Variational Principles In Classical Mechanics

Two dramatically different philosophical approaches to classical mechanics were proposed during the 17th - 18th centuries. Newton developed his vectorial formulation that uses time-dependent differential equations of motion to relate vector observables like force and rate of change of momentum. Euler, Lagrange, Hamilton, and Jacobi, developed powerful alternative variational formulations based on the assumption that nature follows the principle of least action. These variational formulations now play a pivotal role in science and engineering.This book introduces variational principles and their application to classical mechanics. The relative merits of the intuitive Newtonian vectorial formulation, and the more powerful variational formulations are compared. Applications to a wide variety of topics illustrate the intellectual beauty, remarkable power, and broad scope provided by use of variational principles in physics.The second edition adds discussion of the use of variational principles applied to the following topics:(1) Systems subject to initial boundary conditions(2) The hierarchy of related formulations based on action, Lagrangian, Hamiltonian, and equations of motion, to systems that involve symmetries.(3) Non-conservative systems.(4) Variable-mass systems.(5) The General Theory of Relativity.Douglas Cline is a Professor of Physics in the Department of Physics and Astronomy, University of Rochester, Rochester, New York.

Philosophic, less formalistic approach to analytical mechanics offers model of clear, scholarly exposition at graduate level with coverage of basics, calculus of variations, principle of virtual work, equations of motion, more.

DIVHistorical, theoretical survey with many insights, much hard-to-find material. Hamilton’s principle, Hamilton-Jacobi equation, etc. /div

Two dramatically different philosophical approaches to classical mechanics were proposed during the 17th - 18th centuries. Newton developed his vectorial formulation that uses time-dependent differential equations of motion to relate vector observables like force and rate of change of momentum. Euler, Lagrange, Hamilton, and Jacobi, developed powerful alternative variational formulations based on the assumption that nature follows the principle of least action. These variational formulations now play a pivotal role in science and engineering.This book introduces variational principles and their application to classical mechanics. The relative merits of the intuitive Newtonian vectorial formulation, and the more powerful variational formulations are compared. Applications to a wide variety of topics illustrate the intellectual beauty, remarkable power, and broad scope provided by use of variational principles in physics.The second edition adds discussion of the use of variational principles applied to the following topics:(1) Systems subject to initial boundary conditions(2) The hierarchy of related formulations based on action, Lagrangian, Hamiltonian, and equations of motion, to systems that involve symmetries.(3) Non-conservative systems.(4) Variable-mass systems.(5) The General Theory of Relativity.Douglas Cline is a Professor of Physics in the Department of Physics and Astronomy, University of Rochester, Rochester, New York.

Given a conservative dynamical system of classical physics, how does one find a variational principle for it? Is there a canonical recipe for such a principle? The case of particle mechanics was settled by Lagrange in 1788; this text treats continuous systems. Recipes devised are algebraic in nature, and this book develops all the mathematical tools found necessary after the minute examination of the adiabatic fluid dynamics in the introduction. These tools include: Lagrangian and Hamiltonian formalisms, Legendre transforms, dual spaces of Lie algebras and associated 2-cocycles; and linearized and Z2-graded versions of all of these. The following typical physical systems, together with their Hamiltonian structures, are discussed: Classical Magnetohydro-dynamics with its Hall deformation; Multifluid Plasma; Superfluid He-4 (both irrotational and rotating) and 3He-A; Quantum fluids; Yang-Mills MHD; Spinning fluids; Spin Glass; Extended YM Plasma; A Lattice Gas. Detailed motivations, easy-to-follow arguments, open problems, and over 300 exercises help the reader. Request Inspection Copy

Thereareabout500booksonvariationalprinciples. Theyareconcernedmostlywith the mathematical aspects of the topic. The major goal of this book is to discuss the physical origin of the variational principles and the intrinsic interrelations between them. For example, the Gibbs principles appear not as the rst principles of the theory of thermodynamic equilibrium but as a consequence of the Einstein formula for thermodynamic uctuations. The mathematical issues are considered as long as they shed light on the physical outcomes and/or provide a useful technique for direct study of variational problems. Thebookisacompletelyrewrittenversionoftheauthor’smonographVariational Principles of Continuum Mechanics which appeared in Russian in 1983. I have been postponing the English translation because I wished to include the variational pr- ciples of irreversible processes in the new edition. Reaching an understanding of this subject took longer than I expected. In its nal form, this book covers all aspects of the story. The part concerned with irreversible processes is tiny, but it determines the accents put on all the results presented. The other new issues included in the book are: entropy of microstructure, variational principles of vortex line dynamics, va- ational principles and integration in functional spaces, some stochastic variational problems, variational principle for probability densities of local elds in composites with random structure, variational theory of turbulence; these topics have not been covered previously in monographic literature.

Variational principles have proven to be surprisingly fertile. For example, Fermat used variational methods to demonstrate that light follows the fastest route from one point to another, an idea which came to be a cornerstone of geometrical optics. This book explains variational principles and charts their use throughout modern physics. It examines the analytical mechanics of Lagrange and Hamilton, the basic tools of any physicist. The book also offers simple but rich first impressions of Einstein’s General Relativity, Feynman’s Quantum Mechanics, and more that reveal amazing interconnections between various fields of physics.