Variational methods and complementary formulations in dynamics
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 graduate students, as well as for engineers, physicists, and mathematicians engaged in work in various branches of theoretical and applied mechanics.
scientific article; zbMATH DE number 863244