Variational Methods and Complementary Formulations in Dynamics

Not many disciplines can c1aim the richness of creative ideas that make up the subject of analytical mechanics. This is not surprising since the beginnings of analyti cal mechanics mark also the beginnings of the theoretical treatment of other physical sciences, and contributors to analytical mechanics have been many, inc1uding the most brilliant mathematicians and theoreticians in the history of mankind. As the foundation for theoretical physics and the associated branches of the engineering sciences, an adequate command of analytical mechanics is an essential tool for any engineer, physicist, and mathematician active in dynamics. A fascinating dis cipline, analytical mechanics is not only indispensable for the solution of certain mechanics problems but also contributes so effectively towards a fundamental under standing of the subject of mechanics and its applications. In analytical mechanics the fundamental laws are 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 State ments, 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 dis placement formulation and an impulse formulation, for each problem. This duality is an important characteristic of mechanics problems and is given special emphasis in the present book.

Klappentext

Variational methods provide a versatile framework for several branches of theoretical mechanics. For problems in dynamics, variational formulations provide a powerful alternative to vector methods. This approach has a rich legacy of ideas advanced by numerous researchers including such celebrated mathematicians as d'Alembert, Lagrange, Hamilton, Jacobi, Gauss and Euler. In this volume, the subject matter is developed systematically with many worked-out problems. Initially, differential variational formulations are described followed by the integral formulations. A detailed account of the essentials of the calculus of variations is provided. While classical formulations in dynamics have a long history, the complementary formulations are relatively new. This book is the first to provide a detailed development of complementary formulations and also highlights certain dualities that are revealed as a consequence of the two formulations. A chapter on special applications studies problems of small amplitude oscillations about equilibrium and steady state configurations, and the problem of impulsive or spike loads. The book ends with historical sketches of the personalities associated with variational methods in dynamics. For structural, mechanical and aeronautical engineers. This volume can also be recommended as a graduate text in analytic dynamics.

Inhalt
I Fundamentals.- II Differential Variational Formulations.- III Integral Variational Formulations.- IV Canonical Transformations and the Hamilton-Jacobi Equation.- V Rigid Body Dynamics.- VI Special Applications.- Appendix A The Calculus of Variations.- A.1 Functions and Functionals.- A.2 Review of Extremum Values of Functions.- A.3 Stationary Values of Definite Integrals.- A.4 A Note about Weak and Strong Variations.- A.5 An Alternative Expression for a Single Euler-Lagrange Equation.- A.6 The Brachystochrone Problem.- A.7 Path-independent Functionals.- A.8 Several Dependent Functions.- A.9 Variational Notation.- A.10 Constraint Equations.- Lagrange Multipliers.- Algebraic and Differential Equation Constraints.- A.11 Variable End Points.- Suggested Reading.- Appendix B Developments in Mechanics Some Historical Perspectives.- Author Index.