Variational methods and complementary formulations in dynamics
type: Item
instance of: scholarly article
title: Variational methods and complementary formulations in dynamics
zbMATH Open document ID: 0845.70001
published in: Solid Mechanics and Its Applications
publication date: 1996-04-08T00:00:00Z
review text: In analytical mechanics the fundamental laws can be expressed in terms of work done and energy exchanged. The extensive use of mathematics is a consequence of the fact that in analytical mechanics problems can be expressed by variational statements, thus giving rise to the employment of variational methods. Further it can be shown that the independent variables may be either displacements or impulses, thus providing in principle the possibility of two complementary formulations, i.e. a displacement formulation and an impulse formulation, for each problem. This duality is an important characteristic of mechanics problems, and special emphasis in the present book is put on it. The book begins by discussing fundamental concepts such as coordinate transformation, degrees of freedom, virtual displacement and virtual work, holonomic and nonholonomic constraints, work, potential energy, duality, complementary potential energy, kinetic energy, complementary kinetic energy, and generalized variables, all in the first chapter. In the second chapter variational methods are used to develop extremum formulations of differential type for problems in dynamics. The chapter begins with D'Alembert's principle, and then introduces Gauss' principle of least constraint, Hertz' principle of least curvature in configuration space, and ends with Lagrange's equation, in dual representation, i.e. in displacement and impulse variables. The third chapter presents variational methods in integral form. Duality, e.g. formulation in conventional and complementary form, is again emphasized, and Hamilton's principle is discussed, as well as the significance of Lagrange multipliers, and the concept of ignorable coordinates. The fourth chapter deals with canonical transformations, the fifth is devoted to the applications of variational methods in gyrodynamics, and the sixth chapter illustrates some of the foregoing theory in special applications. The text is intended for senior undergraduate or for gr…
Mathematics Subject Classification ID: 70-01
Mathematics Subject Classification ID: 70H30
Mathematics Subject Classification ID: 70Hxx
zbMATH DE Number: 863244
zbMATH Keywords: displacement formulation
zbMATH Keywords: impulse formulation
zbMATH Keywords: duality
zbMATH Keywords: virtual displacement
zbMATH Keywords: virtual work
zbMATH Keywords: holonomic and nonholonomic constraints
zbMATH Keywords: generalized variables
zbMATH Keywords: D'Alembert's principle
zbMATH Keywords: Gauss' principle of least constraint
zbMATH Keywords: Hertz' principle of least curvature
zbMATH Keywords: Lagrange's equation
zbMATH Keywords: Hamilton's principle
zbMATH Keywords: Lagrange multipliers
zbMATH Keywords: concept of ignorable coordinates
zbMATH Keywords: canonical transformations
zbMATH Keywords: gyrodynamics
author: B. Tabarrok
author: F. P. J. Rimrott
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 863244