their supposed photons emission, we found that the circling electrons are emitting the previously absorbed photons only during the spiral path from higher toward lower orbits; this emission avoids their fall. Then we show that the compensation velocity (to restore the resonance source-detector located at different heights) has opposite direction with respect to the one predicted by relativity. [ABSTRACT FROM AUTHOR]

Explanations are proposed for several light behaviors using classical mechanics. The Planck-Einstein relation is the starting point. A careful inspection reveals that this relationship has two dynamic interacting entities. One is a free (moving) photon, fp, represented by its angular frequency, x. The other is a proposed, resting, independent, electromagnetic field denoted by the reduced Planck constant S, which is the product of energy and period. Consistent with these entities are several proposals supported by well-established equations. They include the presence of a hidden factor that allows for the photon's internal dynamics, a more precise description of the electromagnetic interactions of gravity with light, the mechanism behind Fraunhofer diffraction, the role of gravity in the fine structure constant, the role of gravity in the refraction of light and its implications for relativity, and the mechanism of Fermat's principle. [ABSTRACT FROM AUTHOR]

A classical continuum mechanics model no longer fulfills its basic assumptions, when the deformations are not smooth or discontinuous. Peridynamics (PD) as an integral formulation could better overcome these issues. Therefore, it is better suited to solve fracture problems such as crack initiation, its growth and formation of crack patterns. In PD the magnitudes of internal forces at a material point depend on the collective interaction of all material points within a subdomain called horizon. To take advances from the formulation, typically it is implemented in a mesh‐free form. In that case, it is easy to separate interactions between different material points. As consequence, a relatively high resolution is required to describe continuous deformation functions and, consequently computationally expensive. Therefore, the application of mesh‐free PD in undamaged regions requires an unnecessary high effort for receiving sufficient accurate results. In contrast, the finite element method (FEM) as a classical continuum mechanics‐based approach, is very efficient, if continuous stress distributions can be assumed. Consequently, a computational cost reduction can be achieved, if the PD is applied in the local damaged zone only and the remaining area is modeled and analyzed with the classical FEM. Realization of such an approach requires a sufficient accurate coupling of the PD and the FEM subdomains. A brief overview about different coupling approaches is given. Then the general approach of an Arlequin based method is presented. Here the coupling is achieved via an overlapping of PD and FEM areas. The coupling equations are developed and realized with help of the Penalty method. The Penalty method is considered as an alternative procedure to the Lagrange multiplier strategy without creating new unknowns, which essentially simplifies the coupled PD‐FEM approach. The validity and limits of the proposed technique are demonstrated through analysis, where the choice of the Penalty number is discussed as well as artificial wave oscillations, caused by the coupling area. [ABSTRACT FROM AUTHOR]

An experimental and theoretical study of the spectral response of coupled viscoelastic bars subject to axial oscillations is performed. Novel closed formulas for the envelope function and its width are derived. These formulas explicitly show the role played by energy dissipation. They show that the internal friction does not affect the width of the envelope of the individual resonances. The formulation is based on the equations of classical mechanics combined with Voigt's viscoelastic model. The systems studied consist of a sequence of one, two, or three coupled bars, with their central axes collinear. One of the bars is assumed to be much longer than the others. We discuss the connection between our results and the concept of the strength function phenomenon discovered for the first time in nuclear physics. Our formulation is an alternative and exact approach to the approximated studies based on the fuzzy structure theory that has been used by other authors to describe systems consisting of coupled bars. The analytical expressions describe the measurements in the laboratory very well. [ABSTRACT FROM AUTHOR]

Textbook treatments of classical mechanics typically assume that the Lagrangian is nonsingular; that is, the matrix of second derivatives of the Lagrangian with respect to the velocities is invertible. This assumption ensures that (i) Lagrange's equations can be solved for the accelerations as functions of coordinates and velocities, and (ii) the definitions of the conjugate momenta can be inverted to solve for the velocities as functions of coordinates and momenta. This assumption, however, is unnecessarily restrictive—there are interesting classical dynamical systems with singular Lagrangians. The algorithm for analyzing such systems was developed by Dirac and Bergmann in the 1950s. After a brief review of the Dirac–Bergmann algorithm, several examples are presented using familiar components: point masses connected by massless springs, rods, cords, and pulleys. [ABSTRACT FROM AUTHOR]

Peridynamics (PD) is a non-local mechanics theory that overcomes the limitations of classical continuum mechanics (CCM) in predicting the initiation and propagation of cracks. However, the calculation efficiency of PD models is generally lower than that of the traditional finite element method (FEM). Structural idealization can greatly improve the calculation efficiency of PD models for complex structures. This study presents a PD shell model based on the micro-beam bond via the homogenization assumption. First, the deformations of each endpoint of the micro-beam bond are calculated through the interpolation method. Second, the micro-potential energy of the axial, torsional, and bending deformations of the bond can be established from the deformations of endpoints. Finally, the micro moduli of the shell model can be obtained via the equivalence principle of strain energy density (SED). In addition, a new fracture criterion based on the SED of the micro-beam bond is adopted for crack simulation. Numerical examples of crack propagation are provided, and the results demonstrate the effectiveness of the proposed PD shell model. [ABSTRACT FROM AUTHOR]

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Inspired by the “water dancing ball” problem proposed in the International Physicists’ Tournament 2019, we study the small-angle oscillations of a cylinder on a plane subject to a thin sheet of water. The interaction between the water flow and the cylinder is modeled using basic physics principles such as Newton’s laws, momentum, and energy conservation. We found that the flux of water around the cylinder applies a force which is responsible for the oscillations of the cylinder. From the motion equation of the cylinder, the period of oscillation is calculated in the harmonic approximation. Our results are contrasted with the period measurements presented recently by Pagaud and Delance, with a good agreement found, showing an error of only 10%.

We study the motion of a test particle in a conservative force-field. Our aim is to find three-dimensional potentials with symmetrical properties, i.e., V(x,y,z)=P(x,y)+Q(z), or, V(x,y,z)=P(x2+y2)+Q(z) and V(x,y,z)=P(x,y)Q(z), where P and Q are arbitrary C2-functions, which are characterized as semi-separable and they produce a pre-assigned two-parametric family of orbits f(x,y,z) = c1, g(x,y,z) = c2 (c1, c2 = const) in 3D space. There exist two linear PDEs which are the basic equations of the Inverse Problem of Newtonian Dynamics and are satisfied by these potentials. Pertinent examples are presented for all the cases. Two-dimensional potentials are also included into our study. Families of straight lines is a special category of curves in 3D space and are examined separately.

We study the motion of a test particle in a conservative force field. In the framework of the 2D inverse problem of Newtonian dynamics, we find 2D potentials that produce a preassigned monoparametric family of regular orbits f(x,y)=c on the xy-plane (where c is the parameter of the family of orbits). This family of orbits can be represented by the “slope function” γ=fyfx uniquely. A new methodology is applied to the basic equation of the planar inverse problem in order to find potentials of a special form, i.e., V=F(x+y)+G(x−y), V=F(x+iy)+G(x−iy) and V=P(x)+Q(y), and polynomial ones. According to this methodology, we impose differential conditions on the family of orbits f(x,y) = c. If they are satisfied, such a potential exists and it is found analytically. For known families of curves, e.g., circles, parabolas, hyperbolas, etc., we find potentials that are compatible with them. We offer pertinent examples that cover all the cases. The case of families of straight lines is referred to.

Mixed quantum-classical methods are commonly used to calculate infrared spectra for condensed-phase systems. These methods have been applied to study water in a range of conditions from liquid to solid to supercooled. Here, we show that these methods also predict infrared line shapes in excellent agreement with experiments in supercritical water. Specifically, we study the OD stretching mode of dilute HOD in H2O. We find no qualitative change in the spectrum upon passing through the near-critical region (Widom line) or the hydrogen-bond percolation line. At very low densities, the spectrum does change qualitatively, becoming rovibrational in character. We describe this rovibrational spectrum from the perspective of classical mechanics and provide a classical interpretation of the rovibrational line shape for both HOD and H2O. This treatment is perhaps more accessible than the conventional quantum-mechanical treatment. [ABSTRACT FROM AUTHOR]

The probability density function I P i ( I x i ) for a classical harmonic oscillator is defined such that the probability of finding the oscillator in any spatial interval [ I x i , I x i + I dx i ] is given by I P i ( I x i ) I dx i . Probability distribution for a classical harmonic oscillator The probabilistic nature of a classical harmonic oscillator can be analyzed in terms of the probability density function. With the help of these trajectories, the future progress of the system can be predicted.[1] Therefore, a probability density function (PDF) is not required for such systems and is rarely discussed in classical mechanics. [Extracted from the article]

This study aims to highlight the importance of visual representation skills in classroom practice, especially in classical mechanics physics, by discussing (1) free-body diagram representation, (2) resultant force system, (3) types and directions of quantities, (4) representation of results and units. This research was conducted on 30 students in second grade at Beringin Kupang High School. The research method was carried out experimentally using a special image representation test tool for classical mechanical materials. The research data were analyzed by descriptive method. The results showed that (1) more than 50% of students did not draw FBD before solving the problem, (2) the success rate of students in representing the resultant force system was the highest at 33%, (3) determining the type and direction of motion the success rate was 67%, and (4) representation results and units with a success rate of 40%. The results of this study become a reference for increasing the role of visual representation in learning physics.

The classical theory of elasticity relies on the perfect structure of matter. No matter how small a medium is, it always stays homogeneous—but the reality is different. Even though the classical continuum mechanics suffices well for describing phenomena that evolve at super‐microscopic scales, it ceases to reproduce reasonable responses when the size effects are pronounced. It is due to the local constitutive equations of classical continuum mechanics, which are devoid of any length scales associated with the underlying microstructure. The gradient‐dependent theory of elasticity redresses this shortcoming by incorporating the kinematic quantities of the corresponding representative volume element by enriching the differential equations with higher‐order spatial and temporal derivatives. In this study, the scaled boundary finite element method is formulated for the equations of motion with higher‐order inertia terms. To this end, the semi‐discretized scaled boundary finite element equations of the medium are derived by introducing the scaled boundary transformation of geometry to the gradient‐enriched equations of motion and applying the Galerkin method of weighted residuals. It is shown that the well‐established available solution methods are incapable of handling the frequency‐domain representation of the derived formulation. Accordingly, a numerical solution method based on the shooting technique is proposed. The solution procedure is formulated for general numerical integration schemes via the infinite Taylor series. The evolution of the impedance‐diffusion matrix and contribution of inter‐subdomain forces and tractions are extracted. Four different numerical integration methods are employed to describe the solution procedure. In addition, a comparison regarding their computational efficiency is presented. For the sake of verification, the scaled boundary finite element solutions of three numerical examples are compared with reference solutions that are obtained using finite element models with extremely fine meshes. The convergence trends are also presented for both h$$ h $$‐ and p$$ p $$‐refinement methods. The results demonstrate the capability of the proposed formulation in reproducing accurate responses. [ABSTRACT FROM AUTHOR]

We study the dynamics of two bodies moving along an immobile straight line with regard for a finite speed of gravity. It is shown that the escape velocity is higher than the corresponding velocity in the classical celestial mechanics. We present estimates for this velocity. [ABSTRACT FROM AUTHOR]

We revisit in this note the Hénon's isochrone problem. By using the standard Abel inversion technique for one-dimensional motion, we recover in a simple way the Hénon's parabolae and get all isochrone central potentials under mild smoothness assumptions on the potential function. Our approach also allows us to conclude that isochronous radial periods with explicit energy dependence are necessarily Keplerian, i.e., T2 ∝ |E|−3, and that their corresponding orbits can be easily integrated by mapping them into the usual Kepler problem. It can also be employed to study some other inverse central-force problems and, in particular, it provides a proof of Bertrand's theorem. [ABSTRACT FROM AUTHOR]

We define a thermostatic system to be a convex space of states together with a concave function sending each state to its entropy, which is an extended real number. This definition applies to classical thermodynamics, classical statistical mechanics, quantum statistical mechanics, and also generalized probabilistic theories of the sort studied in quantum foundations. It also allows us to treat a heat bath as a thermostatic system on an equal footing with any other. We construct an operad whose operations are convex relations from a product of convex spaces to a single convex space and prove that thermostatic systems are algebras of this operad. This gives a general, rigorous formalism for combining thermostatic systems, which captures the fact that such systems maximize entropy subject to whatever constraints are imposed upon them. [ABSTRACT FROM AUTHOR]

Despite limited tools, Leonardo da Vinci displayed ingenious problem-solving. The authors examine a combination of Leonardo's thought and physical experiments regarding the acceleration of falling objects. Leonardo recorded that if a water-pouring vase moves transversally (sideways), mimicking the trajectory of a vertically falling object, it generates a right (as in orthogonal) triangle with equal leg length, composed of falling material lining up diagonally (forming the hypotenuse) and the vase trajectory forming one of the legs. On the hypotenuse, Leonardo wrote "Equatione di Moti," or equalization of motions, noting the equivalence of the two orthogonal motions, one effected by gravity and the other prescribed by the experimenter. The authors present an analytical solution using Newtonian mechanics to confirm Leonardo's "Equivalence principle." [ABSTRACT FROM AUTHOR]

We provide an alternative approach to a problem posed in the article “Hourglass of Constant Weight” by Becker and Pöschel (Granul Matter 10(3):231–232, 2008. ), i.e. 14 years ago. Our goal to return to this article after so much time is purely pedagogical. Although there is a slight and subtle inaccuracy in the derivation of equation (8) in the work of Becker and Pöschel, which we correct, we use this circumstance only to give a detailed pedagogical exposition of an excellent physical problem and to show the pedagogical usefulness of the concept of momentum flow when considering the mechanics of systems with variable mass. [ABSTRACT FROM AUTHOR]

A gas in a box is perhaps the most important model system studied in thermodynamics and statistical mechanics. Usually, studies focus on the gas, whereas the box merely serves as an idealized confinement. The present article focuses on the box as the central object and develops a thermodynamic theory by treating the geometric degrees of freedom of the box as the degrees of freedom of a thermodynamic system. Applying standard mathematical methods to the thermodynamics of an empty box allows equations with the same structure as those of cosmology and classical and quantum mechanics to be derived. The simple model system of an empty box is shown to have interesting connections to classical mechanics, special relativity, and quantum field theory. [ABSTRACT FROM AUTHOR]

Classical statistical mechanics has long relied on assumptions such as the equipartition theorem to understand the behavior of the complicated systems of many particles. The successes of this approach are well known, but there are also many well-known issues with classical theories. For some of these, the introduction of quantum mechanics is necessary, e.g., the ultraviolet catastrophe. However, more recently, the validity of assumptions such as the equipartition of energy in classical systems was called into question. For instance, a detailed analysis of a simplified model for blackbody radiation was apparently able to deduce the Stefan–Boltzmann law using purely classical statistical mechanics. This novel approach involved a careful analysis of a "metastable" state which greatly delays the approach to equilibrium. In this paper, we perform a broad analysis of such a metastable state in the classical Fermi–Pasta–Ulam–Tsingou (FPUT) models. We treat both the α -FPUT and β -FPUT models, exploring both quantitative and qualitative behavior. After introducing the models, we validate our methodology by reproducing the well-known FPUT recurrences in both models and confirming earlier results on how the strength of the recurrences depends on a single system parameter. We establish that the metastable state in the FPUT models can be defined by using a single degree-of-freedom measure—the spectral entropy (η)—and show that this measure has the power to quantify the distance from equipartition. For the α -FPUT model, a comparison to the integrable Toda lattice allows us to define rather clearly the lifetime of the metastable state for the standard initial conditions. We next devise a method to measure the lifetime of the metastable state t m in the α -FPUT model that reduces the sensitivity to the exact initial conditions. Our procedure involves averaging over random initial phases in the plane of initial conditions, the P 1 - Q 1 plane. Applying this procedure gives us a power-law scaling for t m , with the important result that the power laws for different system sizes collapse down to the same exponent as E α 2 → 0 . We examine the energy spectrum E (k) over time in the α -FPUT model and again compare the results to those of the Toda model. This analysis tentatively supports a method for an irreversible energy dissipation process suggested by Onorato et al.: four-wave and six-wave resonances as described by the "wave turbulence" theory. We next apply a similar approach to the β -FPUT model. Here, we explore in particular the different behavior for the two different signs of β. Finally, we describe a procedure for calculating t m in the β -FPUT model, a very different task than for the α -FPUT model, because the β -FPUT model is not a truncation of an integrable nonlinear model. [ABSTRACT FROM AUTHOR]

The range of motion of a particle with certain energy E confined in a potential is determined from the energy conservation law in classical mechanics. The counterpart of this question in quantum mechanics can be regarded as what is the possible range of expectation values of the position operator 〈 x 〉 of a particle that satisfies E = 〈 H 〉. This range depends on the state of the particle, but the universal upper and lower bounds, which are independent of the state, must exist. In this study, we show that these bounds can be derived by using the bootstrap method. We also point out that the bootstrap method can be regarded as a generalization of the uncertainty relations, meaning that the bounds are determined by the uncertainty relations in a broad sense. Furthermore, the bounds on possible expectation values of various quantities other than position can be determined in the same way. However, in the case of multiple identical particles (bosons and fermions), we find some difficulty in the bootstrap method. Because of this issue, the predictive power of the bootstrap method in multi-particle systems is limited in the derivation of observables including energy eigenstates. In addition, we argue an application of the bootstrap method to thermal equilibrium states. We find serious issues that temperature and entropy cannot be handled. Although we have these issues, we can derive some quantities in micro-canonical ensembles of integrable systems governed by generalized Gibbs ensembles. [ABSTRACT FROM AUTHOR]

In this work, a dynamic model is proposed to simulate the drilling and steering process of an autonomous burrowing mole to access scientific samples from the deep subsurface of the Moon. The locomotive module is idealized as a rigid rod. The characteristic parameters are considered, including the length, cross-section diameter, and centroid of a cylindrical rod. Based on classical Lagrangian mechanics, a 3-DOF dynamic model for the locomotion of this autonomous device is developed. By introducing resistive force theory, the interaction scheme between the locomotive body and the lunar regolith is described. The effects of characteristic parameters on resistive forces and torques are studied and discussed. Proportional-derivative control strategies are introduced to calculate the tracking control forces following a planned trajectory. The simulation results show that this method provides a reliable manipulation of a mole-type robot to avoid obstacles during the tracking control process in layered sediments. Overall, the proposed reduced-order model is able to simulate the operating and controlling scenarios of an autonomous burrowing robot in lunar subsurface environments. This model provides intuitive inputs to plan the space missions of a drilling robot to extract subsurface samples on an extraterrestrial planet such as the Moon or Mars. [ABSTRACT FROM AUTHOR]

This work is devoted to modeling the dynamics of large molecules. The key issue in modeling the dynamics of real molecular systems is to correctly represent the temperature of the system using the available theoretical tools. In most works on molecular dynamics, vibrations of atoms inside a molecule are modeled with enviable persistence, which has nothing to do with physical temperature. These vibrations represent the energy internal to the molecule. Therefore, it should not be present in problems in the dynamics of inert molecular systems. In this work, by means of classical mechanics, it is shown that the simplest system containing only three molecular bodies, due to multiple acts of pair interactions of these bodies, reproduces the temperature even in an extremely complex unstable motion of the system. However, at the same time, it is necessary to separate the stochastic part of the movement from the deterministic one. Calculations also show that translational fluctuations in the motion of molecules make the greatest contribution to temperature. The contribution of rotational energy to the total energy of fluctuation motions is small. It follows from these results that the thermal state of the system is determined only by the translational temperature. The latter, in turn, opens up possibilities for a simplified description of many complex systems composed of carbon molecules such as fullerenes and nanotori. [ABSTRACT FROM AUTHOR]

Bertrand theorem's states that, among central-force potentials with bound orbits, there are only two types of central-force scalar potentials with the property that all bound orbits are also closed orbits: the inverse-square law and Hooke's law. These solutions are considered basic examples in classical mechanics since they help in understanding the regular and predictable motion of bodies and superintegrable dynamical systems. However, there are strong beliefs that other potentials may arise in dynamical systems which are not predicted by Bertrand's theorem. Besides, several dynamical systems such as the solar system are characterized by chaotic and unbounded orbits which are not predicted by Bertrand's theorem. In this work, we prove an extension of Bertrand's theorem by means of non-standard Lagrangians and show the existence of a family of solutions for chaotic unstable periodic orbits. [ABSTRACT FROM AUTHOR]

This study clarifies a doubt on the validity of using roller tests rigs for evaluating the dynamics of rail vehicles running on straight tracks. The tested vehicle has no forward speed and stands on a wheel equipped with rails of which the tangential constant velocity, V, simulates the real vehicle velocity. Using this test method assumes that bogie dynamics depend only on contact forces. Recently, ref. [1] argue that, due to the lack of forward speed, roller test rigs are lacking inertial 'centrifugal' forces. As these forces cannot be calculated on test rigs where the vehicle forward speed is zero, hence comes the validity question. We answer by developing an imaginary scenario where a mass, representing a railway wheelset, is associated to a spring forming a pendulum oscillating along the normal to the vehicle speed. The paper demonstrates that the calculated inertial forces are split into two forces, normal and tangential to the trajectory, and that, although each one of these two forces depends on V, only their resultant, which does not depend on V, has a mechanical meaning and does not contradict classical mechanics which support the validity of test rigs. A numerical example supports this conclusion. [ABSTRACT FROM AUTHOR]

It is advisable to avoid and, even better, demystify such grandiose terms as "infinity" or "singularity" in the description of the cosmos. Its proliferation does not positively contribute to the understanding of key concepts that are essential for an updated account of its origin and evolutionary history. It will be here argued that, as a matter of fact, there are no infinities in physics, in the real world: all that appears, in any given formulation of nature by means of mathematical equations, actually arises from extrapolations, which are made beyond the bounds of validity of the equations themselves. Such a crucial point is rather well known, but too often forgotten, and is discussed in this paper with several examples; namely, the famous Big Bang singularity and others, which appeared before in classical mechanics and electrodynamics, and notably in the quantization of field theories. A brief description of the Universe's history and evolution follows. Special emphasis is put on what is presently known, from detailed observations of the cosmos and, complementarily, from advanced experiments of very high-energy physics. To conclude, a future perspective on how this knowledge might soon improve is given. [ABSTRACT FROM AUTHOR]