A Modern Approach to Classical Mechanics

Harald Iro

• Publisher: WSPC
• Year: 2015
• Edition: 2
• eBook ISBN:
  9789814696302
• Topic: Physical Sciences
• Subtopic: Electromagnetism

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.

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Contents:

• Basic Considerations and Concepts
• Fundamentals of Classical Mechanics
• One-Dimensional Motion of a Particle
• Peculiar Motion in Two Dimensions
• Motion in a Central Force
• Gravitational Force Between Two Bodies
• Collisions of Particles. Scattering
• Changing the Frame of Reference
• Lagrangian Mechanics
• Conservation Laws and Symmetries
• The Rigid Body
• Small Oscillations
• Hamiltonian Mechanics
• Hamilton–Jacobi Theory
• Three-body Systems
• Approximating Non-Integrable Systems

Readership: Advanced undergraduates, academics, and lecturers who may have used the first edition and would recommend their teaching assistants and students to use an updated edition in their own graduate study.
Key Features:

• As compared to other books, integrability and chaotic behavior are considered right from the beginning and are thus naturally incorporated in the text. A short historical background is presented throughout the book either in the form of notes or of references

1

Basic considerations and concepts

1.1 Why classical mechanics is still challenging

In the latter half of the 19th century, physics was widely considered to be complete. The basic pillar of the physics was Newtonian Mechanics, and this was augmented by the theories of Lagrange, Hamilton and Jacobi. So at this time, increased focus was directed at the epistemology of physics, particularly mechanics. E. MACH¹, to mention just one prominent thinker, thoroughly expounded epistemological aspects of mechanics, in his book, “The Science of Mechanics – a Critical Account of its Development” (see Bibliography).

At the beginning of the 20th century two epistemological issues proved to be very fruitful for the future of physics: The existence of the ether and the existence of atoms. The former was settled by Einstein’s theory of relativity, partly motivated by Mach’s criticism of Newton’s absolute space, and the second found its resolution in quantum mechanics.

But there was also a further issue at the turn of the century, one that was to have a major impact on physics only much later. The question of the stability of planetary motion was studied in particular by H. POINCARɲ. In 1889 he submitted to the Swedish Academy a prize-winning paper on the stability of the motion of three gravitationally interacting bodies. In later work, he concluded that the stability question is of a fundamental nature. In “Science and Method” he anticipated an essential characteristic of chaotic behavior: ‘small differences in the initial conditions’ may ‘produce very great ones in the phenomena’.

Half a century passed before the conception of classical mechanics changed fundamentally in the eyes of the majority of physicists³. This occurred with the appearance of chaotic behavior in deterministic equations such as those of Newton. It all started with E. Lorenz’s observation in 1963 that a simplified atmospheric model – derived from hydrodynamic equations and consisting of three coupled nonlinear first-order ordinary differential equations – turned out to show quite different numerical solutions for extremely tiny changes in the initial conditions. This discovery precipitated a tremendous amount of research into dynamical systems worldwide. We mention only two research fields directly related to classical mechanics: chaotic behavior in a particular nonlinear two-dimensional Hamiltonian system and chaos in the restricted three-body systems with gravitational interactions. We discuss both these systems in detail later. At the same time the celebrated Kolmogorov-Arnold-Moser (KAM) theorem (1954–63) solved Poincaré’s convergence problem in the power series treatment of dynamical systems that are not exactly solvable.

As a result of this discovery, the generally accepted presumption of predictability of mechanical systems was overturned. At least since P. S. LAPLACE⁴ expressed that predictability⁵ in 1814, it had been universally believed that given all the initial conditions, and given sufficiently powerful calculational tools, one could predict the future state of any classical system. This is still true in principle, but the fact remains that due to extreme sensitivity to changes in the initial conditions, it is practically not possible to predict the future for many systems, since the initial values are always only known to a certain accuracy.

The new picture also has implications for other fields of physics. For example, in statistical physics, it sheds new light on ergodic theory, and allows a new understanding of the arrow of time – that is, the apparent irreversible direction of time in the macroscopic world, despite the microscopic time reversibility of the fundamental laws.

Chaotic behavior is ubiquitous even in rather simple mechanical systems. Most textbooks on classical mechanics only consider the small minority of so-called integrable systems. In the light of the ‘chaos revolution’, this cannot be considered adequate in a modern approach. Therefore, we introduce concepts and tools necessary to understand integrability and chaotic behavior quite early in the treatment, along with examples of chaotic systems. Presenting this modern view of classical mechanics is our chief goal.

1.2 The birth of classical mechanics

It was an outstanding genius who gave rise not only to mechanics but to physics as the science where starting from basic laws the observed phenomena are derived: I. NEWTON⁶. Newton’s “Philosophiae naturalis principia mathematica” (Mathematical principles of natural philosophy, London 1687; “Principia” for short) terminated a period of about two millennia⁷. Newton’s ‘Magnum Opus’⁸ replaced the dicta of antiquity and the subsequent suppositions about the nature of motion, its properties and causes, by a few axioms, exploited their consequences and compared these successfully with physical reality. Classical mechanics is based on the central axioms (or laws) of the “Principia”.

There were three editions of the “Principia” (1687, 1713, 1726) followed by translations into English (1727 by Andrew Motte), French (1759 by the Marquise Émilie du Châtelet), German (1871 by Jakob Ph. Wolfers), and Russian (1915/16 by Aleksei N. Krylov). The latter three include also instructive commentaries and additions. Another remarkable Latin edition is the so-called ‘Jesuit’ edition by François Jaquier and Thomas Le Seur (Geneva 1739–42) which provides a huge amount of valuable annotations doubling the volume of the “Principia”.

The “Principia” starts with two sections followed by three books. The first section entitled “Definitions” deals with the concepts of mass, motion, force, time, and space. In the second section “Axioms, or Laws of Motion” the famous, three basic laws are presented.

Then follows the first book “Of the Motion of Bodies” dedicated to the motion of a body subject to central forces. In its first section Newton introduces eleven important Lemmata needed for his proofs⁹. Afterwards he starts discussing various aspects of the motion of a body subject to a ‘centripetal force’ (i.e. central force, see Eq. 2.31). Subsequently he considers bodies mutually attracting each other with central forces, switching then to forces between solid bodies¹⁰.

The second book, also entitled “Of the Motion of Bodies”, treats the motion in resisting media. Fluids and the motion of bodies in a fluid are discussed. In particular the circular motion of a fluid is examined, thereby disproving R. DESCARTES’¹¹ vortex theory.

The third book “On the System of the World”, i.e. the planetary system, contains the law of the gravitational force and is devoted to celestial mechanics. Topics are: The shape of the earth and the weight of bodies in different regions of the earth, moon’s motion and its inequalities, the tides, and comets¹². The motion of the moon is considered as a three body system (sun, earth, and moon) and observed time dependencies of the parameters of the lunar orbit are calculated.

Before we proceed, following and extending Newton’s ideas, we consider some general premises.

1.3 Observations and the resulting pictures

The laws of physics inevitably have a mathematical form, since physics aims to be quantitative, and even precise. But behind the mathematical form stands concepts and reasoning. A (theoretical) physicist’s everyday routine is mainly concerned with mathematics. But particularly when new theories are formed or old ones are scrutinized, it is the concepts and their understanding that is important. What is the relation between physical reality and its mathematical image?

Everyone who observes, conceptualizes or even changes the surrounding nature develops an imagination that goes beyond simple cognition of the environment. They construct pictures of nature – for example, suppositions are made about connections between processes; conclusions are drawn about influences between processes. A picture of the world emerges that often also includes metaphysical currents, perhaps even religious ones. Physics is restricted to the rational part of the world view. We quote below views held by outstanding physicists on the relation between the exterior world and the individual’s knowledge of the world. Presumab...

Table of contents

• Cover
• Half Title
• Title
• Copyright
• Preface of the first edition
• Preface to the second edition
• Contents
• 1 Basic considerations and concepts
• 2 Foundations of classical mechanics
• 3 One dimensional motion of a particle
• 4 Peculiar motion in two dimensions
• 5 Motion in a central force
• 6 Gravitational force between two bodies
• 7 Collisions of particles. Scattering
• 8 Changing the frame of reference
• 9 Lagrangian mechanics
• 10 Conservation laws and symmetries
• 11 The rigid body
• 12 Small oscillations
• 13 Hamiltonian mechanics
• 14 Hamilton Jacobi theory
• 15 Three body systems
• 16 Approximating non integrable systems
• In retrospect
• Appendix
• Bibliography
• Index