scholarly journals Equations of Motion in the Theory of Relativistic Vector Fields

2019 ◽  
Vol 83 ◽  
pp. 12-30 ◽  
Author(s):  
Sergey G. Fedosin
Keyword(s):  
Vector Fields ◽  
Equations Of Motion ◽  
Lagrange Function ◽  
Equation Of Motion ◽  
Field Potentials ◽  
Least Action ◽  
Vector Potentials ◽  
Principle Of Least Action ◽  
System Of Particles ◽  
Poynting Theorem

Within the framework of the theory of relativistic vector fields, the covariant expressions are presented for the equations of motion of the matter and the field. These expressions can be written either in terms of the field tensors, that is, the fields’ strengths and solenoidal vectors, or in terms the four-potentials, that is, the fields’ scalar and vector potentials. This state of things is due to the fact that the Lagrange function initially implied the complementarity of description in terms of the strengths and the field potentials. It is shown that the equation for the fields, obtained by taking the covariant derivative in the equation for the metric, has a deeper meaning than the ordinary equation of motion of the matter, found with the help of the principle of least action. In particular, the above-mentioned equation for the fields leads to the generalized Poynting theorem, and after integration over the volume it allows us to introduce for consideration the integral vector as a measure of the energy and the fields’ energy fluxes, associated with a system of particles and fields.

scholarly journals A least action principle for interceptive walking

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Soon Ho Kim ◽  
Jong Won Kim ◽  
Hyun Chae Chung ◽  
MooYoung Choi
Keyword(s):  
Social Systems ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Action Principle ◽  
Lagrange Equations ◽  
Least Action Principle ◽  
Least Action ◽  
The Least Action Principle ◽  
Principle Of Least Action ◽  
Human Participants

AbstractThe 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.

On the propagation of elastic waves in aeolotropic media I. General principles

1954 ◽  
Vol 226 (1166) ◽  
pp. 339-355 ◽  
Keyword(s):  
Continuous Medium ◽  
Equations Of Motion ◽  
Plane Waves ◽  
Elastic Plane ◽  
Wave Surface ◽  
Least Action ◽  
Phase Velocities ◽  
Principle Of Least Action ◽  
Propagation Of Elastic Waves ◽  
General Method

Using the principle of least action the equations of motion and momentum conditions for elastic disturbances in any continuous medium are derived. Consideration of energy flux defines the form of the wave surface as the first negative pedal of the surface of phase velocities for elastic plane waves in the medium . A general method for obtaining the forms of these surfaces and an associated inverse surface is given and general conclusions about the propagation of disturbances are drawn.

scholarly journals A Differential Equation for Mutation Rates in Environmental Coevolution

2021 ◽  
Author(s):  
C E Neal-Sturgess
Keyword(s):  
Population Level ◽  
Equation Of Motion ◽  
Rate Of Change ◽  
Mutation Rates ◽  
Time Base ◽  
Time Rate ◽  
Least Action ◽  
Principle Of Least Action ◽  
Selection For ◽  
Rate Frequency

AbstractIn their paper Natural selection for least action (Kaila and Annila 2008) they depict evolution as a process conforming to the Principle of Least Action (PLA). From this concept, together with the Coevolution model of Lewontin, an equation of motion for environmental coevolution is derived which shows that it is the time rate (frequency) of evolutionary change of the organism (mutations) that responds to changes in the environment. It is not possible to compare the theory with viral or bacterial mutation rates, as these are not measured on a time base. There is positive evidence from population level avian studies where the coefficient of additive evolvability (Cav) and its square (IA) change with environmental favourability in agreement with this model. Further analysis shows that the time rate of change of the coefficient of additive evolvability (Cav) and its square (IA) are linear with environmental favourability, which could help in defining the Lagrangian of the environmental effects.

Motions of Bodies—Newton’s Laws

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Binary Stars ◽  
Body Problem ◽  
Equations Of Motion ◽  
Three Dimensions ◽  
Planetary Motion ◽  
Harmonic Oscillator Potential ◽  
Least Action ◽  
Alternative Approach ◽  
Principle Of Least Action ◽  
Newton’S Laws

Chapter 2 covers Newtonian dynamics, Newton’s law of gravitation and the motion of mutually gravitating bodies. The principle of least action is used to provide an alternative approach to Newton’s laws. Motion of several bodies is described. By analogy the same results are used to describe the motion of a single body in three dimensions. The equations of motion are solved for a harmonic oscillator potential. The general central potential is considered. The equations are solved for an attractive inverse square law force and shown to agree with Kepler’s laws of planetary motion. The Michell–Cavendish experiment to determine Newton’s gravitational constant is described. The centre of mass is defined and the motion of composite bodies described. The Kepler 2-body problem is solved and applied to binary stars. The positions of the five Lagrangian points are calculated. Energy conservation in mechanical systems is discussed, and friction and dissipation are considered.

A Principle in classical mechanics with a ‘relativistic’ path-element extending the principle of least action

1955 ◽  
Vol 51 (3) ◽  
pp. 469-475
Author(s):  
Edgar B. Schieldrop
Keyword(s):  
Kinetic Energy ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Sufficient Condition ◽  
Least Action ◽  
Actual Motion ◽  
Principle Of Least Action ◽  
Terminal Values ◽  
Newtonian Space ◽  
Image Position

1. A particle with mass m and coordinates x1x2, x3 relative to a set of rectangular axes fixed in Newtonian space is moving in a field of conservative forces with a potential energy V(x1, x2, x3) and a kinetic energyThe equations of motion, written(representing the three equations i = l, i = 2, i = 3 in a way to be used in this paper), constitute, as they stand, a sufficient condition in order to ensurein the sense that the Hamiltonian integral has a stationary value if the actual motion is compared with neighbouring motions with the same terminal positions and the same terminal values of the time as in the actual motion.

scholarly journals THE PRINCIPLE OF LEAST ACTION FOR TEST PARTICLES IN A FOUR-DIMENSIONAL SPACE–TIME EMBEDDED IN 5D

2008 ◽  
Vol 23 (04) ◽  
pp. 249-259 ◽  
Author(s):  
J. PONCE DE LEON
Keyword(s):  
Dimensional Space ◽  
Equations Of Motion ◽  
Rest Mass ◽  
Geodesic Equation ◽  
Space Time ◽  
Geodesic Motion ◽  
Least Action ◽  
Test Particles ◽  
Principle Of Least Action ◽  
Dimensional Space Time

It is well known that, in the five-dimensional scenario of braneworld and space–time-mass theories, geodesic motion in 5D is observed to be non-geodesic in 4D. Usually, the discussion is purely geometric and based on the dimensional reduction of the geodesic equation in 5D, without any reference to the test particle whatsoever. In this work we obtain the equation of motion in 4D directly from the principle of least action. So our main thrust is not the geometry but the particle observed in 4D. A clear physical picture emerges from our work. Specifically, that the deviation from the geodesic motion in 4D is due to the variation of the rest mass of a particle, which is induced by the scalar field in the 5D metric and the explicit dependence of the space–time metric on the extra coordinate. Thus, the principle of least action not only leads to the correct equations of motion, but also provides a concrete physical meaning for a number of algebraic quantities appearing in the geometrical reduction of the geodesic equation.

scholarly journals OBTAINING THE EQUATIONS OF MOTION OF A MECHANICAL BODY THROUGH ANALYTICAL METHODS

2021 ◽  
Author(s):  
Armandt Erasmus
Keyword(s):  
Dynamical System ◽  
Analytical Methods ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Lagrangian Mechanics ◽  
Conservation Of Energy ◽  
Least Action ◽  
Principle Of Least Action ◽  
Action Value ◽  
External Forces

The aim of this paper is to obtain the equations of motion in n-dimensional space for the case where no external forces act on a mechanical system using analytical methods. One such method is known as Lagrangian Mechanics. Lagrangian Mechanics is founded on the principle of least action which states that the spontaneous change from one configuration to another of a dynamical system has a minimum action value if the law of conservation of energy holds.

scholarly journals Two Components of the Macroscopic General Field

2017 ◽  
Vol 01 (02) ◽  
pp. 1750002 ◽  
Author(s):  
Sergey G. Fedosin
Keyword(s):  
Vector Fields ◽  
Pressure Field ◽  
Vacuum Field ◽  
Charge Component ◽  
Field Equations ◽  
Least Action ◽  
Principle Of Least Action ◽  
General Field ◽  
Energy And Momentum ◽  
Mass Component

The general field, containing all the macroscopic fields in it, is divided into the mass component, the source of which is the mass four-current, and the charge component, the source of which is the charge four-current. The mass component includes the gravitational field, acceleration field, pressure field, dissipation field, strong interaction and weak interaction fields, other vector fields. The charge component of the general field represents the electromagnetic field. With the help of the principle of least action we derived the field equations, the equation of the matter’s motion in the general field, the equation for the metric, the energy and momentum of the system of matter and its fields, and calibrated the cosmological constant. The general field components are related to the corresponding vacuum field components so that the vacuum field generates the general field at the macroscopic level.

scholarly journals Formulas for Parcel Velocity and Vorticity in a Rotating Cartesian Coordinate System

2015 ◽  
Vol 72 (10) ◽  
pp. 3908-3922 ◽  
Author(s):  
Robert Davies-Jones
Keyword(s):  
Coordinate System ◽  
Equations Of Motion ◽  
Least Action ◽  
Isentropic Flow ◽  
Principle Of Least Action ◽  
Dependent Variables ◽  
Model Equations ◽  
Barotropic Vorticity ◽  
Cartesian Coordinate ◽  
Conservation Theorem

Abstract Formulas in an Eulerian framework are presented for the absolute velocity and vorticity of individual parcels in inviscid isentropic flow. The analysis is performed in a rectangular Cartesian rotating coordinate system. The dependent variables are the Lagrangian coordinates, initial velocities, cumulative temperature, entropy, and a potential. The formulas are obtained in two different ways. The first method is based on finding a matrix integrating factor for the Euler equations of motion and a propagator for the vector vorticity equation. The second method is a variational one. Hamilton’s principle of least action is used to minimize the fluid’s absolute kinetic energy minus its internal energy and potential energy subject to the Lin constraints and constraints of mass and entropy conservation. In the first method, the friction and diabatic heating terms in the governing equations are carried along in integrands so that the generalized formulas lead to Eckart’s circulation theorem. Using them to derive other circulation theorems, the helicity-conservation theorem, and Cauchy’s formula for the barotropic vorticity checks the formulas further. The formulas are suitable for generating diagnostic fields of barotropic and baroclinic vorticity in models if some simple auxiliary equations are added to the model and integrated stably forward in time alongside the model equations.

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