Problems In Classical Mechanics Research Articles

Machian fractons, Hamiltonian attractors, and nonequilibrium steady states

We study the N fracton problem in classical mechanics, with fractons defined as point particles that conserve multipole moments up to a given order. We find that the nonlinear Machian dynamics of the fractons is characterized by late-time attractors in position-velocity space for all N, despite the absence of attractors in phase space dictated by Liouville's theorem. These attractors violate ergodicity and lead to nonequilibrium steady states, which always break translational symmetry, even in spatial dimensions where the Hohenberg-Mermin-Wagner-Coleman theorem for equilibrium systems forbids such breaking. We provide a conceptual understanding of our results using an adiabatic approximation for the late-time trajectories and an analogy with the idea of “order-by-disorder” borrowed from equilibrium statistical mechanics. Altogether, these fracton systems host a paradigm for Hamiltonian dynamics and nonequilibrium many-body physics. Published by the American Physical Society 2024

Magic in generalized Rokhsar-Kivelson wavefunctions

Magic is a property of a quantum state that characterizes its deviation from a stabilizer state, serving as a useful resource for achieving universal quantum computation e.g., within schemes that use Clifford operations. In this work, we study magic, as quantified by the stabilizer Renyi entropy, in a class of models known as generalized Rokhsar-Kivelson systems, i.e., Hamiltonians that allow a stochastic matrix form (SMF) decomposition. The ground state wavefunctions of these systems can be written explicitly throughout their phase diagram, and their properties can be related to associated classical statistical mechanics problems, thereby allowing powerful analytical and numerical approaches that are not usually available in conventional quantum many body settings. As a result, we are able to express the SRE in terms of wave function coefficients that can be understood as a free energy difference of related classical problems. We apply this insight to a range of quantum many body SMF Hamiltonians, which affords us to study numerically the SRE of large high-dimensional systems, and in some cases to obtain analytical results. We observe that the behaviour of the SRE is relatively featureless across quantum phase transitions in these systems, although it is indeed singular (in its first or higher order derivative, depending on the nature of the transition). On the contrary, we find that the maximum of the SRE generically occurs at a cusp away from the quantum critical point, where the derivative suddenly changes sign. Furthermore, we compare the SRE and the logarithm of overlaps with specific stabilizer states, asymptotically realised in the ground state phase diagrams of these systems. We find that they display strikingly similar behaviors, which in turn establish rigorous bounds on the min-relative entropy of magic.

Perturbative stability analysis of deformable charged bodies revealing the mathematical origin of Planck constant

This article considers elementary particles like electrons as finitely sized deformable bodies and analyzes the classical mechanics of their interior continuum. The consequent findings answer a long-standing question in physics by computing Planck constant mathematically. It also explains quantum phenomena like wave-particle duality, uncertainty principle and atomic stability without resorting to phenomenological relations like Planck’s law or Bohr’s postulates. The analysis introduces the criteria for perpetual stability under arbitrary perturbations to compensate for the lack of system-defining initial conditions typically available in a classical mechanics problem. Accordingly, unperturbed leading order field solutions are first constructed from steady rotation and axisymmetric electromagnetic potentials. Then, novel perturbation theories uniquely render the particulate geometry along with mass and charge distributions by exploiting the stability conditions. The approach reveals the perpetually stable body to have surface charges encapsulating a slender disk with a radius-to-thickness ratio 51.36 and a rim velocity of 0.09798c. The volumetric densities for rest-mass and charge are seen to vary with dimensionless radius (r) as (1−r2)−3/2 and (1−r2)−2 . The theory infers that the deformable system with many degrees of freedom would exhibit several pulsating modes. One such vibration, named quantum mode in this paper, causes spontaneous oscillation of the particulate center in the equatorial plane with a unique time period of oscillation. This wavy motion manifests wave-particle duality, whereas its probabilistic amplitude is identified as the reason behind quantum uncertainties. Moreover, in response to external fields, the charge deforms to create dipoles which can nullify radiation-inducing Poynting vector to ensure atomic stability. Finally, the Planck constant is retrieved after dividing the rest-energy by the frequency of the quantum mode. Thus, the paper concludes that a new detailed mechanics can potentially describe subatomic physics if charges are perceived as perpetually stable but readily deformable finite bodies, not rigid shape-less entities.

Chaotic Phenomena for Generalised N-centre Problems

We study a class of singular dynamical systems which generalise the classical N-centre problem of Celestial Mechanics to the case in which the configuration space is a Riemannian surface. We investigate the existence of topological conjugation with the archetypal chaotic dynamical system, the Bernoulli shift. After providing infinitely many geometrically distinct and collision-less periodic solutions, we encode them in bi-infinite sequences of symbols. Solutions are obtained as minimisers of the Maupertuis functional in suitable free homotopy classes of the punctured surface, without any collision regularisation. For any sufficiently large value of the energy, we prove that the generalised N-centre problem admits a symbolic dynamics. Moreover, when the Jacobi-Maupertuis metric curvature is negative, we construct chaotic invariant subsets.

Optimal lamella geometry for mixed flow dryers

Drying harvested grain crops prior to storage is a crucial task in the prevailing climatic conditions of Europe. Drying is an extremely energy-intensive process. Its inappropriate application leads to environmental pollution, quality deterioration, and ultimately significant financial losses. Various methods are available for conducting drying operations, with the mixed flow dryer being one of the most employed approach. The mixed flow dryer utilizes air blower systems to redirect the flow of the granulate. Previous research has indicated that uneven distribution of grain flow around the air blower lamellae can cause drying irregularities. By leveraging insights from a long-established classical mechanical problem (the Brachistochrone problem) and harnessing the explicit dynamical modelling capabilities offered by contemporary computing technology (Discrete element method), we have successfully devised an optimized lamella geometry that minimizes the non-uniformity of particle flow.

Least-action variational continuum description of Darcian permeation of liquids and gases through stone materials

Watertightness of natural or artificial stone materials, like trap rock or concrete, is a basic property for assessing durability of the built heritage to weathering exposure. Fundamental research of the past century has shown that a reliable experimental determination of intrinsic permeability in stone materials can be achieved by uniaxial water-permeation tests under open-flow conditions, together with the opportunity of employing air permeation tests in place of water-tests to obtain important additional information on a given stone material. Correlations of practical relevance in such laboratory applications, among inlet pressure, outlet pressure and flow rate, stem all from solutions to the general continuum modelling problem of the permeation through a non-deformable homogeneous filter of an incompressible or compressible fluid under stationary uniaxial conditions. The present study addresses such class of mechanical problems on a theoretical variational basis to infer general correlations of practical relevance in permeability testing with a specific consideration of applications in which air-permeability testing combined with use of thinner filters makes the hypothesis of isothermal process in the permeating gas not straightforwardly applicable.

Design of cycloidal rays in optical waveguides in analogy to the fastest descent problem

In this work, we present the design of cycloidal waveguides from a gradient refractive index (GRIN) medium in analogy to the fastest descent problem in classical mechanics. Light rays propagate along cycloids in this medium, of which the refractive index can be determined from relating to the descending speed under gravity force. It can be used as GRIN lenses or waveguides, and the frequency specific focusing and imaging properties have been discussed. The results suggest that the waveguide can be viewed as an optical filter. Its frequency response characteristics change with the refractive index profile and the device geometries.

Analysis of ill-conditioned cases of a mass moving on a sphere with friction

Previous work treated the problem of a mass sliding over a rough spherical surface in broad generality, providing both analytic and numerical solutions. This paper examines special cases of 2D motion along a surface meridian when the initial speed is precisely chosen so that the sliding mass nearly stops before speeding up and subsequently leaving the surface. Carrying the solution for these critical cases into the time domain via both an analytical method and numerical integration adds richness that might otherwise be missed. The numerical method raises practical mathematical issues that must be handled carefully to obtain accurate results. Although conceptually simple, this classical mechanics problem is an excellent vehicle for students to gain proficiency with mathematical analysis tools and further their appreciation for how applied mathematics can bring new insight into situations where intuition may fall short.

Dirac equation from stereographic projection of the momentum sphere

The Dirac equation is commonly demonstrated under stringent hypotheses and after considerable math work made in relativistic quantum mechanics and quantum field theory. Here, a purely geometric approach free from hypotheses is suggested. The suggestion draws inspiration from the technique of stereographic projection that was developed before the quantum era to solve gyroscopic problems of classical mechanics. The projected variable is the generalized (or canonical) momentum vector. Its undetermined geometric orientations define a sphere in the momentum space and the projection onto the equatorial plane generates the Pauli matrices as soon as the conventional stereographic matrix is introduced. The associated eigenvalue problem results in the Dirac equation and the eigenvector (bispinor) has components that are related to geometric elements of the momentum space. The procedure has the advantage of revealing the correct form of the Dirac matrices without the mathematical effort that characterizes the presentation of the equation in traditional approaches. The other remarkable advantage is that, unlike the common reduction to the case of free space, the spatial inhomogeneity due to interaction potentials is included in the demonstration from the very beginning. The whole suggestion has an interdisciplinary character (relativity, complex analysis, rotation of rigid bodies, Pauli matrices) and can be useful in teaching the equation to students who lack in sufficient knowledge of quantum mechanics. Students equipped with more advanced education can benefit from the purely geometric perspective of this work if used to complement their studies about the equation.

Selected Problems of Classical and Modern Celestial Mechanics and Stellar Dynamics: II–Modern Studies

Selected Problems of Classical and Modern Celestial Mechanics and Stellar Dynamics: II–Modern Studies

Selected Problems of Classical and Modern Celestial Mechanics and Stellar Dynamics: II–Modern Studies

A review is given, in the modern context of applications, of the major important scientific results obtained by the scientists and graduates of St. Petersburg State University in the field of celestial mechanics and stellar dynamics. The second part of the review covers the following topics: estimates and calculation of the MOID parameter, problems of asteroid–comet hazard, dust complexes in the Solar System, rotational dynamics of planetary satellites, circumbinary dynamics, and methods for the discovery and determination of orbits of exoplanets.

Selected Problems of Classical and Modern Celestial Mechanics and Stellar Dynamics: I–Classical Results

Selected Problems of Classical and Modern Celestial Mechanics and Stellar Dynamics: I–Classical Results

Selected Problems of Classical and Modern Celestial Mechanics and Stellar Dynamics: I–Classical Results

A review is given, in the modern context of applications, of the major important scientific results obtained by scientists and graduates of St. Petersburg State University in the field of celestial mechanics and stellar dynamics. The following topics are discussed: the Antonov laws of stellar dynamics, Abalakin–Batrakov libration points, Kholshevnikov metrics, Agekyan–Anosova homological region, Orlov metastable triple systems, Ogorodnikoff–Milne models, Ossipkov–Merritt models, estimation and calculation of the MOID parameter, photogravitational celestial mechanics and solar sail, problems of asteroid–comet hazard, dust complexes in the Solar System, rotational dynamics of planetary satellites, circumbinary dynamics, and methods for the discovery and determination of orbits of exoplanets. The first part of the review presents the classical results.

Deep energy method in topology optimization applications

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.

A geometric classification of Hamiltonian equilibria by spectral types

We give one algebraic equation and two semialgebraic equations whose corresponding semi-algebraic varieties are associated to definite types of bifurcations for parameter-dependent Hamiltonian systems. These equations provide a geometric classification of the spectral types of equilibria of a given m-degrees of freedom Hamiltonian system. In fact the complement of such varieties is the disconnected union of open domains (in parameter space) in which the spectral type of the equilibrium does not change. We analyse such varieties thoroughly, in the space of invariants, for generic systems with 2,3,4 degrees of freedom. This technique can be applied to any parameter-dependent Hamiltonian system, and it does give the bifurcation-decomposition of parameter space. We conclude with applications to two problems in classical mechanics.

Crises and chaotic scattering in hydrodynamic pilot-wave experiments.

Theoretical foundations of chaos have been predominantly laid out for finite-dimensional dynamical systems, such as the three-body problem in classical mechanics and the Lorenz model in dissipative systems. In contrast, many real-world chaotic phenomena, e.g., weather, arise in systems with many (formally infinite) degrees of freedom, which limits direct quantitative analysis of such systems using chaos theory. In the present work, we demonstrate that the hydrodynamic pilot-wave systems offer a bridge between low- and high-dimensional chaotic phenomena by allowing for a systematic study of how the former connects to the latter. Specifically, we present experimental results, which show the formation of low-dimensional chaotic attractors upon destabilization of regular dynamics and a final transition to high-dimensional chaos via the merging of distinct chaotic regions through a crisis bifurcation. Moreover, we show that the post-crisis dynamics of the system can be rationalized as consecutive scatterings from the nonattracting chaotic sets with lifetimes following exponential distributions.

Solution for Simple Classical Systems Using SCILAB

The use of computers as a learning tool has been a time-tested method, which can attract the interest and motivation of students to solve any complicated system easily. Scilab is an open-source computational software package that provides various tools to simulate and solve problems in classical mechanics. In this paper, two types of classical mechanics problems are solved. Under kinematics section, we solve the Projectile motion considering air resistance and particle falling under the action of gravity, with resistive force dependence on different exponents of velocity. Similarly, under the waves superposition section, we solve the two sinusoidal waves superimposed to give a Lissajous pattern and superposition of two or more waves using Scilab software for a better understanding of the classical problems by visualization of graphs, as Scilab has 2D and 3D plotting facilities.

A novel multi-objective optimization method with local search scheme using shuffled complex evolution applied to mechanical engineering problems

PurposeIn this work, the multi-objective optimization shuffled complex evolution is proposed. The algorithm is based on the extension of shuffled complex evolution, by incorporating two classical operators into the original algorithm: the rank ordering and crowding distance. In order to accelerate the convergence process, a Local Search strategy based on the generation of potential candidates by using Latin Hypercube method is also proposed.Design/methodology/approachThe multi-objective optimization shuffled complex evolution is used to accelerate the convergence process and to reduce the number of objective function evaluations.FindingsIn general, the proposed methodology was able to solve a classical mechanical engineering problem with different characteristics. From a statistical point of view, we demonstrated that differences may exist between the proposed methodology and other evolutionary strategies concerning two different metrics (convergence and diversity), for a class of benchmark functions (ZDT functions).Originality/valueThe development of a new numerical method to solve multi-objective optimization problems is the major contribution.

New Directions in Modeling and Computational Methods for Complex Mechanical Dynamical Systems

This paper presents a new conceptualization of complex nonlinear mechanical systems and develops new and novel computational methods for determining their response to given applied forces and torques. The new conceptualization uses the idea of including particles of zero mass to describe the dynamics of such systems. This leads to simplifications in the development of their equations of motion and engenders a straightforward new computational approach to simulate their behavior. The purpose of the paper is to develop a new analytical and computational methodology to handle complex systems and to illustrate it through the study of an old unsolved problem in classical mechanics, that of a non-uniform rigid spherical shell rolling, without slipping, under gravity on an arbitrary dimpled bowl-shaped rigid surface. The new conceptualization provides the explicit equations of motion for the system, the analytical determination of the reaction forces supplied by the surface, and a straightforward computational approach to simulate the dynamics. Detailed analytical and numerical results are provided. The computations illustrate the complexity of the dynamical behavior of the system and its high sensitivity to the initial orientation of the shell and to the presence of any initial angular velocity normal to the surface.

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.