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This textbook provides a rigorous yet concise introduction to Ordinary Differential Equations (ODEs) and Dynamical Systems, addressing classical topics like Sturm-Liouville boundary value problems and modern qualitative theory. Its development stems from the author's experience in teaching and incorporating feedback from students and peers. The manuscript has evolved through multiple revisions, aiming to meet educational needs and has gained international recognition, evidenced by citations and various online references.
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.
Springer eBooks, 2018
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
2010
Formalism of classical mechanics underlies a number of powerful mathematical methods, widely used in theoretical and mathematical physics . In these lectures we present some selected topics of classical mechanics, which may be useful for graduate level students intending to work in one of the branches of a vast field of theoretical physics. Except for the last chapter, which is devoted to the discussion of singular theories and their local symmetries, the topics selected correspond to the standard course of classical mechanics.
DiBenedetto, Emmanuele
, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
Chinmoy Taraphdar, 2007
All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise, without the prior written permission of the publisher.
Foundations of Physics, 2005
We show that in classical mechanics the momentum may depend only on the coordinates and can thus be considered as a field. We formulate a special Lagrangian formalism as a result of which the momenta satisfy differential equations which depend only on the coordinates. The solutions correspond to all possible trajectories. As a bonus the Hamilton-Jacobi equation results in a very simple way.
This textbook covers all the standard introductory topics in classical mechanics, including Newton's laws, oscillations, energy, momentum, angular momentum, planetary motion, and special relativity. It also explores more advanced topics, such as normal modes, the Lagrangian method, gyroscopic motion, fictitious forces, 4-vectors, and general relativity.
2011
This is not a book! These are personal notes written while preparing lectures on several courses at the UM. They are rather informal and may even contain mistakes. I tried to be as synthetic as I could, without missing the observations that I consider important. I probably will not lecture all I wrote, and did not write all I plan to lecture. So, I included empty or sketched paragraphs, about material that I think should/could be lectured within the same course. The last section, “Bestiario”, is a list of famous examples proposed to students for their projects. References contain some introductory manuals, some classics, and other books where I have learnt things in the past century. Besides, good material and further references can easily be found on the web, for example in Wikipedia. Pictures were made with “Grapher ” on my MacBook. This work is licensed under a
2018
A revision of different first order ODE numerical integration schemes is presented in the ambit of classical mechanics. Their performance is tested on a rescaled SHO, and their traits and efficiency discussed. From these, an RK4 method is chosen to study a Duffing-Holmes oscillator. Its nonlinearity is shown to cause a period-doubling route to chaos through the exploration of a particular range of the forcing amplitude parameter using a bifurcation diagram.
2019
This is an english version of the notes written for my lectures on "Tópicos de Sistemas Dinâmicos" for the "Licenciatura em Matemática" of the University of Minho, during the last decade (available at my page ). Emphasis is on examples, and on the interplay between different areas of mathematics. Some very important parts of the modern theory of dynamical systems, as hyperbolic theory, hamiltonian systems, or the qualitative theory of differential equations, are almost completely missing. Other interesting results or directions are only sketched. Main references and sources are [KH95, HK03], others are suggested along the text. e.g. means exempli gratia, that is, "for example", and is used to introduce important or interesting examples. ex: means "exercise", to be solved at home or in the classroom. indicates the end of a proof. Pictures were made with Grapher on my MacBook, or taken from Wikipedia, or produced with my own Java codes, like the one below.
Classical Mechanics is the most basic part of the Physics. You may Learn about the basic notation in Dynamics. You may learn by yourself with Computer and Mathematica. Rich text with Mathematica examples in Electromagnetic Field.
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