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 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.

scholarly journals Non-Smooth Geodesic Flows and Classical Mechanics

1969 ◽  
Vol 12 (2) ◽  
pp. 209-212 ◽  
Author(s):  
J. E. Marsden
Keyword(s):  
Phase Space ◽  
Kinetic Energy ◽  
Hamiltonian Systems ◽  
Configuration Space ◽  
Smooth Function ◽  
Cotangent Bundle ◽  
Classical Mechanics ◽  
Riemannian Metric ◽  
Geodesic Flows ◽  
Image Position

As is well known, there is an intimate connection between geodesic flows and Hamiltonian systems. In fact, if g is a Riemannian, or pseudo-Riemannian metric on a manifold M (we think of M as q-space or the configuration space), we may define a smooth function Tg on the cotangent bundle T*M (q-p-space, or the phase space). This function is the kinetic energy of q, and locally is given by

scholarly journals Action in Economics: Mathematical Derivation of Laws of Economics from the Principle of Least Action in Physics

2021 ◽  
Author(s):  
Sayan Kombarov
Keyword(s):  
Mathematical Formulation ◽  
Classical Mechanics ◽  
Human Action ◽  
Fundamental Principle ◽  
Austrian School ◽  
Rate Theory ◽  
Main Stream ◽  
Marginal Value ◽  
Least Action ◽  
Principle Of Least Action

The thesis of this paper is mathematical formulation of the laws of Economics with application of the principle of Least Action of classical mechanics. This paper is proposed as the rigorous mathematical approach to Economics provided by the fundamental principle of the physical science – the Principle of Least Action. This approach introduces the principle of Action into main-stream economics and allows reconcile main principles Austrian School of Economics and the laws of market, such Say’s law and marginal value and interest rate theory, with the modern results of mathematical economics, such as Capital Asset Pricing Model (CAPM), game theory and behavioral economics. This principle is well known in classical mechanics as the law of conservation of action that governs any system as a whole and all its components. It led to the revolution in physics, as it allows to derive the laws of Newtonian and quantum mechanics and probability. Ludwig von Mises defined Economics is the science of Human Action. Action is introduced into Economics by the founder of Austrian School of Economic, Carl Menger. Production or acquisition of any goods, services and assets are results of purposeful acts in the form of expenditure of work and energy in the form of flow of money and material resources. Humans take them to achieve certain desired goals with given resources and time. Any economic good and service, financial, productive, or real estate asset is the result of such action.

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

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Xiaobo Zhai ◽  
Changyu Huang ◽  
Gang Ren
Keyword(s):  
General Relativity ◽  
Quantum Physics ◽  
Harmonic Maps ◽  
Classical Mechanics ◽  
Harmonic Mapping ◽  
Geodesic Equation ◽  
Lagrange Equations ◽  
Least Action ◽  
Principle Of Least Action ◽  
Fundamental Equations

Abstract 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.

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 The Principle of Least Action for Reversible Thermodynamic Processes and Cycles

Entropy ◽  
2018 ◽  
Vol 20 (7) ◽  
pp. 542 ◽  
Author(s):  
Tian Zhao ◽  
Yu-Chao Hua ◽  
Zeng-Yuan Guo
Keyword(s):  
Heat Release ◽  
Optimization Problems ◽  
Optimal Path ◽  
Classical Mechanics ◽  
Carnot Cycle ◽  
Heat Absorption ◽  
Least Action ◽  
Principle Of Least Action ◽  
Thermodynamic Processes ◽  
Release Processes

The principle of least action, which is usually applied to natural phenomena, can also be used in optimization problems with manual intervention. Following a brief introduction to the brachistochrone problem in classical mechanics, the principle of least action was applied to the optimization of reversible thermodynamic processes and cycles in this study. Analyses indicated that the entropy variation per unit of heat exchanged is the mode of action for reversible heat absorption or heat release processes. Minimizing this action led to the optimization of heat absorption or heat release processes, and the corresponding optimal path was the first or second half of a Carnot cycle. Finally, the action of an entire reversible thermodynamic cycle was determined as the sum of the actions of the heat absorption and release processes. Minimizing this action led to a Carnot cycle. This implies that the Carnot cycle can also be derived using the principle of least action derived from the entropy concept.

The whole of physics

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Physical Chemistry ◽  
General Relativity ◽  
Electromagnetic Waves ◽  
Materials Science ◽  
Classical Mechanics ◽  
Least Action ◽  
Principle Of Least Action ◽  
Theory Of Gravitation ◽  
Mathematical Equations ◽  
Deep Significance

How the Principle of Least Action underlies all physics (all physics that can be reduced to mathematical equations) is explained at a qualitative, semi-popular level. It even applies to smartphones. The domains of classical mechanics, continuum mechanics, materials science, light and electromagnetic waves, special and general relativity (Einstein’s Theory of Gravitation), electrodynamics, quantumelectrodynamics (QED), hydrodynamics, physical chemistry, statistical mechanics, and the quantum world, are examined. It is shown that the Principles of Least Time, Least Resistance, and Maximal Ageing, and Lenz’s Law are, in fact, examples of the Principle of Least Action. It is also shown how Planck’s constant is a measure of “absolute smallness,” and its units are the units of action. Never again, post quantum mechanics, can there be any doubt about the deep significance of action in physics.

Lagrangian Mechanics

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Kinetic Energy ◽  
Quadratic Form ◽  
Lagrangian Mechanics ◽  
Conservation Of Energy ◽  
Curved Space ◽  
Generalized Coordinates ◽  
Least Action ◽  
Principle Of Least Action ◽  
External Conditions ◽  
Definition Of

It is demonstrated how d’Alembert’s Principle can be used as the basis for a more general mechanics – Lagrangian Mechanics. How this leads to Hamilton’s Principle (the Principle of Least Action) is shown mathematically and in words. It is further explained why Lagrangian Mechanics is so general, why forces of constraint may be ignored, and how external conditions lead to “curved space.” Also, it is explained why the Lagrangian, L, has the form L = T − V (where T is the kinetic energy and V is the potential energy), and why T is in “quadratic form” (T = 1/2mv2). It is shown how Noether’s Theorem leads to a more fundamental definition of energy and links the conservation of energy to the homogeneity of time. The ingenious Lagrange multipliers are explained, and also generalized forces and generalized coordinates.

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.

II.—On the Equations of Motion of a Single Particle

1922 ◽  
Vol 41 ◽  
pp. 26-33
Author(s):  
J. H. M. Wedderburn
Keyword(s):  
Kinetic Energy ◽  
Single Particle ◽  
Equations Of Motion ◽  
Christoffel Symbol ◽  
Lagrangian Equations ◽  
Second Derivatives ◽  
The Matrix ◽  
Image Position

§ 1. When solved for the second derivatives, the Lagrangian equations of motion for a system in which there are no extraneous forces have the formor, disregarding the parameter t, being the second Christoffel symbol † of the matrix associated with the kinetic energy. If there are extraneous forces Fk and, denoting t by xn+1, we add to the set of equations, the equations of motion are

Sign in / Sign up

Export Citation Format

Share Document