On quadratic first integrals of a particular natural system in classical mechanics

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.

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Impulse control of a two-link manipulation robot

10.35634/2226-3594-2021-57-02 ◽

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.

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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

Advanced Instrument Engineering ◽

10.4018/978-1-4666-4165-5.ch013 ◽

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.

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Higher order first integrals in classical mechanics

Journal of Mathematical Physics ◽

10.1063/1.2789555 ◽

2007 ◽

Vol 48 (10) ◽

pp. 102902 ◽

Cited By ~ 3

Author(s):

Joshua T. Horwood

Keyword(s):

Classical Mechanics ◽

Higher Order ◽

First Integrals

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Vector Calculus

10.1093/oso/9780198822370.003.0034 ◽

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.

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

Strategic Applications of Measurement Technologies and Instrumentation - Advances in Chemical and Materials Engineering ◽

10.4018/978-1-5225-5406-6.ch002 ◽

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.

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