Problems In Classical Mechanics Research Articles

A tale of two rhythms: Locked clocks and chaos in biology.

The fundamental mechanisms that control and regulate biological organisms exhibit a surprising level of complexity. Oscillators are perhaps the simplest motifs that produce time-varying dynamics and are ubiquitous in biological systems. It is also known that such biological oscillators interact with each other-for instance, circadian oscillators affect the cell cycle, and somitogenesis clock proteins in adjacent cells affect each other in developing embryos. Therefore, it is vital to understand the effects that can emerge from non-linear interaction between oscillations. Here, we show how oscillations typically arise in biology and take the reader on a tour through the great variety in dynamics that can emerge even from a single pair of coupled oscillators. We explain how chaotic dynamics can emerge and outline the methods of detecting this in experimental time traces. Finally, we discuss the potential role of such complex dynamical features in biological systems.

MODIFICATIONS OF THE FORMULA E = MC2 FOR SOLVING ENGINEERING PROBLEMS OF CLASSICAL MECHANICS

The formula E = mC2 is fundamental in Einsteinʼs special theory of relativity. It is given a new interpretation, where the operator C of transitions, which characterizes their speed, has a more complex mathematical structure and is controlled in contrast to a constant number, as is customary in relativistic mechanics. The possibility of using its proposed modifications related to the transition of mass to energy and energy to the mass of a physical body for solving engineering problems of classical mechanics, which is a special case of relativistic mechanics, is shown. As an example, the process of functioning of a machine which is a system of devices for converting energy and material, is considered. The efficiency of its functioning is evaluated through the operators of transitions of structural elements.

Isogeometric analysis of cracks with peridynamics

Isogeometric analysis (IGA) is an important mesh-free method that provides the technique for the integration of computer-aided design and analysis. However, its formulation is based on classical continuum mechanics and is not suitable for crack propagation problems. The peridynamics (PD) theory is based on the non-local integral equation, which avoids the discontinuous problem of classical continuum mechanics and can solve the crack problems. In this paper, a coupling model of IGA and bond-based PD (IGA-PD) is proposed to provide the solution to crack problem of model with exact geometry. We prove that in the IGA-PD coupling model, the PD nodes can be constructed by using the control net from IGA. The implicit and explicit formulas of the PD model are combined with IGA formulations based on the force balance principle. Through setting the isogeometric model on the domain near the boundaries, the surface effect of the PD model can be relieved. The IGA-PD model fully utilizes the advantages of non-local continuum theory and improves the precision and computational efficiency in calculation of crack problems. The coupling method is straightforward and effective, and can be integrated within existing IGA codes. Examples of static problems and crack propagations are given to prove the effectiveness of the proposed method.

On the semiclassical JWKB and MAF solutions of the uniformly lengthening pendulum via change of dependent variable in the Bessel’s equation

Semiclassical JWKB method and method of modified Airy functions (MAF) are the fundamental conventional tools in quantum mechanical and optical waveguide systems and yet their classical mechanical applicability has somehow escaped much attention. In this work, semiclassical JWKB and MAF solution of a classical mechanical problem regarding the Lengthening Pendulum (LP) is being studied to reinforce the power of the semiclassical methods in this aspect. Here, JWKB and MAF solutions of the associated Bessel’s equation is being obtained by transforming it first into one of the normal forms via change of dependent variable on the contrary to our previous work where change of independent variable was used for the JWKB solution. By the same motivation for the suggested change of dependent variable here, we once again, obtain advantageous results in comparison to the conventional numerical exact solutions given in the literature where adiabatic LP systems with very low lengthening rates are required. We obtain both JWKB and MAF approximated Bessel functions successfully and our semiclassical solutions for the LP system works with a great accuracy even for much higher lengthening rates in comparison to these conventional numerical methods. Here we study under what conditions our semiclassical solutions can give accurate results by using the conventional JWKB and MAF applicability criteria. In effect, the semiclassical JWKB and MAF methods which is normally known to be used in quantum mechanical and optical waveguide systems are being used to solve the pure classical-mechanical LP system successfully and advantageously here and the applicability criteria of these semiclassical methods in terms of the physical LP system parameters are being determined and presented.

Machine learning assisted stochastic-XFEM for stochastic crack propagation and reliability analysis

We present a novel stochastic extended finite element (S-XFEM) method for solving fracture mechanics problems under uncertainty. The proposed S-XFEM couples XFEM and a novel multi-output Gaussian process based machine learning algorithm, referred to as the hybrid polynomial correlated function expansion (H-PCFE). With this approach, the underlying stochastic fracture mechanics problem is decoupled into multiple classical fracture mechanics problems. Since solution of a classical fracture mechanics problem is computationally expensive, we utilize the H-PCFE model as a surrogate to emulate the behavior of the system. Training samples required for training the H-PCFE model are generated by using the XFEM. Because of the intrinsic capability of XFEM in providing a mesh insensitive solution, the proposed approach can provide reasonable result from a coarse mesh. On the other hand, H-PCFE is capable of providing highly accurate solution from very few training samples. Overall, the proposed S-XFEM is highly efficient in solving stochastic fracture mechanics problems. The proposed approach is used for solving three stochastic fracture mechanics problems. Different case studies involving reliability analysis and stochastic fracture propagation have been reported. The procedure yields highly accurate results for all problems, indicating its possible applications to other large scale systems.

Неупругое растяжение медного однопроволочного проводника при неограниченных местных деформациях и положительной температуре

Results of experimental and desktop studies of the single-wire copper conductor deformation under the overcurrent action are given in the article. The conductor was studied using the JSM-6390L reflection electron microscope. A mathematical model of the stress-strain state of the copper bar in tension and under a temperature below 700 °С has been developed on the basis of the classical nonlinear problem of the structural mechanics. The mechanical forces in a single-wire copper conductor, which cause neck formation during overcurrent flow, have been determined. The mathematical model has been simplified to simple analytical dependences providing their use in forensic fire and technical investigations.

Developing an application for stress analysis in the OpenFOAM software

The article contains the results of studying stress analysis in the OpenFOAM software and a process of developing a software GUI application for this analysis. The author defines the urgency for the development, compares the available software products in this field. A stack of technologies to develop a graphical interface source code is defined. The study resulted in the desktop GUI application for stress analysis in the OpenFOAM software. The results of the application tests using a classical continuum mechanics problem included in the platform distributive are given. The statements that have scientific novelty of the development and its practical use are formulated.

On the analytic representation of Newtonian systems

We show that the theory of self-adjoint differential equations can be used to provide a satisfactory solution of the inverse variational problem in classical mechanics. A Newtonian equation when transformed to the self-adjoint form allows one to find an appropriate Lagrangian representation (direct analytic representation) for it. On the other hand, the same Newtonian equation in conjunction with its adjoint provides a basis to construct a different Lagrangian representation (indirect analytic representation) for the system. We obtain the time-dependent Lagrangian of the damped Harmonic oscillator from the self-adjoint form of the equation of motion and at the same time identify the adjoint of the equation with the so called Bateman image equation with a view to construct a time-independent indirect Lagrangian representation. We provide a number of case studies to demonstrate the usefulness of the approach derived by us. We also present similar results for a number of nonlinear differential equations by using an integral representation of the Lagrangian function and make some useful comments.

A three-dimensional damage-based contact element model for simulating the interfacial behaviors of rocks and its validation and applications

To describe the mechanical behavior of interfaces existing in rocks, a three-dimensional contact element model with a certain thickness was proposed that could simultaneously consider the heterogeneity, damage evolution and state variation of interfaces. The Weibull distribution function was employed to describe the rock heterogeneity. The damage evolution of the contact element was analyzed by using the equivalent damage method. Two criteria—the Mohr–Coulomb criterion and the maximum tensile stress criterion—were employed to assess the damage occurrence. Once the element was identified as being completely damaged, the state variation of the contact element was determined. Three types of element states—contact without sliding, contact with sliding, and separation—were taken into consideration. The model was then implemented and programmed into the finite element method and validated with experimental measurements and theoretical solutions for three classical rock mechanical problems, including one static problem (i.e., a direct shear test) and two dynamic problems (the seismic responses of rock joints and split-Hopkinson pressure bar tests), in an attempt to examine the applicability of the proposed contact element model. The results demonstrated that the contact element model can effectively model the rock interfacial problems and can therefore help solve problems in discontinuous rock masses.

Boosting Monte Carlo simulations of spin glasses using autoregressive neural networks.

The autoregressive neural networks are emerging as a powerful computational tool to solve relevant problems in classical and quantum mechanics. One of their appealing functionalities is that, after they have learned a probability distribution from a dataset, they allow exact and efficient sampling of typical system configurations. Here we employ a neural autoregressive distribution estimator (NADE) to boost Markov chain Monte Carlo (MCMC) simulations of a paradigmatic classical model of spin-glass theory, namely, the two-dimensional Edwards-Anderson Hamiltonian. We show that a NADE can be trained to accurately mimic the Boltzmann distribution using unsupervised learning from system configurations generated using standard MCMC algorithms. The trained NADE is then employed as smart proposal distribution for the Metropolis-Hastings algorithm. This allows us to perform efficient MCMC simulations, which provide unbiased results even if the expectation value corresponding to the probability distribution learned by the NADE is not exact. Notably, we implement a sequential tempering procedure, whereby a NADE trained at a higher temperature is iteratively employed as proposal distribution in a MCMC simulation run at a slightly lower temperature. This allows one to efficiently simulate the spin-glass model even in the low-temperature regime, avoiding the divergent correlation times that plague MCMC simulations driven by local-update algorithms. Furthermore, we show that the NADE-driven simulations quickly sample ground-state configurations, paving the way to their future utilization to tackle binary optimization problems.

Quantum Optimal Transport is Cheaper

We compare bipartite (Euclidean) matching problems in classical and quantum mechanics. The quantum case is treated in terms of a quantum version of the Wasserstein distance. We show that the optimal quantum cost can be cheaper than the classical one. We treat in detail the case of two particles: the equal mass case leads to equal quantum and classical costs. Moreover, we show examples with different masses for which the quantum cost is strictly cheaper than the classical cost.

Likely cavitation and radial motion of stochastic elastic spheres

The cavitation of solid elastic spheres is a classical problem of continuum mechanics. Here, we study this problem within the context of ‘stochastic elasticity’ where the constitutive parameters are characterised by probability density functions. We consider homogeneous spheres of stochastic neo-Hookean material, composites with two concentric stochastic neo-Hookean phases, and inhomogeneous spheres of locally neo-Hookean material with a radially varying parameter. In all cases, we show that the material at the centre determines the critical load at which a spherical cavity forms there. However, while under dead-load traction, a supercritical bifurcation, with stable cavitation, is obtained in a static sphere of stochastic neo-Hookean material, for the composite and radially inhomogeneous spheres, a subcritical bifurcation, with snap cavitation, is also possible. For the dynamic spheres, oscillatory motions are produced under suitable dead-load traction, such that a cavity forms and expands to a maximum radius, then collapses again to zero periodically, but not under impulse traction. Under a surface impulse, a subcritical bifurcation is found in a static sphere of stochastic neo-Hookean material and also in an inhomogeneous sphere, whereas in composite spheres, supercritical bifurcations can occur as well. Given the non-deterministic material parameters, the results can be characterised in terms of probability distributions.

Collisions in Classical Mechanics in Terms of Mass-Momentum “Vectors” with Galilean Transformations

We present the usefulness of mass-momentum “vectors” to analyze the collision problems in classical mechanics for both one and two dimensions with Galilean transformations. The Galilean transformations connect the mass-momentum “vectors” in the center-of-mass and the laboratory systems. We show that just moving the two systems to and fro, we obtain the final states in the laboratory systems. This gives a simple way of obtaining them, in contrast with the usual way in which we have to solve the simultaneous equations. For one dimensional collision, the coefficient of restitution is introduced in the center-of-mass system. This clearly shows the meaning of the coefficient of restitution. For two dimensional collisions, we only discuss the elastic collision case. We also discuss the case of which the target particle is at rest before the collision. In addition to this, we discuss the case of which the two particles have the same masses.

Unsolved problem in classical mechanics

Unsolved problem in classical mechanics

Definition of work: unsolved problem in classical mechanics

The critical analysis of the foundations of classical mechanics is proposed. Methodological basis of the analysis is the unity of formal logic and rational dialectics. The main results of the analysis of concepts “force”, “work” and “work of force” are the following statements: (1) the standard concept “work of force” contradicts to dialectics (practice) and formal logic because the concept “force” is a general abstract concept. The concept “force of interaction of material objects” is a general concrete concept because this concept expresses the property (feature, sign, quality) of the interaction of material objects. But the property does not perform the work because the property is not a material object. Work is performed by a controllable (operated) material object. Consequently, the standard mathematical definition of work is an incorrect relationship; (2) the concept “force of interaction of material objects” can only be defined in a genetic way: one should measure the force of interaction with the help of a dynamometer. The dynamometer reading represents the value of the measured quantity (force). If the measured quantity (force) is a variable quantity, then a change of this quantity (force) is called the movement of this quantity (force). But if the force is a constant quantity, then the movement of the force does not exist. Consequently, the standard mathematical definition of work is an incorrect relationship; (3) the position of the force in the coordinate system cannot be determined because a force is not a material object, and the quantity of the force does not have the dimension (measure) “meter”. This signifies that: (a) a force does not exist in the material coordinate system; (b) a force does not exist on the material path; the path length (distance) for force does not exist. Consequently, force cannot move over a distance that has the dimension “meter”. This signifies that the standard mathematical definition of work is an incorrect relationship; (4) in point of view of dialectics, work and energy are nonidentity concepts. Work is the activity of man, a robot, a controllable machine created by man. Work represents a rational, reasonable, well-directed, purposeful, purposive activity which manifested in the informational and material change of reality. This activity is the impossible if intellectual, informational, energy, and material resources do not exist. Thus, the standard mathematical definition of the work of force is a formal-logical, dialectical (practical), and physical errors.

Definition of Work: Unsolved Problem in Classical Mechanics

Definition of Work: Unsolved Problem in Classical Mechanics

One-Dimensional Theoretical Solution and Two-Dimensional Numerical Simulation for Functionally-Graded Piezoelectric Cantilever Beams with Different Properties in Tension and Compression.

The existing studies indicate polymers will present obviously different properties in tension and compression (bimodular effect) which is generally ignored because of the complexity of the analysis. In this study, a functionally graded piezoelectric cantilever beam with bimodular effect was investigated via analytical and numerical methods, respectively, in which a one-dimensional theoretical solution was derived by neglecting some unimportant factors and a two-dimensional numerical simulation was performed based on the model of tension-compression subarea. A full comparison was made to show the rationality of one-dimensional theoretical solution and two-dimensional numerical simulation. The result indicates that the layered model of tension-compression subarea also makes it possible to use numerical technique to simulate the problem of functionally graded piezoelectric cantilever beam with bimodular effect. Besides, the modulus of elasticity E* and the bending stiffness D* proposed in the one-dimensional problem may succinctly describe the piezoelectric effect on the classical mechanical problem without electromechanical coupling, which shows the advantages of one-dimensional solution in engineering applications, especially in the analysis and design of energy harvesting/sensing/actuating devices made of piezoelectric polymers whose bimodular effect is relatively obvious.

Non-integrability of generalised Charlier and Saint-Germain problem

Certain generalisation of a classical problem of celestial mechanics is considered. A material point moves in the potential force field that is a superposition of a radial force and a constant force. The potential of the central forces is proportional to an integer power −n of the distance from the origin. For all positive integers n except n=1 non-integrability in meromorphic functions is proved. Case n=1 is integrable. For negative integer n analysis is much more complex: for n=−1 non-integrability is proved, for n=−2 additional first integral is found, for n=−3 although the system meets the necessary integrability conditions Poincaré cross-sections show presence of chaotic layers in phase space.

Ascending Disk: Theoretical and Numerical Study

In this work we analyzed a classical mechanical problem: a disc with a circular hole rotating and ascending on a inclined plane. The principal idea is to show the basic equations in both Lagrangian and Newtonian formalisms and to present numerical solutions. The equations are not solvable analytically, then we proceed to integrate numerically using the standardized Runge-Kutta method of order 4. We present several phase space showing the time evolution of the system. Additionally we presented an extension of the system considering a damping factor that decays exponentially in time.

Giant disparity and a dynamical phase transition in large deviations of the time-averaged size of stochastic populations.

We study large deviations of the time-averaged size of stochastic populations described by a continuous-time Markov jump process. When the expected population size N in the steady state is large, the large deviation function (LDF) of the time-averaged population size can be evaluated by using a Wentzel-Kramers-Brillouin (WKB) method, applied directly to the master equation for the Markov process. For a class of models that we identify, the direct WKB method predicts a giant disparity between the probabilities of observing an unusually small and an unusually large values of the time-averaged population size. The disparity results from a qualitative change in the "optimal" trajectory of the underlying classical mechanics problem. The direct WKB method also predicts, in the limit of N→∞, a singularity of the LDF, which can be interpreted as a second-order dynamical phase transition. The transition is smoothed at finite N, but the giant disparity remains. The smoothing effect is captured by the van-Kampen system size expansion of the exact master equation near the attracting fixed point of the underlying deterministic model. We describe the giant disparity at finite N by developing a different variant of WKB method, which is applied in conjunction with the Donsker-Varadhan large-deviation formalism and involves subleading-order calculations in 1/N.