Principle Of Least Action Research Articles

Effective field theory for distorted photonic crystals

Photonic crystals are periodic structure of dielectric materials that can control light propagations in the media because of their photonic-dispersion led by the well-ordered lattice-points arrangements. We here study the behavior of light propagation in distorted-photonic crystals (D-PCs), which possess the gradual spatial distortion of lattice-points positions, as the effective field theory in terms of the differential geometry. In order to investigate the trajectory of light ray in the D-PC, we derive the geodesics equation that is given from the principle of least action, with defining the metric tensor in terms of the lattice-positions distortion. The geodesics equation indicates that the trajectory can be bent by just introducing lattice-positions distortion. We show some exact solutions of the trajectory in the case of simple distortion and that those results well agree with the results of the Finite Difference Time Domain (FDTD) simulations.

An extended Hamilton principle as unifying theory for coupled problems and dissipative microstructure evolution

An established strategy for material modeling is provided by energy-based principles such that evolution equations in terms of ordinary differential equations can be derived. However, there exist a variety of material models that also need to take into account non-local effects to capture microstructure evolution. In this case, the evolution of microstructure is described by a partial differential equation. In this contribution, we present how Hamilton’s principle provides a physically sound strategy for the derivation of transient field equations for all state variables. Therefore, we begin with a demonstration how Hamilton’s principle generalizes the principle of stationary action for rigid bodies. Furthermore, we show that the basic idea behind Hamilton’s principle is not restricted to isothermal mechanical processes. In contrast, we propose an extended Hamilton principle which is applicable to coupled problems and dissipative microstructure evolution. As example, we demonstrate how the field equations for all state variables for thermo-mechanically coupled problems, i.e., displacements, temperature, and internal variables, result from the stationarity of the extended Hamilton functional. The relation to other principles, as the principle of virtual work and Onsager’s principle, is given. Finally, exemplary material models demonstrate how to use the extended Hamilton principle for thermo-mechanically coupled elastic, gradient-enhanced, rate-dependent, and rate-independent materials.

Analysis of cracks development in rock massif with the use of dynamic destruction resistance

The object of research: the process of development of main cracks in the massif of rocks under the action of a wave impulse taking into account the dynamic destruction resistance.
 Investigated problem: Description of the dynamics of crack development in the impulse mode of increasing their size, as well as the transition to an unstable mode of crack growth.
 The main scientific results: In the present article uses an approach based on the principle of least action and N-characteristic of dynamic crack growth resistance. This allowed to obtain analytical equations of the crack growth trajectory in the rock massif and the main characteristics of the process.
 The area of practical use of the research results: The considered approach allows to predict the formation of the shear surface in quarries and natural slopes. This approach also allows to set a time-varying probability of their steady state and adjust, if necessary, the parameters of blasting.
 Innovative technological product: Analytical regularities for the driving force of the crack and dynamic destruction resistance are obtained, which take into account the velocity and acceleration of the crack, as well as the time and nonstationarity of the impulse load.
 Scope of the innovative technological product: The proposed approach can be used in the design of blasting operations in quarries and in the calculation of the probabilities of the steady state of natural massifs under the influence of impulse loads.

Complex scenarios with competing factors: A conception paper applied to the COVID-19 case.

A theory to analyze complex scenarios facing threats with competing factors and limited resources has been introduced. The scenarios are modeled as closed systems. Hamilton’s principle of stationary action is used to conceive a theory in which competing factors dispute available resources to minimize undesirable outcomes. The result indicates that the minimum response is obtained by a combination of the competing factors weighted by their corresponding criticalities. The theory has been applied to the COVID-19 pandemic with two competing factors: Health and Economy. As main result, to minimize the total number of deaths, the recommendation is to balance the emphasis on both factors. This implies to give more emphasis to the economic factor, by avoiding restrict interventions like lockdowns and business closures. The model may evolve from a qualitative to a quantitative status, allowing for computational simulations aimed at validations and forecasting. As such, this approach may become a useful tool for strategic decision-making regarding resources allocations to reduce guessing in scenarios full of uncertainties.

A continuum model to study fluid dynamics within oscillating elastic nanotubes

We derive equations of motion that describe the dynamics of a fluid confined within an elastic nanotube subject to periodic bending deflections. We use the principle of least action applied to a continuous open system at constant temperature. We solve the equations analytically in two limiting situations: when the tube oscillations are so small that they do not affect the fluid motion, but this one affects the tube dynamics; and when the flow magnitude is so small that it has no influence on the tube dynamics, but this one affects fluid motion. In the first case, we find out that the characteristic bending frequency spectrum of the tube depends not only on the magnitude of flow velocity, as previously stated in the literature, but also on the fluid velocity profile. This could constitute the basis of a strategy for indirect determination of the slip length in carbon nanotubes conveying flow via measurement of the buckling speed. In the second case, we find that tube vibrations can modify the dynamics of the fluid. Particularly, for a fluid subject to a constant pressure gradient, the tube motion induces an oscillatory motion in the fluid with twice the frequency of the tube. Moreover, the amplitude of the oscillatory fluid motion persists at high frequencies. This could constitute a strategy to generate high-frequency flows at nanoscales. Our results open up a panorama to control flow across nanotubes via tube vibrations, which could be complementary to chemical functionalization of nanostructures.

Optimal transportation of grain boundaries: A forward model for predicting migration mechanisms

It has been hypothesized that the most likely atomic rearrangement mechanism during grain boundary (GB) migration is the one that minimizes the lengths of atomic displacements in the dichromatic pattern. In this work, we recast the problem of atomic displacement minimization during GB migration as an optimal transport (OT) problem. Under the assumption of a small potential energy barrier for atomic rearrangement, the principle of stationary action applied to GB migration is reduced to the determination of the Wasserstein metric for two point sets. In order to test the minimum distance hypothesis, optimal displacement patterns predicted on the basis of a regularized OT based forward model are compared to molecular dynamics (MD) GB migration data for a variety of GB types and temperatures. Limits of applicability of the minimum distance hypothesis and interesting consequences of the OT formulation are discussed in the context of MD data analysis for twist GBs, general Σ3 twin boundaries and a tilt GB that exhibits shear coupling. The forward model may be used to predict atomic displacement patterns for arbitrary disconnection modes and a variety of metastable states, facilitating the analysis of multimodal GB migration data.

Effect of polymer chains entanglements, crosslinks and finite extensibility on the nonlinear dynamic oscillations of dielectric viscoelastomer actuators

Soft materials exhibiting large deformation under external stimuli have gained an increasing attention in the recent past because of their potential applications in soft transducers aimed at achieving biomimetic actuation. This paper theoretically analyzes the effect of internal properties, including entanglements, crosslinks and finite extensibility of polymer chains along with inherent viscoelastic properties of polymers on the performance of Dielectric Elastomer actuator (DEA) in dynamic modes of actuation. A physics based nonaffine material model proposed by Davidson and Goulbourne is used to model the polymer chains entanglements, crosslinks and finite extensibility. To incorprate the viscoelastic properties, a rheological material model based on the additive decomposition of the isotropic strain energy density into equilibrium and viscous parts is implemented. A computationally efficient method, which relies on the principle of least action is used for extracting the governing equation representing the dynamic motion of the DE actuator. The results demonstrate that DEAs with strong entanglements and crosslinks along with small finite extensibility of polymer chains exhibits lower deformation level in DC dynamic modes of actuation. It is inferred that the strong entanglements and crosslinks in polymer chains enhances the resonant frequency, but debilitate the intensity of vibration of viscoelastic DEAs. Further, the periodicity and stability of the nonlinear oscillations exhibited by the viscoelastic DEAs are assessed by employing the Poincare maps and phase portraits. The results of the present investigation can be applied effectively in bridging the mechanism between the microcosmic polymer chains and macroscopic dynamic behavior of viscoelastic DE actuators.

On a strong minimum condition of a fractal variational principle

Variational theory is the theoretical basis for various numerical methods. A minimum variational principle plays an important role in guaranteeing the convergence of the numerical algorithm, and the Weierstrass theorem is used to judge the minimum. This paper suggests a fractal modification of Weierstrass function for fractal variational principles, and a strong minimum condition is given. The Hamilton least-action principle for the Duffing oscillator in a fractal space is used as an example to show the correctness of the suggested strong minimum condition.

A generalization of the Lagrange–Hamilton formalism with application to non-conservative systems and the quantum to classical transition

This work has two aims. The first is to develop a Lagrange–Hamilton framework for the analysis of multi-degree-of-freedom nonlinear systems in which non-conservative effects are included in the variational principle of least action from the outset. The framework is a generalization of the Bateman approach in which a set of adjoint coordinates is introduced. A function termed the M-function is introduced as the Fourier transform over the momenta of the joint probability density function (JPDF) of the displacements and momenta, and it is shown that for statistical systems, this function can be written as an expectation involving the new principle function and a general dimensional constant ℏ. This leads to a concise derivation of the Fokker–Planck–Kolmogorov equation. It is found that the equation governing the M-function can be expressed in terms of the new Hamiltonian by replacing momenta by differential operators, meaning that the function satisfies the same equation as the quantum wave function. This gives rise to the second aim of this work: to explore relations between the developed classical framework and quantum mechanics. It is shown that for an undamped linear system, the solution of the M-function equation yields the response JPDF as a sum of Wigner functions. This classical analysis leads to a number of well-known results from quantum mechanics as ℏ → 0, and the extension of this result to nonlinear systems is discussed. The quantum wave function associated with the Hamiltonian is then considered, and the relevance of this function to the physical system is discussed.

A semi-discrete scheme derived from variational principles for global conservative solutions of a Camassa–Holm system

We define a kinetic and a potential energy such that the principle of stationary action from Lagrangian mechanics yields a Camassa–Holm system (2CH) as the governing equations. After discretizing these energies, we use the same variational principle to derive a semi-discrete system of equations as an approximation of the 2CH system. The discretization is only available in Lagrangian coordinates and requires the inversion of a discrete Sturm–Liouville operator with time-varying coefficients. We show the existence of fundamental solutions for this operator at initial time with appropriate decay. By propagating the fundamental solutions in time, we define an equivalent semi-discrete system for which we prove that there exists unique global solutions. Finally, we show how the solutions of the semi-discrete system can be used to construct a sequence of functions converging to the conservative solution of the 2CH system.

A least action principle for interceptive walking

The principle of least effort has been widely used to explain phenomena related to human behavior ranging from topics in language to those in social systems. It has precedence in the principle of least action from the Lagrangian formulation of classical mechanics. In this study, we present a model for interceptive human walking based on the least action principle. Taking inspiration from Lagrangian mechanics, a Lagrangian is defined as effort minus security, with two different specific mathematical forms. The resulting Euler–Lagrange equations are then solved to obtain the equations of motion. The model is validated using experimental data from a virtual reality crossing simulation with human participants. We thus conclude that the least action principle provides a useful tool in the study of interceptive walking.

Bayesian mechanics of perceptual inference and motor control in the brain

The free energy principle (FEP) in the neurosciences stipulates that all viable agents induce and minimize informational free energy in the brain to fit their environmental niche. In this study, we continue our effort to make the FEP a more physically principled formalism by implementing free energy minimization based on the principle of least action. We build a Bayesian mechanics (BM) by casting the formulation reported in the earlier publication (Kim in Neural Comput 30:2616–2659, 2018, https://doi.org/10.1162/neco_a_01115) to considering active inference beyond passive perception. The BM is a neural implementation of variational Bayes under the FEP in continuous time. The resulting BM is provided as an effective Hamilton’s equation of motion and subject to the control signal arising from the brain’s prediction errors at the proprioceptive level. To demonstrate the utility of our approach, we adopt a simple agent-based model and present a concrete numerical illustration of the brain performing recognition dynamics by integrating BM in neural phase space. Furthermore, we recapitulate the major theoretical architectures in the FEP by comparing our approach with the common state-space formulations.

Natural Selection as the Sum over all Histories

As evolution can be connected to the principle of least action, and if it is depicted in evolution-space versus time then it corresponds to the direction of ultimate causation. As an organism evolves and follows a path of proximate causation, if the vector is closely parallel to that of the Ultimate Causation then the changes will confer desirable attributes which will lead to further development. If, however, the variations do not occur in a direction close to that of the ultimate causation vector the evolved organism will quickly die out. Therefore Natural Selection may be viewed as similar to Feynman’s “sum over all histories”. This approach is compatible with both Neutral Theory and Selection, as it includes both positive and negative mutations and selection. Therefore, the principle of least action gives a direction, but not a purpose, to evolution.

Testing a conjecture on the origin of the standard model

A simple Planck-scale model of nature appears to yield the entire standard model of particle physics, including its fundamental constants. The conjecture derives from Dirac’s proposal to describe fermions as tethered objects and models elementary particles as rational tangles. The tangle model appears to explain the Dirac equation, the principle of least action, the observed particle spectrum of fermions and bosons, and the three observed gauge interactions with their Lie groups and all their other properties. In a natural way, the specific tangles for each elementary particle define spin, quantum numbers and all other properties. No aspect of the standard model remains unexplained. Rational tangles appear to imply all the observed propagators and interaction vertices in Feynman diagrams. Other propagators or vertices are excluded. The tangle model thus yields each term of the full Lagrangian of the standard model. Over 100 predictions and tests about physics beyond the standard model are deduced from the conjecture. The predictions cover magnetic monopoles, the weak interaction, the quark model, non-perturbative effects, glueballs, effects of gravity, and more. The predictions agree with all observations performed so far. The conjectured tangles for the elementary particles imply specific Planck-scale processes that occur during propagation and at interaction vertices. These processes determine particle masses, mixing angles and coupling constants. Approximate estimates are possible; ways to improve the calculations are pointed out.

The Boas-Lewontin law: evidence & theory

In Walters (2018) I proposed the Boas-Lewontin Law. In the first part of this Report, I provide a precis of the empirical evidence for it. Consequences of the Second Law of Thermodynamics and the Principle of Least Action allow a very simple derivation of the law, which makes up the second part of the Report.

Time-Neutrality of Natural Laws Challenged: Time Is Not an Illusion but Ongoing Energy-Driven Information Loss

It is shown that the time of entropy increase, here called action time, is caused by a dynamically understood energy. It drives time by decreasing its presence per state, that is by abandoning order, information, and creating entropy. This mechanism can be derived from basic principles via the Lagrange-Euler formalism, just considering the properties of really experienced, oriented time and thus abandoning the paradigm of time neutrality. It describes nature driven by a dynamically understood principle of least action, which is identified as manifestation of fundamental irreversibility in nature. This readily explains the second law of thermodynamics and also yields the entropy law for non-linear irreversible thermodynamics: maximum entropy production within the restraints of the system. Dynamic energy-driven time, action time, and time asymmetry is generated via the process of erasing information and liberating its energy irreversibly as heat. It is not an illusion but information-based reality. It is the loss of information to the past and different from clock-time, which is just an artificial scale, using information for tracking real time, action time. Energy-driven fundamental irreversibility of nature can better describe experienced reality and opens the way to understand and finally imitate the self-organizing creativity in nature. It also draws far reaching consequences for understanding quantum physics, gravitation and cosmology as well as biology. From the point of view of irreversibility, nature turns out to be more elegant, simpler and rationally understandable. For the first time, it can be explained in a few words what energy and nature basically represent and why it must have been information, which has started the universe.

Nonlinear vibrations and time delay control of an extensible slowly rotating beam

Nonlinear dynamics of a rotating flexible slender beam with embedded active elements is studied in the paper. Mathematical model of the structure considers possible moderate oscillations thus the motion is governed by the extended Euler–Bernoulli model that incorporates a nonlinear curvature and coupled transversal–longitudinal deformations. The Hamilton’s principle of least action is applied to derive a system of nonlinear coupled partial differential equations (PDEs) of motion. The embedded active elements are used to control or reduce beam oscillations for various dynamical conditions and rotational speed range. The control inputs generated by active elements are represented in boundary conditions as non-homogenous terms. Classical linear proportional (P) control and nonlinear cubic (C) control as well as mixed (P-C) control strategies with time delay are analyzed for vibration reduction. Dynamics of the complete system with time delay is determined analytically solving directly the PDEs by the multiple timescale method. Natural and forced vibrations around the first and the second mode resonances demonstrating hardening and softening phenomena are studied. An impact of time delay linear and nonlinear control methods on vibration reduction for different angular speeds is presented.

Exploring 1:3 internal resonance for broadband piezoelectric energy harvesting

Recent literature has shown that the spectral characteristics of a harvester can be significantly improved by invoking internal resonance between the modes. Taking motivation from this premise, the present work analyzes the nonlinear dynamics and harvesting performance of a 1:3 internally resonant piezoelectric cantilever beam with a lumped mass under harmonic excitation. The governing modal equations are derived using Galerkin’s approach and Hamilton’s principle of least action. The method of multiple scales has been used to study the dynamics of the harvester in the proximity of primary resonances. The steady state periodic responses of the harvester are obtained using a pseudo-arc continuation technique and subsequently, their implications on the harvested power are discussed. The frequency responses of the harvested power are shown to boast significantly high magnitudes across a wide frequency band due to the energy transfer between the modes near the primary resonances. Results presented in the study show that the higher mode invoked through internal resonance plays a significant role in broadband power generation. The transfer of energy between the modes is found to occur only within a certain threshold of excitation level which in turn affects the magnitude of harvested power. Numerical simulations have also been carried out the results of which are compared against the analytical results. The outcomes of this first-hand analysis shall aid in the proposition of an efficient design for a broadband energy harvester.

Derivation of the Equations of Electrodynamics and Gravitation from the Principle of Least Action

The problem of justification of gravitation and electrodynamic equations with the help of the minimal action principle is a classical one. A derivation of the Vlasov–Maxwell–Einstein equation is proposed from the classical principle, but the little more general principle of minimal action. So we get new derivation of the right-hand side of the Maxwell and Einstein equations and we get a closed system of equations for gravitation and electrodynamics.

Influence of a Liquid Film Covering a Solid on Liquid Jet Impact Against Its Surface

The impact of a cylindrical liquid jet with a hemispherical tip on an elastic-plastic body uniformly covered in the impact domain by a liquid film has been studied, depending on the thickness of the film. The jet is directed orthogonally to the body surface, which is plane. The liquid in the jet and the film is water; the material of the body is Monel 400. The radius of the jet is $$R$$ , its velocity is $$V=300$$ m/s. The film thickness $$d$$ is varied in the range $$0-0.2R$$ . The action of the jet is considered until it becomes nearly stationary. A similar configuration can occur during the collapse of cavitation bubbles nearby a solid. The dynamics of the liquid (in the jet and the film) and the surrounding gas is calculated by the CIP-CUP method without explicit interface tracking using dynamically adaptive Soroban grids. The body is modeled by an ideal elastic-plastic semi-space, in which the deformations are considered to be small; the plasticity effect is realized according to the Mises condition. The dynamics of the body is calculated by a UNO-modification of the Godunov method of the second order of accuracy. Some features of the body surface loading and the corresponding response of the body (the deformation of its surface, the changes in its stress state) resulting from varying the liquid film thickness have been revealed. It is shown, in particular, that the body surface pressure profiles, initially very different for various $$d$$ , after the time moment $$t\approx R/V$$ become approximately the same (with a small exception at the periphery) and closely correspond to the profile arising under the stationary action of a similar jet of incompressible liquid. The jet impact results in the formation of a very shallow pit on the body surface (with a depth of about $$10^{-3}R$$ ). At $$d=0$$ , the depth of the pit in its center is approximately two times greater than at $$d=0.2R$$ .