scholarly journals Power and beauty of the Lagrange equations

2020 ◽  
Vol 17 (1 Jan-Jun) ◽  
pp. 47
Author(s):  
Luis De la Peña ◽  
Ana María Cetto ◽  
Andrea Valdés-Hernández
Keyword(s):  
Variational Principle ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Integrals Of Motion ◽  
Lagrange Equations ◽  
Holonomic Constraints ◽  
Point Particles ◽  
Lagrange Formulation ◽  
Classical Particles ◽  
Advanced Treatments

The Lagrangian formulation of the equations of motion for point particles isusually presented in classical mechanics as the outcome of a series ofinsightful algebraic transformations or, in more advanced treatments, as theresult of applying a variational principle. In this paper we stress two mainreasons for considering the Lagrange equations as a fundamental descriptionof the dynamics of classical particles. Firstly, their structure can benaturally disclosed from the existence of integrals of motion, in a waythat, though elementary and easy to prove, seems to be less popular--or less frequently made explicit-- than others insupport of the Lagrange formulation. The second reason is that the Lagrangeequations preserve their form in \emph{any} coordinate system --even in moving ones, if required. Their covariant nature makes themparticularly suited to deal with dynamical problems in curved spaces orinvolving (holonomic) constraints. We develop the above and related ideas inclear and simple terms, keeping them throughout at the level of intermediatecourses in classical mechanics. This has the advantage of introducing sometools and concepts that are useful at this stage, while they may also serveas a bridge to more advanced courses.

scholarly journals The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories

2016 ◽  
Vol 24 (2) ◽  
pp. 173-193
Author(s):  
Jana Musilová ◽  
Stanislav Hronek
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Conservation Laws ◽  
Calculus Of Variations ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Stationary Points ◽  
General Description ◽  
Lagrange Equations ◽  
Jet Bundles

Abstract As widely accepted, justified by the historical developments of physics, the background for standard formulation of postulates of physical theories leading to equations of motion, or even the form of equations of motion themselves, come from empirical experience. Equations of motion are then a starting point for obtaining specific conservation laws, as, for example, the well-known conservation laws of momenta and mechanical energy in mechanics. On the other hand, there are numerous examples of physical laws or equations of motion which can be obtained from a certain variational principle as Euler-Lagrange equations and their solutions, meaning that the \true trajectories" of the physical systems represent stationary points of the corresponding functionals.It turns out that equations of motion in most of the fundamental theories of physics (as e.g. classical mechanics, mechanics of continuous media or fluids, electrodynamics, quantum mechanics, string theory, etc.), are Euler-Lagrange equations of an appropriately formulated variational principle. There are several well established geometrical theories providing a general description of variational problems of different kinds. One of the most universal and comprehensive is the calculus of variations on fibred manifolds and their jet prolongations. Among others, it includes a complete general solution of the so-called strong inverse variational problem allowing one not only to decide whether a concrete equation of motion can be obtained from a variational principle, but also to construct a corresponding variational functional. Moreover, conservation laws can be derived from symmetries of the Lagrangian defining this functional, or directly from symmetries of the equations.In this paper we apply the variational theory on jet bundles to tackle some fundamental problems of physics, namely the questions on existence of a Lagrangian and the problem of conservation laws. The aim is to demonstrate that the methods are universal, and easily applicable to distinct physical disciplines: from classical mechanics, through special relativity, waves, classical electrodynamics, to quantum mechanics.

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.

Elements of Classical Field Theory

2021 ◽  
pp. 24-34
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Classical Field ◽  
Lagrange Equations ◽  
Lie Group Of Transformations ◽  
Infinitesimal Transformations ◽  
Conserved Currents

The purpose of this chapter is to recall the principles of Lagrangian and Hamiltonian classical mechanics. Many results are presented without detailed proofs. We obtain the Euler–Lagrange equations of motion, and show the equivalence with Hamilton’s equations. We derive Noether’s theorem and show the connection between symmetries and conservation laws. These principles are extended to a system with an infinite number of degrees of freedom, i.e. a classical field theory. The invariance under a Lie group of transformations implies the existence of conserved currents. The corresponding charges generate, through the Poisson brackets, the infinitesimal transformations of the fields as well as the Lie algebra of the group.

Weyl–Euler–Lagrange equations on twistor space for tangent structure

2016 ◽  
Vol 13 (07) ◽  
pp. 1650095
Author(s):  
Zeki Kasap
Keyword(s):  
Mathematical Modeling ◽  
Dynamical System ◽  
Classical Mechanic ◽  
Twistor Space ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Equation System ◽  
Lagrange Equations ◽  
Four Manifolds ◽  
Riemannian Geometries

Twistor spaces are certain complex three-manifolds, which are associated with special conformal Riemannian geometries on four-manifolds. Also, classical mechanic is one of the major subfields for mechanics of dynamical system. A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space for classical mechanic. Euler–Lagrange equations are an efficient use of classical mechanics to solve problems using mathematical modeling. On the other hand, Weyl submitted a metric with a conformal transformation for unified theory of classical mechanic. This paper aims to introduce Euler–Lagrage partial differential equations (mathematical modeling, the equations of motion according to the time) for the movement of objects on twistor space and also to offer a general solution of differential equation system using the Maple software. Additionally, the implicit solution of the equation will be obtained as a result of a special selection of graphics to be drawn.

Introduction

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Mirror Symmetry ◽  
Complex Geometry ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Hamiltonian Formalism ◽  
Theoretical Physics ◽  
Lagrange Equations ◽  
Low Dimensional Topology ◽  
The Subject ◽  
Low Dimensional

Symplectic topology has a long history. It has its roots in classical mechanics and geometric optics and in its modern guise has many connections to other fields of mathematics and theoretical physics ranging from dynamical systems, low-dimensional topology, algebraic and complex geometry, representation theory, and homological algebra, to classical and quantum mechanics, string theory, and mirror symmetry. One of the origins of the subject is the study of the equations of motion arising from the Euler–Lagrange equations of a one-dimensional variational problem. The Hamiltonian formalism arising from a Legendre transformation leads to the notion of a ...

Hamiltonian Formulation for Continuous Third-order Systems Using Fractional Derivatives

2021 ◽  
Vol 14 (1) ◽  
pp. 35-47
Keyword(s):  
Variational Principle ◽  
Fractional Derivatives ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Lagrange Equations ◽  
Third Order ◽  
Derivative Operator ◽  
Hamilton Equations ◽  
Liouville Fractional Derivative ◽  
Continuous Field

Abstract: We constructed the Hamiltonian formulation of continuous field systems with third order. A combined Riemann–Liouville fractional derivative operator is defined and a fractional variational principle under this definition is established. The fractional Euler equations and the fractional Hamilton equations are obtained from the fractional variational principle. Besides, it is shown that the Hamilton equations of motion are in agreement with the Euler-Lagrange equations for these systems. We have examined one example to illustrate the formalism. Keywords: Fractional derivatives, Lagrangian formulation, Hamiltonian formulation, Euler-lagrange equations, Third-order lagrangian.

scholarly journals DIRECT QUANTIZATION OF EQUATIONS OF MOTION: FROM CLASSICAL DYNAMICS TO TRANSITION AMPLITUDES VIA STRINGS

2010 ◽  
Vol 07 (08) ◽  
pp. 1385-1405
Author(s):  
DENIS KOCHAN
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Equations Of Motion ◽  
Functional Integral ◽  
Classical Dynamics ◽  
World Sheet ◽  
Lagrange Equations ◽  
Standard Quantum ◽  
New Type ◽  
Fundamental Physical

New method of quantization is presented. It is based on classical Newton–Lagrange equations of motion (representing the fundamental physical law of mechanics) rather than on their traditional Lagrangian and/or Hamiltonian precursors. It is shown that classical dynamics is governed by canonical two-form Ω, which embodies kinetic energy and forces acting within the system. New type of variational principle employing differential two-form Ω and "umbilical strings" is introduced. The Feynman path integral over histories of the system is then rearranged to "umbilical world-sheet" functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, world-sheet approach reduces to the standard quantum mechanics. As an example Quantum Mechanics with friction is analyzed in detail.

FRACTIONAL TIME ACTION AND PERTURBED GRAVITY

Fractals ◽  
2011 ◽  
Vol 19 (02) ◽  
pp. 243-247 ◽  
Author(s):  
MADHAT SADALLAH ◽  
SAMI I. MUSLIH ◽  
DUMITRU BALEANU ◽  
EQAB RABEI
Keyword(s):  
Variational Principle ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Variational Problems ◽  
Integral Function ◽  
Lagrange Equations ◽  
Action Integral

In this paper, we used the scaling concepts of Mandelbrot of fractals in variational problems of mechanical systems in order to re-write the action integral function as an integration over the fractional time. In addition, by applying the variational principle to this new fractional action, we obtained the modified Euler-Lagrange equations of motion in any fractional time of order 0 < α ≤ 1. Two examples are investigated in detail.

scholarly journals Solving Equations of Motion by Using Monte Carlo Metropolis: Novel Method Via Random Paths Sampling and the Maximum Caliber Principle

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 916
Author(s):  
Diego González Diaz ◽  
Sergio Davis ◽  
Sergio Curilef
Keyword(s):  
Monte Carlo ◽  
Energy State ◽  
Equations Of Motion ◽  
Free Particle ◽  
Classical Mechanics ◽  
Canonical System ◽  
Path Space ◽  
Lagrange Equations ◽  
Solving Equations ◽  
Novel Method

A permanent challenge in physics and other disciplines is to solve Euler–Lagrange equations. Thereby, a beneficial investigation is to continue searching for new procedures to perform this task. A novel Monte Carlo Metropolis framework is presented for solving the equations of motion in Lagrangian systems. The implementation lies in sampling the path space with a probability functional obtained by using the maximum caliber principle. Free particle and harmonic oscillator problems are numerically implemented by sampling the path space for a given action by using Monte Carlo simulations. The average path converges to the solution of the equation of motion from classical mechanics, analogously as a canonical system is sampled for a given energy by computing the average state, finding the least energy state. Thus, this procedure can be general enough to solve other differential equations in physics and a useful tool to calculate the time-dependent properties of dynamical systems in order to understand the non-equilibrium behavior of statistical mechanical systems.

scholarly journals Weyl-Euler-Lagrange Equations of Motion on Flat Manifold

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Zeki Kasap
Keyword(s):  
Riemannian Manifold ◽  
Partial Differential Equations ◽  
Euclidean Space ◽  
Conformal Transformation ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Unified Theory ◽  
Flat Manifold ◽  
Lagrange Equations ◽  
Symbolic Algebra

This paper deals with Weyl-Euler-Lagrange equations of motion on flat manifold. It is well known that a Riemannian manifold is said to be flat if its curvature is everywhere zero. Furthermore, a flat manifold is one Euclidean space in terms of distances. Weyl introduced a metric with a conformal transformation for unified theory in 1918. Classical mechanics is one of the major subfields of mechanics. Also, one way of solving problems in classical mechanics occurs with the help of the Euler-Lagrange equations. In this study, partial differential equations have been obtained for movement of objects in space and solutions of these equations have been generated by using the symbolic Algebra software. Additionally, the improvements, obtained in this study, will be presented.

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