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.

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 Pareto ratio and Pareto principle

2021 ◽  
Author(s):  
Gennady Grachev
Keyword(s):  
21St Century ◽  
Alternative Explanation ◽  
Wealth Inequality ◽  
Pareto Principle ◽  
Late 19Th Century ◽  
Least Action ◽  
Generalized Pareto ◽  
Principle Of Least Action ◽  
Empirical Regularity ◽  
Definition Of

In the late 19th century, Vilfredo Pareto published the results of his observations of wealth inequality in Italy as a ratio in which the numerator is equal to the share of population ranked in descending order of wealth, and the denominator is the share of wealth of the population. Both the form of presentation of the concentration of inequality in the form of a visual ratio (the Pareto ratio), and an unexpectedly large inequality (80/20), which became known as the Pareto principle. In the 21st century, a new concept was introduced - the generalized Pareto principle, which is understood as any proportion in which the sum of the numerator and denominator is 100%. As a result, an extraordinary situation arose, when the empirical regularity began to be identified with the measure of its measurement. To resolve this confusion, the definition of Pareto ratio is provided in this work, based on a generally accepted concept of "wealthy" element of the system, as also an alternative explanation of the Pareto principle is proposed with the use of isoperimetric inequalities conceptually related with the principle of least action.

Lagrangian mechanics

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Inertial Frame ◽  
Free Particle ◽  
Extremal Length ◽  
Gradient Force ◽  
Lagrangian Mechanics ◽  
Least Action ◽  
Straight Line ◽  
Principle Of Least Action ◽  
The One ◽  
Line Path

This chapter shows how the Newtonian law of motion of a particle subject to a gradient force derived from a ‘potential energy’ can always be obtained from an extremal principle, or ‘principle of least action’. According to Newton’s first law, the trajectory representing the motion of a free particle between two points p1 and p2 is a straight line. In other words, out of all the possible paths between p1 and p2, the trajectory effectively followed by a free particle is the one that minimizes the length. However, even though the use of the principle of extremal length of the paths between two points gives the straight line joining the points, this does not mean that the straight-line path is traced with constant velocity in an inertial frame. Moreover, the trajectory describing the motion of a particle subject to a force is not uniform and rectilinear.

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 MATHEMATICAL MODEL OF THE DYNAMICS OF A SIX-MASS SYSTEM OF A PARALLEL STRUCTURE MANIPULATOR MECHANISM

2021 ◽  
pp. 32-37
Author(s):  
V. V. Dyashkin-Titov ◽  
N. S. Vorob’eva ◽  
V. V. Zhoga
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Kinetic Energy ◽  
Quadratic Form ◽  
Degrees Of Freedom ◽  
Parallel Structure ◽  
Mass System ◽  
Generalized Coordinates ◽  
Comparative Results

The paper is devoted to the construction of a mathematical model of the dynamics of a parallel structure manipulator with three controlled degrees of freedom, based on the reduction of the kinetic energy of the manipulator to a quadratic form relative to three independent generalized coordinates, comparative results of mathematical modeling are presented.

scholarly journals From the principle of least action to the conservation of quantum information in chemistry. Can one generalize the periodic table?

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Quantum Information ◽  
Energy Conservation ◽  
Periodic Table ◽  
Chemical Elements ◽  
Chemical Change ◽  
Conservation Of Energy ◽  
Least Action ◽  
Principle Of Least Action ◽  
Smooth Transformation ◽  
As If

In fact, the first law of conservation (that of mass) was found in chemistry and generalized to the conservation of energy in physics by means of Einstein’s famous “E=mc2”. Energy conservation is implied by the principle of least action from a variational viewpoint as in Emmy Noether’s theorems (1918): any chemical change in a conservative (i.e. “closed”) system can be accomplished only in the way conserving its total energy. Bohr’s innovation to found Mendeleev’s periodic table by quantum mechanics implies a certain generalization referring to the quantum leaps as if accomplished in all possible trajectories (according to Feynman’s interpretation) and therefore generalizing the principle of least action and needing a certain generalization of energy conservation as to any quantum change. The transition from the first to the second theorem of Emmy Noether represents well the necessary generalization: its chemical meaning is the generalization of any chemical reaction to be accomplished as if any possible course of time rather than in the standard evenly running time (and equivalent to energy conservation according to the first theorem).The problem: If any quantum change is accomplished in all possible “variations (i.e. “violations) of energy conservation” (by different probabilities), what (if any) is conserved?An answer: quantum information is what is conserved. Indeed, it can be particularly defined as the counterpart (e.g. in the sense of Emmy Noether’s theorems) to the physical quantity of action (e.g. as energy is the counterpart of time in them). It is valid in any course of time rather than in the evenly running one. That generalization implies a generalization of the periodic table including any continuous and smooth transformation between two chemical elements.

On the Shoulders of Giants

2018 ◽  
Author(s):  
David D. Nolte
Keyword(s):  
Eighteenth Century ◽  
Robert Hooke ◽  
New Approach ◽  
Isaac Newton ◽  
Least Action ◽  
New Science ◽  
Principle Of Least Action ◽  
Parabolic Trajectory ◽  
Leonhard Euler ◽  
New Generation

Galileo’s parabolic trajectory launched a new approach to physics that was taken up by a new generation of scientists like Isaac Newton, Robert Hooke and Edmund Halley. The English Newtonian tradition was adopted by ambitious French iconoclasts who championed Newton over their own Descartes. Chief among these was Pierre Maupertuis, whose principle of least action was developed by Leonhard Euler and Joseph Lagrange into a rigorous new science of dynamics. Along the way, Maupertuis became embroiled in a famous dispute that entangled the King of Prussia as well as the volatile Voltaire who was mourning the death of his mistress Emilie du Chatelet, the lone female French physicist of the eighteenth century.

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

2020 ◽  
Author(s):  
Jerzy Warminski ◽  
Lukasz Kloda ◽  
Jaroslaw Latalski ◽  
Andrzej Mitura ◽  
Marcin Kowalczuk
Keyword(s):  
Time Delay ◽  
Control Strategies ◽  
Vibration Reduction ◽  
Least Action ◽  
Active Elements ◽  
Principle Of Least Action ◽  
Second Mode ◽  
Delay Control ◽  
Speed Range ◽  
System With Time Delay

AbstractNonlinear 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 ($$P-C$$ 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.

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