Caltagirone - Discrete mechanics: concepts and applications

Series Editor

Roger Prudhomme

Jean-Paul Caltagirone

First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Jean-Paul Caltagirone to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2018962477

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-78630-283-0

Nature is complex; we must accept it. A concept or a theory might initially develop as a series of reproducible experiments, previously established principles, carefully conducted thought experiments and so on. Any theory must then be subjected to repeated verification, usually of an experimental nature, to validate or disprove its hypotheses, axioms and principles. Most key milestones in the history of science were intuitively understood by their authors before they were formalized in terms of equations; the equations describing a theory can be adjusted over time as the state of mathematics progresses. This was, for example, the case with Newtons laws of dynamics, which were only rephrased into the modern formulation later, by Euler.

Some phenomena continue to defy our efforts to fully understand them, such as turbulence; we are not currently capable of giving an accurate a priori description of the mechanisms underlying the interactions of vortices, or the processes by which they are created and dissipated. In some of the more accessible cases, simulations are our only tool for understanding turbulence at every scale and analyzing the physical behavior of turbulent flows in more depth. Simulation-based approaches rely on the equations of fluid mechanics, which are derived from Newtons second law. This perspective is expanded into a method: first, an equation is derived, which is validated in simple cases; this equation is then solved to explore the behavior of more complex phenomena. Over time, with enough simulation and exploration, the model is refined and our understanding of the phenomenon improves alongside it. The equation becomes more intelligent than its author.

But, in mechanics, the laws of dynamics and their modern counterparts the equations of motion still remain shrouded in some mystery, with hidden surprises for the attentive observer. The equations of motion are capable of describing the behavior of many problems involving fluids, solids and waves. However, unification has not yet been achieved; for example, the formulations used for solids and fluids do not coincide, even though performing mechanics on an underlying continuum was supposed to guarantee precisely this type of consistency. The equations of continuum mechanics hold hidden structural flaws that are unraveled in this book; the origins of each issue vary, but our most important objection will be raised against the concept of a continuum itself. The results themselves offered by continuum mechanics are usually not invalid, given that they are perfectly consistent with experimental observations; however, the theory is afflicted by deeper problems that are effectively covered up by redundancy in its principles and laws.

Discrete mechanics abandons the notion of a continuum, the principle of local thermodynamic equilibrium, the state equations and constitutive equations, second-order tensors, differentiation at a point, analysis and so on. Building from the ideas proposed by Galileo, the equivalence principle and the notion of relativity, the discrete approach to mechanics begins by defining a geometric basis of space that cannot be reduced to a single point. The discrete equations of motion are then formally derived from a set of hypotheses, axioms and principles.

Jean-Paul CALTAGIRONE

November 2018

The laws of physics were established from experimental observations by finding simple relations to describe these observations; for example, the hypothesis of linearity between the flux and the force was established as a general principle of thermodynamics. Once a relation has been chosen, a coefficient of proportionality is selected. This coefficient is typically viewed as an intrinsic physical property, even though it depends directly on the choice of equations. If the law is overly simplistic, inconsistencies will be encountered when we attempt to generalize to parameter intervals larger than those originally considered. Fortunately, the complexity of the observed physical behavior typically allows us to reconstruct enough consistency to cover a broad spectrum of the parameter space. If only the thermodynamic coefficients themselves have any true physical significance, why should we need to introduce an ideal gas law between the variables, other than for convenience?

The laws of mechanics are some of the oldest laws in physics; throughout history, Galileo, Issac Newton and Albert Einstein each left a prodigious mark on our understanding of the universe around us. The concepts of force, mass and acceleration linked together by Newtons second law have not changed in modern mechanics. Over the past three centuries, a great succession of physicists, mathematicians and engineers gradually perfected the laws and equations of mechanics, which today seem inexorably set in stone. Leonhard Euler, Maurice Couette, Daniel Bernoulli, Ernst Mach, George Gabriel Stokes, Henri Navier, Clifford Truesdell and many others contributed to formalizing the laws of mechanics within a modern mathematical framework. Much has been written on the epistemology of the connections between the theories of these famous individuals, and occasionally the personal relationships between them. The path taken by scientific thought over time seems natural and logical if we consider the path taken by its equations, from classical mechanics to relativistic mechanics. Sometimes, it is not entirely clear where exactly new ideas came from, especially during periods where publication could not always be taken for granted and religion sometimes interfered with the propagation of new theories. Together, this scientific work led to equations of mechanics that have been frozen in time for decades or perhaps even centuries; today, these equations allow us to simulate all kinds of motion of solids, fluids, propagating waves, etc., extremely realistically. Yet, the equations of each type of motion are very different, even though continuum mechanics was supposed to establish a unified representation of all physical phenomena within the same set of equations.

How might we have established the laws of mechanics from scratch if we had access to every observation in history? This book attempts to give an answer. We start with the following question: why did Isaac Newton formulate his fundamental laws of dynamics as an equality of forces, given that he was aware of the equivalence between inertial mass and gravitational mass? This equivalence principle is one of the cornerstones of Albert Einsteins theory of general relativity. Discrete mechanics also relies heavily on this principle, using it to eliminate the concept of mass, which is not required to describe an equality between accelerations in mechanics. The notions of rest mass and relativistic mass at the heart of relativistic mechanics clearly characterize mass as equivalent to energy. Similarly, the fundamental law of dynamics can be formulated as an equality between the acceleration experienced by a particle and the sum of the accelerations applied to this particle.