The objective of this monograph is to present some methodological foundations of theoretical mechanics that are recommendable to graduate students prior to, or jointly with, the study of more advanced topics such as statistical mechanics, thermodynamics, and elementary particle physics. A program of this nature is inevitably centered on the methodological foundations for Newtonian systems, with particular reference to the central equations of our theories, that is, Lagrange's and Hamilton's equations. This program, realized through a study of the analytic representations in terms of Lagrange's and Hamilton's equations of generally nonconservative Newtonian systems (namely, systems with Newtonian forces not necessarily derivable from a potential function), falls within the context of the so-called Inverse Problem, and consists of three major aspects: l. The study of the necessary and sufficient conditions for the existence of a Lagrangian or Hamiltonian representation of given equations of motion with arbitrary forces; 2. The identification of the methods for the construction of a Lagrangian or Hamiltonian from given equations of motion verifying conditions 1; and 3 The analysis of the significance of the underlying methodology for other aspects of Newtonian Mechanics, e. g. , transformation theory, symmetries, and first integrals for nonconservative Newtonian systems. This first volume is devoted to the foundations of the Inverse Problem, with particular reference to aspects I and 2.

This book provides a concise description of the current status of a fascinating scientific problem -- the inverse variational problem in classical mechanics. No advanced methods of contemporary differential geometry are used, and all functions are assumed to be piecewise continuous and sufficiently many times differentiable. The book is well suited to be used as lecture notes in a university course for physicists.

In this second volume on the Foundations of Quantum Mechanics we shall show how it is possible, using the methodology presented in Volume I, to deduce some of the most important applications of quantum mechanics. These deductions are concerned with the structures of the microsystems rather than the technical details of the construction of preparation and registration devices. Accordingly. the only new axioms (relative to Volume I) which are introduced are concerned with the relationship between ensemble operators W, effect operators F, and certain construction principles of the preparation and registration devices. The applications described here are concerned with the measurement of atomic and molecular structure and of collision experiments. An additional and essential step towards a theoretical description of the preparation and registration procedures is carried out in Chapter XVII. Here we demonstrate how microscopic collision processes (that is, processes which can be described by quantum mechanics) can be used to obtain novel preparation and registration procedures if we take for granted the knowledge of only a few macroscopic preparation and registration procedures. By clever use of collision processes we are often able to obtain very precise results for the operators Wand F which describe the total procedures from a very imprecise knowledge of the macroscopic parts of the preparation and regis tration processes. In this regard experimental physicists have done brilliant work. In this sense Chapter XVII represents a general theoretical foundation for the procedures used by experimental physicists.

This book is a compilation of different methods of formulating and solving inverse problems in physics from classical mechanics to the potentials and nucleus-nucleus scattering. Mathematical proofs are omitted since excellent monographs already exist dealing with these aspects of the inverse problems.The emphasis here is on finding numerical solutions to complicated equations. A detailed discussion is presented on the use of continued fractional expansion, its power and its limitation as applied to various physical problems. In particular, the inverse problem for discrete form of the wave equation is given a detailed exposition and applied to atomic and nuclear scattering, in the latter for elastic as well as inelastic collision. This technique is also used for inverse problem of geomagnetic induction and one-dimensional electrical conductivity. Among other topics covered are the inverse problem of torsional vibration, and also a chapter on the determination of the motion of a body with reflecting surface from its reflection coefficient.

In this book we describe the evolution of Classical Mechanics from Newton's laws via Lagrange's and Hamilton's theories with strong emphasis on integrability versus chaotic behavior.In the second edition of the book we have added historical remarks and references to historical sources important in the evolution of classical mechanics.

This textbook — appropriate for a one-semester course in classical mechanics at the late undergraduate or early graduate level — presents a fresh, modern approach to mechanics. About 150 exercises, covering a wide variety of topics and applications, have solutions roughly outlined for enhanced understanding. Unique to this text is the versatile application of programming language Mathematica™ throughout to analyze systems and generate results. Coverage is also devoted to the topic on one dimensional continuum systems. The extensive discussions on inverse problems of mechanical systems and the detailed analysis of stability of classical systems certainly make this an outstanding textbook.

A first consequence of this difference in texture concerns the attitude we must take toward some (or perhaps most) investigations in "applied mathe matics," at least when the mathematics is applied to physics. Namely, those investigations have to be regarded as pure mathematics and evaluated as such. For example, some of my mathematical colleagues have worked in recent years on the Hartree-Fock approximate method for determining the structures of many-electron atoms and ions. When the method was intro duced, nearly fifty years ago, physicists did the best they could to justify it, using variational principles, intuition, and other techniques within the texture of physical reasoning. By now the method has long since become part of the established structure of physics. The mathematical theorems that can be proved now (mostly for two- and three-electron systems, hence of limited interest for physics), have to be regarded as mathematics. If they are good mathematics (and I believe they are), that is justification enough. If they are not, there is no basis for saying that the work is being done to help the physicists. In that sense, applied mathematics plays no role in today's physics. In today's division of labor, the task of the mathematician is to create mathematics, in whatever area, without being much concerned about how the mathematics is used; that should be decided in the future and by physics.

This book is the first volume of a two-volume work on the Foundations of Quantum Mechanics, and is intended as a new edition of the author's book Die Grundlagen der Quantenmechanik [37] which was published in 1954. In this two-volume work we will seek to obtain an improved formulation of the interpretation of quantum mechanics based on experiments. The second volume will appear shortly. Since the publication of [37] there have been several attempts to develop a basis for quantum mechanics which is, in the large part, based upon the work of J. von Neumann [38]. In particular, we mention the books ofG. W. Mackey [39], J. Jauch [40], C. Piron [41], M. Drieschner [9], and the original work ofS. P. Gudder [42], D.J. Foulis and C.H. Randall [43], and N. Zierler [44]. Here we do not seek to compare these different formulations of the foundations of quantum mechanics. We refer interested readers to [45] for such comparisons.