Principle Of Least Action Research Articles

Coupling dynamic behaviors of flexible stretching hub-beam system

In the 1st stage deployment of the IKAROS solar sail, the flexible beam with comparable mass relative to the mass of the rotor is stretching actively in the axial direction, which brings new challenge in the dynamic analysis for the 1st stage deployment of the IKAROS solar sail. Considering the active axial stretching of the flexible beam, a simplified coupling dynamic model for the 1st stage deployment of the IKAROS solar sail is proposed based on the non-holonomic Hamilton least-action principle firstly. And then, a structure-preserving approach combining the generalized multi-symplectic method and the symplectic Runge-Kutta method is constructed to simulate the evolution of the transverse vibration of the beam as well as the evolution of the rotation of the hub. Finally, the associated numerical results in three stages are reported. From the numerical results, it can be found that the coupling effects between the deformation of the beam, the active stretching of the beam and the rotation of the hub are reflected in the stretching stage. In this stage, the transverse vibration of the beam is enhanced by the stretching effect of the beam, and the increase of the energy of the beam in this stage is derived from the decrease of the rotational energy of the hub. In addition, the structure-preserving properties and the validity of the numerical results are verified by the tiny relative energy dissipation of the flexible stretching hub-beam system in the stretching stage.

Extended harmonic mapping connects the equations in classical, statistical, fluid, quantum physics and general relativity

One potential pathway to find an ultimate rule governing our universe is to hunt for a connection among the fundamental equations in physics. Recently, Ren et al. reported that the harmonic maps with potential introduced by Duan, named extended harmonic mapping (EHM), connect the equations of general relativity, chaos and quantum mechanics via a universal geodesic equation. The equation, expressed as Euler–Lagrange equations on the Riemannian manifold, was obtained from the principle of least action. Here, we further demonstrate that more than ten fundamental equations, including that of classical mechanics, fluid physics, statistical physics, astrophysics, quantum physics and general relativity, can be connected by the same universal geodesic equation. The connection sketches a family tree of the physics equations, and their intrinsic connections reflect an alternative ultimate rule of our universe, i.e., the principle of least action on a Finsler manifold.

Viscous Maxwell-Chern-Simons theory for topological electromagnetic phases of matter

We present the fundamental model of a topological electromagnetic phase of matter: viscous Maxwell-Chern-Simons theory. Our model applies to a quantum Hall fluids with viscosity. We solve both continuum and lattice regularized systems to demonstrate that this is the minimal (exactly solvable) gauge theory with a nontrivial photonic Chern number ($C\neq 0$) for electromagnetic waves coupled to a quantum Hall fluid. The interplay of symmetry and topology is also captured by the spin-1 representations of a photonic skyrmion at high-symmetry points in the Brillouin zone. To rigorously analyze the topological physics, we introduce the viscous Maxwell-Chern-Simons Lagrangian and derive the equations of motion, as well as the boundary conditions, from the principle of least action. We discover topologically-protected chiral (unidirectional) edge states which minimize the surface variation and correspond to massless photonic excitations costing an infinitesimal amount of energy. Physically, our predicted electromagnetic phases are connected to a dynamical photonic mass in the integer quantum Hall fluid. This arises from viscous (nonlocal) Hall conductivity and we identify the nonlocal Chern-Simons coupling with the Hall viscosity. The electromagnetic phase is topologically nontrivial $C\neq 0$ when the Hall viscosity inhibits the total bulk Hall response. Our work bridges the gap between electromagnetic and condensed matter topological physics while also demonstrating the central role of spin-1 quantization in nontrivial photonic phases.

Reconstruction of a neural network and alteration of the operators’ strategies during the recognition of facial images

Functional magnetic-resonance imaging was used to study the dynamics of information processing concerning a face in the neural networks of the brain. It is shown that large-scale neural networks of the operators’ brains were reconstructed during the second half of the study. Thus, when an instruction conflicts with the contents of the image so that it is impossible to find the correct solution, the operators transfer to a less energy-intensive behavioral strategy—they simulate the solution of the problem. When the assigned task is carried out under specified conditions, so that it is easy to distinguish the rotation of the head, the subject quickly learns to solve the problem, develops the knack, and adopts an automatic-motion regime—another neural network is activated, called the brain’s default-mode network. It is shown that the medial prefrontal cortex plays a role in developing and altering the strategy of the operator’s activity. The reconstruction of large-scale neural networks is probably caused by an unconscious mechanism that simplifies purposeful activity. The results agree well with the well-known principle of least action.

Verifying Fundamental Solution Groups for Lossless Wave Equations via Stationary Action and Optimal Control

A representation of a fundamental solution group for a class of wave equations is constructed by exploiting connections between stationary action and optimal control. By using a Yosida approximation of the associated generator, an approximation of the group of interest is represented for sufficiently short time horizons via an idempotent convolution kernel that describes all possible solutions of a corresponding short time horizon optimal control problem. It is shown that this representation of the approximate group can be extended to longer horizons via a concatenation of such short horizon optimal control problems, provided that the associated initial and terminal conditions employed in concatenating trajectories are determined via a stationarity rather than an optimality based condition. The construction is illustrated by its application to the approximate solution of a two point boundary value problem.

Indeterminism in Quantum Mechanics: Beyond and/or Within Causation

The problem of indeterminism in quantum mechanics usually being considered as a generalization determinism of classical mechanics and physics for the case of discrete (quantum) changes is interpreted as an only mathematical problem referring to the relation of a set of independent choices to a well-ordered series therefore regulated by the equivalence of the axiom of choice and the well-ordering “theorem”. The former corresponds to quantum indeterminism, and the latter, to classical determinism. No other premises (besides the above only mathematical equivalence) are necessary to explain how the probabilistic causation of quantum mechanics refers to the unambiguous determinism of classical physics. The same equivalence underlies the mathematical formalism of quantum mechanics. It merged the well-ordered components of the vectors of Heisenberg’s matrix mechanics and the non-ordered members of the wave functions of Schrodinger’s undulatory mechanics. The mathematical condition of that merging is just the equivalence of the axiom of choice and the well-ordering theorem implying in turn Max Born’s probabilistic interpretation of quantum mechanics. Particularly, energy conservation is justified differently than classical physics. It is due to the equivalence at issue rather than to the principle of least action. One may involve two forms of energy conservation corresponding whether to the smooth changes of classical physics or to the discrete changes of quantum mechanics. Further both kinds of changes can be equated to each other under the unified energy conservation as well as the conditions for the violation of energy conservation to be investigated therefore directing to a certain generalization of energy conservation.

A visual framework to create photorealistic retinal vessels for diagnosis purposes

The methods developed in recent years for synthesising an ocular fundus can be been divided into two main categories. The first category of methods involves the development of an anatomical model of the eye, where artificial images are generated using appropriate parameters for modelling the vascular networks and fundus. The second type of method has been made possible by the development of deep learning techniques and improvements in the performance of hardware (especially graphics cards equipped with a large number of cores). The methodology proposed here to produce high-resolution synthetic fundus images is intended to be an alternative to the increasingly widespread use of generative adversarial networks to overcome the problems that arise in producing slightly modified versions of the same real images. This will allow the simulation of pathologies and the prediction of eye-related diseases. The proposed approach is based on the principle of least action and correctly places the vessels on the simulated eye fundus without using real morphometric information. An a posteriori analysis of the average characteristics such as the size, length, bifurcations, and endpoint positioning confirmed the substantial accuracy of the proposed approach compared to real data. A graphical user interface allows the user to make any changes in real time by controlling the positions of control points.

Rankine–Hugoniot conditions for fluids whose energy depends on space and time derivatives of density

By using the Hamilton principle of stationary action, we derive the governing equations and Rankine–Hugoniot conditions for continuous media where the specific energy depends on the space and time density derivatives. The governing system of equations is a time reversible dispersive system of conservation laws for the mass, momentum and energy. We obtain additional relations to the Rankine–Hugoniot conditions coming from the conservation laws and discuss the well-founded of shock wave discontinuities for dispersive systems.

Rubber-Liquid Resonator

The acoustic characteristics of a rubber-liquid resonator are calculated, combining the properties of an empty rubber cavity, a Helmholtz resonator, and a water–air resonator, gas bubble in a viscoelastic medium and in a shell, and a bubble in a liquid. The equation of the forced oscillations of the resonator in the sound wave field is obtained by applying the principle of least action. The eigenfrequency of the resonator is calculated. The following sound energy dissipation mechanisms are considered: due to the shear viscosity of rubber, the viscosity of liquid in the neck, heat loss in the air chamber, and radiation loss. Experimental data are presented. Possible resonator applications are discussed.

Natural oscillations of yarn balloons in ring spinning

Ring spinning is the most relevant production process for high quality short staple yarn. Recent technological advances using a twisting system involving frictionless superconducting magnetic bearings motivate a renewed interest in the dynamics of the process.A new deduction of the equations of motion for stationary and oscillatory movement is presented using Hamilton’s Principle of Least Action, taking into account axial transport, air drag and boundary conditions at the bearing. By application of Ritz’s method, system matrices are defined to enable the study of natural frequencies and mode shapes. These are validated by comparing to experimental results from literature, and case studies for industrially relevant parameter variations are performed.

Tolerance and asymptotic modelling of dynamic thermoelasticity problems for thin micro-periodic cylindrical shells

The problem of linear dynamic thermoelasticity in Kirchhoff–Love-type circular cylindrical shells having properties periodically varying in circumferential direction (uniperiodic shells) is considered. In order to describe thermoelastic behaviour of such shells, two mathematical averaged models are proposed—the non-asymptotic tolerance and the consistent asymptotic models. Considerations are based on the known Kirchhoff–Love theory of elasticity combined with Duhamel-Neumann thermoelastic constitutive relations and on Fourier’s theory of heat conduction. The non-asymptotic tolerance model equations depending on a cell size are derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. The consistent asymptotic model equations being independent on a microstructure size are obtained by means of the consistent asymptotic approach. Governing equations of both the models have constant coefficients, contrary to starting shell equations with periodic, non-continuous and oscillating coefficients. As examples, two special length-scale problems will be analysed in the framework of the proposed models. The first of them deals with investigation of the effect of a cell size on the shape of initial distributions of temperature micro-fluctuations. The second one deals with study of the effect of a microstructure size on the distribution of total temperature field approximated by sum of an averaged temperature and temperature fluctuations.

Vortex motion of a continuous medium depending on the pressure change

Abstract In this paper, we study the vortex motion of a continuous medium, which is described by forces obtained from the principle of least action. It is shown that in a continuous medium the vortex force components are proportional to the velocity and pressure gradient components. This article gives a description of the 2D vortex motion of air in zones of high and low pressure. If the pressure decreases, the angular velocity of rotation of the continuous medium increases, whereas if the pressure increases, the angular velocity fades. The lifting force is obtained due to the vortex movement of air in the form of a funnel. It is shown that the vortex force contains a vortex term of the Euler hydrodynamic equations with a relative factor equal to the velocity of the continuous medium squared divided by the sound velocity squared. To describe the motion of a continuous medium correctly it is necessary to replace the forces obtained by Euler with the forces obtained from the minimum of action in the equations of motion. It is concluded that vortex motions and turbulence are described by the obtained equations of motion, and not by the Navier–Stokes equations. Most likely, this is related to the Problem of the Millennium description of turbulence announced at the International Congress of Mathematics in 2000.

Long range frequency chirping of Alfvén eigenmodes

A theoretical framework has been developed for an NBI scenario to model the hard non-linear evolution of global Alfvén eigenmodes (GAEs) where the adiabatic motion of phase-space structures (holes and clumps), associated with frequency chirping, occurs in generalised phase-space of slowing down energetic particles. The radial profile of the GAE is expanded using finite elements which allows update of the mode structure as the mode frequency chirps. Constants of motion are introduced to track the dynamics of energetic particles during frequency chirping by implementing proper action-angle variables and canonical transformations which reduce the dynamics essentially to 1D. Consequently, we specify whether the particles are drifting inward/outward as the frequency deviates from the initial MHD eigenfrequency. Using the principle of least action, we have derived the non-linear equation describing the evolution of the radial profile by varying the total Lagrangian of the system with respect to the weights of the finite elements. For the choice of parameters in this work, it is shown that the peak of the radial profile is shifted and also broadens due to frequency chirping. The time rate of frequency change is also calculated using the energy balance and we show that the adiabatic condition remains valid once it is satisfied. This model clearly illustrates the theoretical treatment to study the long range adiabatic frequency sweeping events observed for Alfvén gap modes in real experiments.

Principles of Least Action in Urban Traffic Flow

Drawing on the strong connection between traffic modelling and physics we apply the principle of least action to traffic flow in an urban setting.

A theory of gravitation

The paper presents a theory of gravitation as a continuum dynamical theory, i.e. the equations of motion are first order in the time derivatives. The theory satisfies the laws of conservation of total energy, momentum and mass in the standard sense, i.e. as applications of Stokes’ theorem. A model in this theory is defined by the specification of an energy density function that accounts for the total mechanical and gravitational energy of the system and which in turn defines an action function. The derivation of the equations of motion is based on Hamilton’s principle of least action with the “same” action function as used for the derivation of the Einstein equation of the theory of general relativity, however, not with respect to variations of the metric but with respect to variations induced by local displacements of space, i.e. variations of the conjugate variable of the momentum and variations of the momentum density. Entropy density is introduced as a variable. This makes it possible to determine the thermodynamic equilibrium conditions for a model. Solutions of the equilibrium conditions are the Schwarzschild and Minkowski metrics on space-time, and the metrics defining the spherical and hyper spherical spaces.

Maupertuis-Hamilton least action principle in the space of variational parameters for Schrödinger dynamics; A dual time-dependent variational principle

Time-dependent variational principle (TDVP) provides powerful methods in solving the time-dependent Schröinger equation. As such Kan developed a TDVP (Kan 1981 Phys. Rev. A 24, 2831) and found that there is no Legendre transformation in quantum variational principle, suggesting that there is no place for the Maupertuis reduced action to appear in quantum dynamics. This claim is puzzling for the study of quantum–classical correspondence, since the Maupertuis least action principle practically sets the very basic foundation of classical mechanics. Zambrini showed within the theory of stochastic calculus of variations that the Maupertuis least action principle can lead to the Nelson stochastic quantization theory (Zambrini 1984 J. Math. Phys. 25, 1314). We here revisit the basic aspect of TDVP and reveal the hidden roles of Maupertuis-Hamilton least action in the Schrödinger wavepacket dynamics. On this basis we propose a dual least (stationary) action principle, which is composed of two variational functionals; one responsible for ‘energy related dynamics’ and the other for ‘dynamics of wave-flow’. The former is mainly a manifestation of particle nature in wave-particle duality, while the latter represents that of matter wave. It is also shown that by representing the TDVP in terms of these inseparably linked variational functionals the problem of singularity, which is inherent to the standard TDVPs, is resolved. The structure and properties of this TDVP are also discussed.

Minimal Geodesics of the Isosceles Three Body Problem

The isosceles three-body problem with nonnegative energy is studied from a variational point of view based on the Jacobi–Maupertuis metric. The solutions are represented by geodesics in the two-dimensional configuration space. Since the metric is singular at collisions, an approach based on the theory of length spaces is used. This provides an alternative to the more familiar approach based on the principle of least action. The emphasis is on the existence and properties of minimal geodesics, that is, shortest curves connecting two points in configuration space. For any two points, even singular points, a minimal geodesic exists and is nonsingular away from the endpoints. For the zero energy case, it is possible to use knowledge of the behavior of the flow on the collision manifold to see that certain solutions must be minimal geodesics. In particular, the geodesic corresponding to the collinear homothetic solution turns out to be a minimal for certain mass ratios.

Adaptive FLC-based UPQC in distribution power systems for power quality problems

This paper proposes UPQC to enhance all Power Quality issues. The UPQC is outlined by consolidating a series APF and shunt APF. The adaptable FLC together with Series APF control the voltage distortion. Shunt APF operation is controlled by d−q axes current from load current and DC-interface voltage is maintained through adaptable FLC. The stationary and transient action of the controller circuit in various load current and variable voltage conditions is discussed. A versatile FLC is planned for the plant when its parameters are changing. The UPQC is used with static and switching nonlinear burdens. It can reduce voltage sag/swell, voltage variations and eliminates harmonics and so on. The proposed controller is implemented using MATLAB or Simulink.

Ж.-Л. ЛАГРАНЖ КАК ОДИН ИЗ ОСНОВОПОЛОЖНИКОВ ТЕОРИИ ЭКСТРЕМУМОВ ФУНКЦИЙ МНОГИХ ПЕРЕМЕННЫХ

In the article we carried out a detailed analysis of the results obtained by J.-L. Lagrange in his first. The theory of extrema of functions of many variables, as part of mathematical analysis, refers to the mathematical foundations of the study of operations. In turn, many optimization problems are actually problems on the conditional extremum of the function of many variables. The relevance of this topic is determined by the fact that the methods for solving problems on the extremum of the function of many variables obtained in the mid 18th - early 20th centuries are used in solving modern problems. A special place here is occupied by L. Euler and J.-L. Lagrange. The aim of the article is to study the conditions for the maximum and minimum functions of many variables obtained by J.-L. Lagrange, and a comparison of its results with the presentation of this topic in modern textbooks on higher mathematics and mathematical analysis. It was established that in his first printed work he first formulated and proved sufficient conditions for the existence of an extremum of the function of many variables by actually establishing a criterion for the positive (negative) definiteness of quadratic forms, long before it appeared in J. Sylvester in the mid-19th century. A comparative analysis of the results of L. Euler and J.-L. Lagrange. It was found that sufficient conditions for the existence of an extremum of functions of many variables obtained in the first printed work of the young Lagrange are included in all modern textbooks in virtually the same form. The examples shown illustrate his theory. These are tasks of geometric and physical content. Special cases are considered in detail: functions of two and three variables. It is noted that this article became programmatic for the young Lagrange, although it remained unnoticed by his contemporaries. Subsequently, based on the method he obtained, he created the variational calculus, using the principle of least action and the theory of extrema, derived the basic laws of mechanics, the rule of factors for finding the conditional extremum of functions of many variables, which is named after him.

‘Uncertainty’ principle in two fluid–mechanics

Hamilton’s principle (or principle of stationary action) is one of the basic modelling tools in finite-degree-of-freedom mechanics. It states that the reversible motion of mechanical systems is completely determined by the corresponding Lagrangian which is the difference between kinetic and potential energy of our system. The governing equations are the Euler-Lagrange equations for Hamil- ton’s action. Hamilton’s principle can be naturally extended to both one-velocity and multi-velocity continuum mechanics (infinite-degree-of-freedom systems). In particular, the motion of multi–velocity continuum is described by a coupled system of ‘Newton’s laws’ (Euler-Lagrange equations) for each component. The introduction of dissipative terms compatible with the second law of thermodynamics and a natural restriction on the behaviour of potential energy (convexity) allows us to derive physically reasonable and mathematically well posed governing equations. I will consider a simplest example of two-velocity fluids where one of the phases is incompressible (for example, flow of dusty air, or flow of compressible bubbles in an incompressible fluid). A very surprising fact is that one can obtain different governing equations from the same Lagrangian. Different types of the governing equations are due to the choice of independent variables and the corresponding virtual motions. Even if the total momentum and total energy equations are the same, the equations for individual components differ from each other by the presence or absence of gyroscopic forces (also called ‘lift’ forces). These forces have no influence on the hyperbolicity of the governing equations, but can drastically change the distribution of density and velocity of components. To the best of my knowledge, such an uncertainty in obtaining the governing equations of multi- phase flows has never been the subject of discussion in a ‘multi-fluid’ community.