scholarly journals CONNEXION OF FIRST INTEGRALS WITH PARTICULAR SOLUTIONS OF THE VARIATIONAL EQUATIONS FOR SYSTEMS OF GENERALIZED CLASSICAL MECHANICS

2001 ◽  
Vol 50 (11) ◽  
pp. 2059
Author(s):  
ZHANG YI
Keyword(s):  
Classical Mechanics ◽  
First Integrals ◽  
Variational Equations ◽  
Particular Solutions

An Observation on Least Action Principle in Classical Mechanics Oriented on the Evaluation of Relativistic Error Concerning the Measurements of Light Propagating in a Liquid

2011 ◽  
Vol 1 (3) ◽  
pp. 14-24 ◽  
Author(s):  
Alessandro Massaro ◽  
Piero Adriano Massaro
Keyword(s):  
Euler Equations ◽  
Classical Mechanics ◽  
First Integrals ◽  
Action Principle ◽  
Classical Approach ◽  
Limit Case ◽  
Least Action Principle ◽  
Calculus Of Variation ◽  
Least Action ◽  
Approximate First Integrals

The authors prove that the standard least action principle implies a more general form of the same principle by which they can state generalized motion equation including the classical Euler equation as a particular case. This form is based on an observation regarding the last action principle about the limit case in the classical approach using symmetry violations. Furthermore the well known first integrals of the classical Euler equations become only approximate first integrals. The authors also prove a generalization of the fundamental lemma of the calculus of variation and we consider the application in electromagnetism.

scholarly journals Impulse control of a two-link manipulation robot

2021 ◽  
Vol 57 ◽  
pp. 77-90
Author(s):  
Yu.F. Dolgii ◽  
I.A. Chupin
Keyword(s):  
Computer Simulation ◽  
Configuration Space ◽  
Impulse Control ◽  
Nonlinear Differential Equations ◽  
Classical Mechanics ◽  
First Integrals ◽  
Initial Moment ◽  
Analytical Integration ◽  
Impulse Controls ◽  
Manipulation Robot

A nonlinear problem of controlling the movements of a two-link manipulation robot is considered. The free mechanical system has two first integrals in involution. Methods of classical mechanics are used for analytical integration of the system of nonlinear differential equations. A trajectory connecting the initial and final positions of the two-link manipulation robot in the configuration space is found. Impulse controls at the initial moment of time impart the necessary energy to the robot to enter this trajectory. Impulse controls are also used to damp the speeds of the robot at the end position. In a computer simulation of the proposed procedure for moving the robot, generalized impulse controls are approximated by rectangular impulses.

An Observation on Least Action Principle in Classical Mechanics Oriented on the Evaluation of Relativistic Error Concerning the Measurements of Light Propagating in a Liquid

2013 ◽  
pp. 166-176
Author(s):  
Alessandro Massaro ◽  
Piero Adriano Massaro
Keyword(s):  
Euler Equations ◽  
Classical Mechanics ◽  
First Integrals ◽  
Action Principle ◽  
Classical Approach ◽  
Limit Case ◽  
Least Action Principle ◽  
Calculus Of Variation ◽  
Least Action ◽  
Approximate First Integrals

The authors prove that the standard least action principle implies a more general form of the same principle by which they can state generalized motion equation including the classical Euler equation as a particular case. This form is based on an observation regarding the last action principle about the limit case in the classical approach using symmetry violations. Furthermore the well known first integrals of the classical Euler equations become only approximate first integrals. The authors also prove a generalization of the fundamental lemma of the calculus of variation and we consider the application in electromagnetism.

Vector Calculus

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Differential Equations ◽  
Classical Mechanics ◽  
Fourier Method ◽  
Mathematical Physics ◽  
First Integrals ◽  
Series Solutions ◽  
Initial Value ◽  
Common Trend ◽  
Vector Calculus ◽  
Integral Curves

This chapter gives a non-technical overview of differential equations from across mathematical physics, with particular attention to those used in the book. It is a common trend in physics and nature, or perhaps just the way numbers and calculus come together, that, to describe the evolution of things, most theories use a differential equation of low order. This chapter is useful for those with no prior knowledge of the differential equations and explains the concepts required for a basic exposition of classical mechanics. It discusses separable differential equations, boundary conditions and initial value problems, as well as particular solutions, complete solutions, series solutions and general solutions. It also discusses the Cauchy–Lipschitz theorem, flow and the Fourier method, as well as first integrals, complete integrals and integral curves.

VECTOR FIELDS, VARIATIONAL EQUATIONS AND COMMUTATORS

2012 ◽  
Vol 22 (08) ◽  
pp. 1250190
Author(s):  
WILLI-HANS STEEB ◽  
YORICK HARDY ◽  
IGOR TANSKI
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Fixed Points ◽  
Lie Algebra ◽  
Autonomous System ◽  
Vector Fields ◽  
Autonomous Systems ◽  
First Integrals ◽  
Variational Equations ◽  
First Order

We study autonomous systems of first order ordinary differential equations, their corresponding vector fields and the autonomous system corresponding to the vector field of the commutator of two such autonomous systems. These vector fields form a Lie algebra. From the variational equations of these autonomous systems, we form new vector fields consisting of the sum of the two vector fields. We show that these new vector fields also form a Lie algebra. Results about fixed points, first integrals and the divergence of the vector fields are also presented.

From Einstein Theories to Least Action Principle, a Relativistic Error of a Limit Case of Classical Mechanics

2019 ◽  
pp. 22-46
Author(s):  
Alessandro Massaro
Keyword(s):  
Light Propagation ◽  
Classical Mechanics ◽  
First Integrals ◽  
Action Principle ◽  
Limit Case ◽  
Least Action Principle ◽  
Motion Error ◽  
Least Action ◽  
Approximate First Integrals ◽  
Error Condition

The authors prove that the standard least action principle implies a more general form of the same principle by which they can state generalized motion equation including the classical Euler equation as a particular case. This form is based on an observation regarding the last action principle about the limit case in the classical approach using symmetry violations. Furthermore, the well-known first integrals of the classical Euler equations become only approximate first integrals. The authors also prove a generalization of the fundamental lemma of the calculus of variation and they consider the application in electromagnetism. This chapter is an enhanced version of a published work. It proves the existence of particular relativistic error condition in classical mechanics, potentially significant on experiments of light propagation in matters. The work includes a discussion of applications potentially correlated with the found particle motion error condition.

scholarly journals Complex Dynamics of Some Hamiltonian Systems: Nonintegrability of Equations of Motion

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Jingjia Qu
Keyword(s):  
Galois Group ◽  
Dynamic Behavior ◽  
Hamiltonian Systems ◽  
Complex Dynamics ◽  
Large Range ◽  
Equations Of Motion ◽  
Differential Galois Group ◽  
Variational Equations ◽  
Particular Solutions ◽  
Double Well Potential

The main purpose of this paper is to study the complexity of some Hamiltonian systems from the view of nonintegrability, including the planar Hamiltonian with Nelson potential, double-well potential, and the perturbed elliptic oscillators Hamiltonian. Some numerical analyses show that the dynamic behavior of these systems is very complex and in fact chaotic in a large range of their parameter. I prove that these Hamiltonian systems are nonintegrable in the sense of Liouville. My proof is based on the analysis of normal variational equations along some particular solutions and the investigation of their differential Galois group.

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