Problems In Classical Mechanics Research Articles

Physics-constrained Bayesian inference of state functions in classical density-functional theory.

We develop a novel data-driven approach to the inverse problem of classical statistical mechanics: Given the experimental data on the collective motion of a classical many-body system, how does one characterize the free energy landscape of that system? By combining non-parametric Bayesian inference with physically motivated constraints, we develop an efficient learning algorithm that automates the construction of approximate free-energy functionals. In contrast to optimization-based machine learning approaches, which seek to minimize a cost function, the central idea of the proposed Bayesian inference is to propagate a set of prior assumptions through the model, derived from physical principles. The experimental data are used to probabilistically weigh the possible model predictions. This naturally leads to humanly interpretable algorithms with full uncertainty quantification of predictions. In our case, the output of the learning algorithm is a probability distribution over a family of free energy functionals, consistent with the observed particle data. We find that surprisingly small data samples contain sufficient information for inferring highly accurate analytic expressions of the underlying free-energy functionals, making our algorithm highly data efficient. In particular, we consider classical particle systems with excluded volume interactions, which are ubiquitous in nature, while being highly challenging in terms of free energy modeling. We validate our approach on the paradigmatic case of one-dimensional fluid and develop inference algorithms for the canonical and grand-canonical statistical-mechanical ensembles. Extensions to higher dimensional systems are conceptually straightforward, while standard coarse-graining techniques allow one to easily incorporate attractive interactions.

Hilbert space fragmentation produces an effective attraction between fractons

Fracton systems exhibit restricted mobility of their excitations due to the presence of higher-order conservation laws. Here we study the time evolution of a one-dimensional fracton system with charge and dipole moment conservation using a random unitary circuit description. Previous work has shown that when the random unitary operators act on four or more sites, an arbitrary initial state eventually thermalizes via a universal subdiffusive dynamics. In contrast, a system evolving under three-site gates fails to thermalize due to strong ``fragmentation'' of the Hilbert space. Here we show that three-site gate dynamics causes a given initial state to evolve toward a highly nonthermal state on a timescale consistent with Brownian diffusion. Strikingly, the dynamics produces an effective attraction between isolated fractons or between a single fracton and the boundaries of the system, as in the Casimir effect of quantum electrodynamics. We show how this attraction can be understood by exact mapping to a simple classical statistical mechanics problem, which we solve exactly for the case of an initial state with either one or two fractons.

Задача устойчивости криволинейной арки малого подъема из пористого материала при случайном ее нагружении

Classical mechanics of deformable bodies is based on the continuity hypothesis. However, many structural elements are made of porous materials. Porous natural materials (soil, rocks) have invariable porosity. Porous synthetic materials (ceramics, concrete, graphite, and pressed powder metals) have controlled porosity. To calculate the strength and hardness of the structure, the material is assumed to be conditionally continuous with the adjusted porosity. Nowadays, there are many available works presenting mechanical characteristics of materials with different porosities. This paper proposes a new class of problems in mechanics of deformable solids. Considering a low arch stability problem, which is important in construction practice, the problem of optimal arch design is solved by controlling the properties of the material. The solution to the problem of stability of the low arch made of porous material is presented. The flat arch with a rectangular crosssection is exposed to equally distributed loading. The near-rational law of the porosity distribution over the cross-section is used. The load is considered as a random variable. The solution to the problem is obtained using the theory of stationary random processes. A comparative analysis of the reliability and material consumption is carried out for the arch with continuous and porous sections. The calculation shows that the porous structure of the arch reduces the material consumption by 13.3% without stability and reliability losses.

Modeling the dynamics of a wedge pair under the action of a constant force

The subject of research is the dynamics of a wedge pair under the influence of a constant force. The aim of the article is to derive a differential equation of longitudinal vibrations of a wedge pair; a partial solution to this equation of longitudinal vibrations of a wedge pair under the influence of a constant force corresponding to the given initial and boundary conditions; a general solution of the differential equation of longitudinal vibrations of a wedge pair under the action of a constant force. Research methods are based on the d’Alembert principle, which is a method for solving problems of classical mechanics; on the method of mathematical modeling and the analytical method for their solution. The following aspects were obtained in the article for the longitudinal vibrations of a wedge pair: the differential equation of the longitudinal vibrations of a wedge pair by the method of force addition between two infinitesimal sections of the rod in accordance with the d’Alembert principle; a partial solution of the differential equation of longitudinal vibrations of a wedge pair under the action of a constant force corresponding to the given initial and boundary conditions; a general solution of the differential equation of longitudinal vibrations of a wedge pair under the action of a constant force.

Contradictions in the Plane Contact Problem of the Theory of Elasticity on the Compression of Cylinders in Contact with Parallel Generators

The inapplicability of the reference formula for determining the convergence of two statically compressed parallel cylinders made of a homogeneous, isotropic and physically linear material has been proved due to a well-known logarithmic feature in the plane classical problem of mechanics of elastic solids. In the special case of the elastic interaction of a cylinder with a half-plane, when one of the radii has an infinite length, it has been found that the convergence also becomes equal to infinity. This paradoxical result contradicts not only the physical and mechanical meaning of the process under study, but also confirms the inadequacy of Flamant model of a simple radial stress state in determining displacements. The authors have proposed an algorithm for eliminating the contradictions based on the solution of Fredholm integral equation of the first kind. In the future, it can be considered as a new fundamental and applied problem of the theory of elasticity, which is of a great importance for a refined assessment of the contact strength and stiffness of the cylindrical parts of load-bearing structures taking into account the general and local deformations (cylindrical rollers, gears, road surfaces, when they are compacted with steel rollers, etc.) on the basis of a flat Flamant calculation scheme considering three stress components and the width of the cylinder contact area previously developed and mathematically approximated by the authors.

An exact solution for a particle in a velocity-dependent force field

We revisit the classical mechanics problem of a particle moving under the influence of a force that depends on its velocity. Using the properties of the rotation matrix and associated operators, we show that it is possible to find an exact analytical solution to a number of problems where the differential equation of motion depends on the velocity. First, we apply our method to the well-known cases of a particle under the influence of the Lorentz force and Coriolis force, providing the complete analytical solution in each case for the motion of the particle in three dimensions. We also show that the complete solution can be obtained when the centrifugal force is included, showing the applicability to cases where there is simultaneous dependence on the position and velocity. This method, which is not currently discussed in a typical course in elementary mechanics, provides an alternative approach to the traditional methods that are used to solve these types of problems.

Solution of mass-spring-damper fractional systems using Caputo derivative and orthogonal collocation

PurposeIn this contribution, the solution of Mass-Spring-Damper Systems in the fractional context by using Caputo derivative and Orthogonal Collocation Method is investigated. For this purpose, different case studies considering constant and periodic sources are evaluated. The dimensional consistency of the model is guaranteed by introducing an auxiliary parameter. The obtained results are compared with those found by using both the analytical solution and the predictor-corrector method of Adams–Bashforth–Moulton type. The influence of the fractional order on the mechanical system is evaluated.Design/methodology/approachIn the present contribution, an extension of the Orthogonal Collocation Method to solve fractional differential equations is proposed.FindingsIn general, the proposed methodology was able to solve a classical mechanical engineering problem with different characteristics.Originality/valueThe development of a new numerical method to solve fractional differential equations is the major contribution.

Exact Closed-Form Solutions of the Motion in Non-Inertial Reference Frames, Using the Properties of Lie Groups SO3 and SE3

The paper offers a general symbolic method to study the motion in a non-inertial reference frame. In order to achieve this, we use the algebraic and geometric properties of the Lie group of special orthogonal tensors, SO3, and the Lie group of the rigid body displacements, SE3. We obtain a simplified form of the initial value problem that models the non-inertial motion using a tensor instrument introduced in this paper. Thus, the study of the motion in a non-inertial reference frame is transferred into the study of a classical motion in an inertial reference frame. The applications of this method refer to solving the relative motion problem and deriving the straightforward solution to classical theoretical mechanics problems. The motion in a uniform gravitational force field in a rotating reference frame, the motion of a charged particle in non-stationary electric and magnetic fields, the exact solution of the relative rigid body motion in the non-inertial reference frame are studied. Using this symbolic method in studying the motion in a non-inertial reference frame reduces the number of computations. In addition, it offers, in some essential particular cases, exact closed-form coordinate-free analytical solutions.

Analyzing the motion of a mass sliding on a sphere with friction

A well-known problem in classical mechanics that is often presented for pedagogical purposes involves a small mass that slides without friction under a gravitational force on the surface of a sphere. Commonly, students are asked to find the angular position where a mass with no azimuthal motion leaves the spherical surface, and this question is easily within the reach of most intermediate physics students. However, a complete solution for more general motion of the mass on the spherical surface (including friction) may be suitable for many advanced undergraduates. Without friction, the problem including azimuthal motion is really an inverted version of an ideal spherical pendulum. This problem is also useful for extending discussion in classical mechanics to more sophisticated topics beyond solving Newton's laws of motion, such as the importance of conservation laws and constants of motion as they relate to symmetry, conservative versus dissipative forces, and the role of constraints.

Analysis of Cycloid features by video analysis

We analyze and discuss the agreement between the experimental data obtained with video analysis and the theoretical calculations for the classical problem of brachistochrone and tautochrone cycloids. We use the open source software Tracker to examine video segments with simultaneous records of both straight and cycloid ramp to taken experimental data and compare them to the theoretical study. The results are in accordance with the theoretical description. The study of this classical problem is a proof of concept of the reliability and accuracy of the method that can be applied in other classical mechanics problems.

On the topological enrichment for crack modeling via the generalized/extended FEM: a novel discussion considering smooth partitions of unity

Purpose It has been usual to prefer an enrichment pattern independent of the mesh when applying singular functions in the Generalized/eXtended finite element method (G/XFEM). This choice, when modeling crack tip singularities through extrinsic enrichment, has been understood as the only way to surpass the typical poor convergence rate obtained with the finite element method (FEM), on uniform or quasi-uniform meshes conforming to the crack. Then, the purpose of this study is to revisit the topological enrichment strategy in the light of a higher-order continuity obtained with a smooth partition of unity (PoU). Aiming to verify the smoothness' impacts on the blending phenomenon, a series of numerical experiments is conceived to compare the two GFEM versions: the conventional one, based on piecewise continuous PoU's, and another which considers PoU's with high-regularity. Design/methodology/approach The stress approximations right at the crack tip vicinity are qualified by focusing on crack severity parameters. For this purpose, the material forces method originated from the configurational mechanics is employed. Some attempts to improve solution using different polynomial enrichment schemes, besides the singular one, are discussed aiming to verify the transition/blending effects. A classical two-dimensional problem of the linear elastic fracture mechanics (LEFM) is solved, considering the pure mode I and the mixed-mode loadings. Findings The results reveal that, in the presence of smooth PoU's, the topological enrichment can still be considered as a suitable strategy for extrinsic enrichment. First, because such an enrichment pattern still can treat the crack independently of the mesh and deliver some advantage in terms of convergence rates, under certain conditions, when compared to the conventional FEM. Second, because the topological pattern demands fewer degrees of freedom and impacts conditioning less than the geometrical strategy. Originality/value Several outputs are presented, considering estimations for the J–integral and the angle of probable crack advance, this last computed from two different strategies to monitoring blending/transition effects, besides some comments about conditioning. Both h- and p-behaviors are displayed to allow a discussion from different points of view concerning the topological enrichment in smooth GFEM.

Entanglement of local operators and the butterfly effect

We study the robustness of quantum and classical information to perturbations implemented by local operator insertions. We do this by computing multipartite entanglement measures in the Hilbert space of local operators in the Heisenberg picture. The sensitivity to initial conditions that we explore is an illuminating manifestation of the butterfly effect in quantum many-body systems. We present a ``membrane theory'' in Haar random unitary circuits to compute the mutual information, logarithmic negativity, and reflected entropy in the local operator state by mapping to a classical statistical mechanics problem and find that any local operator insertion delocalizes information as fast as is allowed by causality after taking the large local Hilbert space dimension limit. Identical behavior is found for conformal field theories admitting holographic duals where the bulk geometry is described by the eternal black hole with a local object situated at the horizon. In contrast to these maximal scramblers, only an $O(1)$ amount of information is found to be delocalized by local operators in free fermionic systems and random Clifford circuits.

Pedal coordinates, solar sail orbits, Dipole drive and other force problems

Pedal coordinates provide a natural framework for studying force problems of classical mechanics in the plane. The trajectory of a test particle under the influence of central and Lorentz-like forces can be translated into pedal coordinates at once without the need of solving any differential equation. We generalize this result to cover more general force laws and also show the advantage of pedal coordinates in certain variational problems. These enable us to link together many dynamical systems as well as problems of the calculus of variation. Finally, we apply these results to compute orbits of Solar sail and Dipole drive.

Development of USA method (understanding, sketching, analysing) as practical way to resolving classical mechanics problems for physics lesson

This study aims to develop a problem solving method known as the USA method (Understanding, Sketching and analysing) as a practical method to solving classical mechanics problems. This research method uses the ADDIE Research and Development model (Analysis, Design, Develop, Implementation and Evaluation) with data analysis techniques using Aiken’s validation and reliability analysis. Research subjects in several physics subject teachers and science class students in West of Papua with the research stages used Problem Analysis and learning needs, USA method design, method development, implementation and evaluation of method. Researchers used 21 validators consisting of 13 validators from physics and mathematics experts who teach at universities and 8 physics teacher participants as practitioners who teach in schools. The results of this development indicate that the level of validity of all assessment indicators reaches 0.75 to 0.928 with a reliability of 0.82 where this interval is called high validity and reliability. The final step is to apply the USA method and then analyse the effectiveness and practicality of this method. The important results obtained by the researcher were 92.5% of respondents stated that the USA method was effective as a better method to supporting the quality of learning physics in classical mechanics lessons. Meanwhile 94.3% of respondents said the USA method was practical as an easy way to understand classical mechanics.

AdS/Deep-Learning made easy: simple examples * *Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2017R1A2B4004810, NRF-2021R1A2C1006791) and the GIST Research Institute (GRI) grant funded by GIST in 2021

Deep learning has been widely and actively used in various research areas. Recently, in gauge/gravity duality, a new deep learning technique called AdS/DL (Deep Learning) has been proposed. The goal of this paper is to explain the essence of AdS/DL in the simplest possible setups, without resorting to knowledge of gauge/gravity duality. This perspective will be useful for various physics problems: from the emergent spacetime as a neural network to classical mechanics problems. For prototypical examples, we choose simple classical mechanics problems. This method is slightly different from standard deep learning techniques in the sense that we not only have the right final answers but also obtain physical understanding of learning parameters.

Semiclassical limit of an inverse problem for the Schrödinger equation

It is a classical derivation that the Wigner equation, derived from the Schrödinger equation that contains the quantum information, converges to the Liouville equation when the rescaled Planck constant $$\varepsilon \rightarrow 0$$ . Since the latter presents the Newton’s second law, the process is typically termed the (semi-)classical limit. In this paper, we study the classical limit of an inverse problem for the Schrödinger equation. More specifically, we show that using the initial condition and final state of the Schrödinger equation to reconstruct the potential term, in the classical regime with $$\varepsilon \rightarrow 0$$ , becomes using the initial and final state to reconstruct the potential term in the Liouville equation. This formally bridges an inverse problem in quantum mechanics with an inverse problem in classical mechanics.

Atomistic two-, three- and four-body potentials. Spatial and material settings

In molecular dynamics or molecular statics (MD/MS) multi-body potentials empirically capture the energetic interactions in atomistic systems enabling the computation of the corresponding atomistic forces as energetic conjugates to the atomistic positions. We distinguish here between spatial and material atomistic positions and consequently between the corresponding spatial and material atomistic forces. In quasi-statics, i.e. MS, the former, also denoted as deformational atomistic forces, contribute to the classical deformational mechanics (i.e., equilibrium) problem that seeks to minimise the total potential energy of an atomistic system with respect to the atomistic positions relative to the ambient space. The latter, also denoted as configurational atomistic forces, contribute to the configurational mechanics (i.e., non-equilibrium) problem that determines the release of total potential energy of an atomistic system upon variation of the atomistic positions relative to the ambient material, i.e., due to perturbations of the material (initial) atomistic configuration. The importance of material atomistic forces is that they drive energetically favourable re-organisations of the material atomistic configuration, thereby characterising the tendency of generic atomistic defects to propagate. In this contribution we focus on two-, three-, and four-body potentials, whereby we distinguish between novel stretch- and classical angle-based potentials for the two latter cases. Taken together, as the main contribution, we derive expressions for the corresponding spatial and, for the first time, material atomistic forces and highlight their striking formal similarity. The derivations are detailed but the final expression compact and well-suited for numerical implementation.

Superintegrability of (2n + 1)-body choreographies, n = 1,2,3,…,∞ on the algebraic lemniscate by Bernoulli (inverse problem of classical mechanics)

For one 3-body and two 5-body planar choreographies on the same algebraic lemniscate by Bernoulli we found explicitly a maximal possible set of (particular) Liouville integrals, 7 and 15, respectively, (including the total angular momentum), which Poisson commute with the corresponding Hamiltonian along the trajectory. Thus, these choreographies are particularly maximally superintegrable. It is conjectured that the total number of (particular) Liouville integrals is maximal possible for any odd number of bodies [Formula: see text] moving choreographically (without collisions) along given algebraic lemniscate, thus, the corresponding trajectory is particularly, maximally superintegrable. Some of these Liouville integrals are presented explicitly. The limit [Formula: see text] is studied: it is predicted that one-dimensional liquid with nearest-neighbor interactions occurs, it moves along algebraic lemniscate and it is characterized by infinitely many constants of motion.

Rare extinction events in cyclic predator–prey games

In the May–Leonard model of three cyclically competing species, we analyze the statistics of rare events in which all three species go extinct due to strong but rare fluctuations. These fluctuations are from the tails of the probability distribution of species concentrations. They render a coexistence of three populations unstable even if the coexistence is stable in the deterministic limit. We determine the mean time to extinction (MTE) by using a WKB-ansatz in the master equation that represents the stochastic description of this model. This way, the calculation is reduced to a problem of classical mechanics and amounts to solving a Hamilton–Jacobi equation with zero-energy Hamiltonian. We solve the corresponding Hamilton’s equations of motion in six-dimensional phase space numerically by using the Iterative Action Minimization Method. This allows to project on the optimal path to extinction, starting from a parameter choice where the three-species coexistence-fixed point undergoes a Hopf bifurcation and becomes stable. Specifically for our system of three species, extinction events can be triggered along various paths to extinction, differing in their intermediate steps. We compare our analytical predictions with results from Gillespie simulations for two-species extinctions, complemented by an analytical calculation of the MTE in which the remaining third species goes extinct. From Gillespie simulations we also analyze how the distributions of times to extinction (TE) change upon varying the bifurcation parameter. Our results shed some light on the sensitivity of the TE to system parameters. Even within the same model and the same dynamical regime, which allows a stable coexistence of species in the deterministic limit, the MTE depends on the distance from the bifurcation point in a way that contains the system size dependence in the exponent. It is challenging and worthwhile to quantify how rare the rare events of extinction are.

Space\u2013time VMS isogeometric analysis of the Taylor\u2013Couette flow

The Taylor–Couette flow is a classical fluid mechanics problem that exhibits, depending on the Reynolds number, a range of flow patterns, with the interesting ones having small-scale structures, and sometimes even wavy nature. Accurate representation of these flow patterns in computational flow analysis requires methods that can, with a reasonable computational cost, represent the circular geometry accurately and provide a high-fidelity flow solution. We use the Space–Time Variational Multiscale (ST-VMS) method with ST isogeometric discretization to address these computational challenges and to evaluate how the method and discretization perform under different scenarios of computing the Taylor–Couette flow. We conduct the computational analysis with different combinations of the Reynolds numbers based on the inner and outer cylinder rotation speeds, with different choices of the reference frame, one of which leads to rotating the mesh, with the full-domain and rotational-periodicity representations of the flow field, with both the convective and conservative forms of the ST-VMS, with both the strong and weak enforcement of the prescribed velocities on the cylinder surfaces, and with different mesh refinements. The ST framework provides higher-order accuracy in general, and the VMS feature of the ST-VMS addresses the computational challenges associated with the multiscale nature of the flow. The ST isogeometric discretization enables exact representation of the circular geometry and increased accuracy in the flow solution. In computations where the mesh is rotating, the ST/NURBS Mesh Update Method, with NURBS basis functions in time, enables exact representation of the mesh rotation, in terms of both the paths of the mesh points and the velocity of the points along their paths. In computations with rotational-periodicity representation of the flow field, the periodicity is enforced with the ST Slip Interface method. With the combinations of the Reynolds numbers used in the computations, we cover the cases leading to the Taylor vortex flow and the wavy vortex flow, where the waves are in motion. Our work shows that all these ST methods, integrated together, offer a high-fidelity computational analysis platform for the Taylor–Couette flow and for other classes of flow problems with similar features.