The Shaggy Steed of Physics

The Shaggy Steed of Physics Second Edition David Oliver The Shaggy Steed of Physics Mathematical Beauty in the Physical World Second Edition With 74 Illustrations David Oliver The Fields of Michael and Gabriel Meadows of Dan, VA 24120 USA [email protected] Cover illustration: Such disparate natural objects as a planet orbiting the sun and the hydrogen atom manifest the same hidden symmetry: the symmetry of the three-sphere. The three-sphere has the remarkable ability to turn itself into a family of nested tori as shown here in this illustration by Huseyin Koçak and David Laidlaw. Library of Congress Cataloging-in-Publication Data Oliver, David. The shaggy steed of physics : mathematical beauty in the physical world / David Oliver.— [2nd ed.]. p. cm. Includes index. ISBN 0-387-40307-8 (alk. paper) 1. Physics. I. Title. QC21.2.O45 2003 530—dc21 2003052882 ISBN 0-387-40307-8 Printed on acid-free paper. 2004, 1994 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), 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. Printed in the United States of America. 987654321 SPIN 10935989 www.springer-ny.com Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH To Duncan Foley who is himself a prince This page intentionally left blank keen as midsummer’s keen beyond conceiving mind of sun will stand, so strictly (over utmost him so hugely) stood my father’s dream —eecummings This page intentionally left blank Preface to the Second Edition I have taken the occasion of a second edition for enhancements and refine- ments that preserve the spirit of the original. I have more fully portrayed the remarkable relationships between symmetry group generators and their Lie bracket and the dynamical invariants of those symmetries and their Poisson bracket in Chapter 3. The radical departure of quantum waves from classical waves is more sharply drawn in Chapter 5. The spectacular way in which the states of the hydrogen atom unfold from its symmetries is the centerpiece of quantum Kepler motion. I now lead readers more deliberately through this unfolding by showing how the master quantum number orchestrating four- dimensional rotational symmetry orchestrates the hydrogen atom states. I offer readers in Chapter 7 a more descriptive portrait of the break-up of regular Kepler motion on tori into periodic resonant points when the motion becomes chaotic. And I now include a sketch of KAM Theory— the greatest achievement in celestial mechanics of the twentieth century—in properly filling out the story of chaotic Kepler motion. Lastly, a new edition has given me the opportunity to correct errors. The Fields of Michael and Gabriel David Oliver Meadows of Dan, Virginia, USA Autumn 2002 ix This page intentionally left blank Preface to the First Edition The universe embraces with sensual presence. No thought, no word is needed to behold sky, sun, moon, stars. It is not in reflection but in experience that one inhales the earthy smells of damp woods, becomes dizzy under the sweep of the starry night sky, basks in the sun’s warmth, rolls in ocean breakers, tastes salt. The sensual universe is open, available, immediate. But beneath its sensuality, the universe has an interior landscape that lies hidden. Though veiled, the inner nature of the cosmos draws us. It invites us to find the underlying source of the motion of sun, moon, and stars, search out the primal form of all matter, and discover the unity beneath the diversity of the earthy substances that excite the senses. The inner nature of the cosmos is a landscape of exquisite beauty which appropriately underlies the exterior splendor of our experience. This interior landscape of matter and motion which lies both in and beyond the senses—the physical face of the universe—is mathematical. That the world we see, smell, and touch is deeply mathematical is a fact of unceasing amazement to me. The natural world presents us mathematical symmetries of compelling beauty; and a rich vein of mathematics, inspired by these raw materials from the natural world, is potently physical. The physical and mathematical resonate, creatively nourishing one another. Although the sensual face of the world presents itself to us quite openly, its physical face, hidden as it is in mathematics, must be gained by following a path of reflection and study. The physical world opens to us through mechanics, the science of motion. Mechanics is the wellspring from which physics flows, the source out of which a multitude of applications and elaborations from astrophysics to elementary particles emerges. Insight into the interior nature of the physical world begins with mechanics, with a vision of the motion that underlies all the objects of the world and the subject of this book. For whom is this book intended? It is intended for lovers. As a young man, I was drawn by love of the physical world and the things one could make from it through knowledge of its interior, mechanical behavior. I followed my fascination into the study of physics and engineering. I found xi xii Preface to the First Edition myself not only conceiving and developing nuclear and electric rockets and sources for power in space with this knowledge but also sharing that vision by teaching other young men and women like myself. Then, slowly, as flickering symptoms scarcely cross across one’s con- sciousness but steadily rise up in the body to reveal a raging disease, the things I was doing with the physical world rose up to confront me. I had begun with delight in scientific discovery and engineering craft. I ended up developing sources of massive energy and power for a new generation of space weaponry—well before the movie and the reality of Star Wars. I hadbeendrawnbythebeautyofthestarrynightsky.Iendedupinthe first ambitious attempt of the human race to fill that sky with devasta- tion. My innocence gave way to the realization that mechanics is a science both wonderful and terrible. Mechanics is the vehicle of all physical theory. Mechanics is the vehicle of war. The two have been inseparable. Mechanics originated in the earliest myths of human origins. Cain was a man born into matter, a worker of the soil given to wonder about force and motion. He took a piece of the natural world and mechanically fashioned it into a plowshare. Then he took that plowshare, split Abel’s head, and simultaneously became the inventor of the sword. Physics, a reveler in du- alities, has never escaped its primal duality in the sword and the plowshare. In the time of the Greeks our science received its name: mechanics (µηχαν˘ η´)—a device for delusion in warfare. The practitioners of Cain’s art had moved on in mechanical prowess. They beat their plowshares into the armed bowels of mechanical horses that could simultaneously split a multitude of heads. Mechanics, originating as delight in force and motion, suffered to become the handmaid of humanly wrought power and delusion in warfare. SeventeenthcenturyVenice,anearlysuperpower,showedtheworldhow to organize mechanical wondering and tinkering under the minions of the arsenal; and the arsenal proliferated beyond renaissance imagination to become the superarsenals of the modern world. The arsenal claimed the most distinguished physicists and engineers of the mid-twentieth century in our own recapitulation of the plowshare and the sword. These men and women hammered Albert Einstein’s beautiful plowshare of nature’s fields of motion, E = mc2, into the most hideous transformation of the sword yet to appear. Is it possible to have a celebration of mechanics with that dreadful her- itage? Let me answer with a story from the mountains in which I now live. A notorious local man was known as “the awfulest man.” Not only was he scheming and exploitative; he could be dangerous. When he died, folks came to the funeral more to hear what the preacher would have to say about the awfulest man than to mourn. The preacher began with silence. Then, looking squarely at the congregation, he broke into a dreamy smile and said, “Didn’t you just love to hear that man whistle!” As surely as a wondrous side of the awfulest man could be recognized Preface to the First Edition xiii by the melliflous trilling that announced his presence, mechanics too can be recognized as a wondrous, if blighted science. Mechanics has a voice in praise of the creation. Mechanics can dance—other than to the tune of the piper. After fleeing the work of the arsenal, I remembered and longed for the ethereal whistle of mechanics; and I began to write. Looking to the redemption of the dreaded science, I turn from humanly fabricated delusion to naturally created wonder. Look to mechanics as the trance of nature rather than the delusion of invincible safety. Think of mechanics as a gift of nature rather than the purveyor of devices to threaten and dominate. So set aside the monstrosities into which the plowshare has been beaten and gather round.

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