Classical Mechanics (2009)

Gregory, R. Douglas

Gregory's Classical Mechanics is a major new textbook for undergraduates in mathematics and physics. It is a thorough, self-contained and highly readable account of a subject many students find difficult. The author's clear and systematic style promotes a good understanding of the subject: each concept is motivated and illustrated by worked examples, while problem sets provide plenty of practice for understanding and technique. Computer assisted problems, some suitable for projects, are also included. The book is structured to make learning the subject easy; there is a natural progression from core topics to more advanced ones and hard topics are treated with particular care. A theme of the book is the importance of conservation principles. These appear first in vectorial mechanics where they are proved and applied to problem solving. They reappear in analytical mechanics, where they are shown to be related to symmetries of the Lagrangian, culminating in Noether's theorem. • Suitable for a wide range of undergraduate mechanics courses given in mathematics and physics departments: no prior knowledge of the subject is assumed • Profusely illustrated and thoroughly class-tested, with a clear direct style that makes the subject easy to understand: all concepts are motivated and illustrated by the many worked examples included • Good, accurately-set problems, with answers in the book: computer assisted problems and projects are also provided. Model solutions for problems available to teachers from cambridge university press Cambridge,

This paper presents an alternative classical mechanics which is invariant under transformations between reference frames and which can be applied in any reference frame without the necessity of introducing fictitious forces. Additionally, a new principle of conservation of energy is also presented.

2019

The first postulate of the classical mechanics, stating that the position and the time are independent, is demonstrated as false, and replaced by a theorem stating that the position and the time are always related by a bijection, accordingly to the experiment. Introducing this theorem instead of the first postulate inside the calculus of variation, provides new equations of motion, close to those of Lagrange, but giving more information on the allowed trajectories. The velocity of a classical mobile appears as the addition of one or many of only two elementary uniform velocities, of rotation and translation, in a typical Fourier series fashion. The addition of a single elementary rotation and a single elementary translation, leads to the Keplerian motion, as expected. This approach can be used for any physical parameter, an illustration is given by the forecast of the Boltzmann’s entropy, the ideal gas law and the equations driving the chemical kinetics.

Index xiv Statistical Mechanics Made Simple xx Classical Mechanics dΩ solid angle 4.5 ω, Ω angular velocity 5.1, 9.2, 9.7 ω angular frequency 2.2, 11.3 ω i natural angular frequencies 14.1 ∇ vector differential operator A.5 a • b scalar product of a and b A.2 a ∧ b vector product of a and b A.3 Vectors It is often convenient to use a notation which does not refer explicitly to a particular set of coordinate axes. Instead of using Cartesian coordinates x, y, z we may specify the position of a point P with respect to a given origin O by the length and direction of the line OP. A quantity which is specified by a magnitude and a direction is called a vector; in this case the F i = F i1 + F i2 + • • • + F iN = N j=1 F ij , (1.2) where of course F ii = 0, since there is no force on the ith body due to itself. Note that since the sum on the right side of (1.2) is a vector sum, this equation incorporates the 'parallelogram law' of composition of forces.

Journal of functional Programming, 2003

We would like to thank the many people who have helped us to develop this book and the curriculum that it is designed to support. We have had substantial help from the wonderful students who studied with us in our classical mechanics class. They have forced us to be clear; they have found bugs that we had to fix, in the software, in the presentation, and in our thinking. We have had considerable technical help in the development and presentation of the subject matter from Harold Abelson. Abelson is one of the developers of the Scmutils software system. He put mighty effort into some sections of the code. We also consulted him when we were desperately trying to understand the logic of mechanics. He often could propose a direction to lead out of an intellectual maze. Matthew Halfant started us on the development of the Scmutils system. He encouraged us to get into scientific computation, using Scheme and functional style as an active way to explain the ideas, without the distractions of imperative languages such as C. In the 1980's he wrote some of the early Scheme procedures for numerical computation that we still use. Dan Zuras helped us with the invention of the unique organization of the Scmutils system. It is because of his insight that the system is organized around a generic extension of the chain rule for taking derivatives. He also helped in the heavy lifting that was required to make a really good polynomial GCD algorithm, based on ideas that we learned from Richard Zippel. This book, and a great deal of other work of our laboratory, could not have been done without the outstanding work of Chris Hanson. Chris developed and maintained the Scheme system underlying this work. In addition, he took us through a pass of reorganization of the Scmutils system that forced the clarification of many of the ideas of types and of generic operations that make our system as good as it is. Guillermo Juan Rozas, co-developer of the Scheme system, made major contributions to the Scheme compiler, and imple-xviii Acknowledgments mented a number of other arcane mechanisms that make our system efficient enough to support our work. Besides contributing to some of the methods for the solution of linear equations in the Scmutils system, Jacob Katzenelson has provided valuable feedback that improved the presentation of the material. Julie Sussman, PPA, provided careful reading and serious criticism that forced us to reorganize and rewrite major parts of the text. She also developed and maintained Gerald Jay Sussman over these many years. Cecile Wisdom, saint, is a constant reminder, by her faith and example, of what is really important. This project would not have been possible without the loving support and unfailing encouragement she has given Jack Wisdom. Their children, William, Edward, Thomas, John, and Elizabeth Wisdom, each wonderfully created, daily enrich his life with theirs. Meinhard Mayer wants to thank Rita Mayer, for patient moral support, particularly during his frequent visits to MIT during the last 12 years; Niels Mayer for introducing him to the wonderful world of Scheme (thus sowing the seeds for this collaboration), as well as Elma and the rest of the family for their love. Many have contributed to our understanding of dynamics over the years. Michel Henon and Boris Chirikov have had particular influence. Stan Peale, Peter Goldreich, Alar Toomre, and Scott Tremaine have been friends and mentors. We thank former students Jihad Touma and Matthew Holman, for their collaboration and continued friendship. We have greatly benefited from associations with many in the nonlinear dynamics community: Tassos