The Shaggy Steed of Physics: Mathematical Beauty in the Physical World

310 Pages·2004·2.989 MB·English

by  David Oliver (auth.)

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
DavidOliver
TheFieldsofMichaelandGabriel
MeadowsofDan,VA24120
USA
[email protected]
Coverillustration:Suchdisparatenaturalobjectsasaplanetorbitingthesunandthehydrogenatom
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
HuseyinKoc¸akandDavidLaidlaw.
LibraryofCongressCataloging-in-PublicationData
Oliver,David.
Theshaggysteedofphysics:mathematicalbeautyinthephysicalworld/DavidOliver.—
[2nded.].
p.cm.
Includesindex.
ISBN0-387-40307-8(alk.paper)
1.Physics. I.Title.
QC21.2.O45 2003
530—dc21 2003052882
ISBN0-387-40307-8 Printedonacid-freepaper.
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To
Duncan Foley
who is himself a prince
keen as midsummer’s keen beyond
conceiving mind of sun will stand,
so strictly (over utmost him
sohugely)stoodmyfather’sdream
—e e cummings
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
theremarkablerelationshipsbetweensymmetrygroupgeneratorsandtheir
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 quan-
tum 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—
thegreatestachievementincelestialmechanicsofthetwentiethcentury—in
properlyfillingoutthestoryofchaoticKeplermotion. Lastly,anewedition
has given me the opportunity to correct errors.
The Fields of Michael and Gabriel David Oliver
Meadows of Dan, Virginia, USA
Autumn 2002
ix
Preface to the First Edition
The universe embraces with sensual presence. No thought, no word is
neededtobeholdsky,sun,moon,stars. Itisnotinreflectionbutinexperi-
encethatoneinhalestheearthysmellsofdampwoods,becomesdizzyunder
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.
Butbeneathitssensuality,theuniversehasaninteriorlandscapethatlies
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
natureofthecosmosisalandscapeofexquisitebeautywhichappropriately
underliestheexteriorsplendorofourexperience. Thisinteriorlandscapeof
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 mathe-
matical symmetries of compelling beauty; and a rich vein of mathematics,
inspired by these raw materials from the natural world, is potently phys-
ical. 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.
Thephysicalworldopenstousthroughmechanics,thescienceofmotion.
Mechanics is the wellspring from which physics flows, the source out of
which a multitude of applications and elaborations from astrophysics to
elementaryparticlesemerges. Insightintotheinteriornatureofthephysical
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
had been drawn by the beauty of the starry night sky. I ended up in the
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
bothwonderfulandterrible. Mechanicsisthevehicleofallphysicaltheory.
Mechanics is the vehicle of war. The two have been inseparable.
Mechanicsoriginatedintheearliestmythsofhumanorigins. Cainwasa
manbornintomatter,aworkerofthesoilgiventowonderaboutforceand
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,hasneverescapeditsprimaldualityintheswordandtheplowshare.
In the time of the Greeks our science received its name: mechanics
(µηχα˘νη´)—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,
folkscametothefuneralmoretohearwhatthepreacherwouldhavetosay
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
mechanicsasagiftofnatureratherthanthepurveyorofdevicestothreaten
and dominate. So set aside the monstrosities into which the plowshare has
been beaten and gather round. Let me tell the story of the inner nature of
the world from the scale of the solar system down to the scale of the atom.
My story speaks of both the physical world and human imagination, of the
heavens and the elements and the individual human beings past and living
entranced with the nature of the universe.
Tellingaboutphysicsisapartofstory-tellingappropriatetothemodern
world. Itisinthestorythatthecollectedknowledgeofacultureisgathered,
celebrated, andpassedon—andphysicsisnoexception. Itismyhopethat
this story of the physical world may touch and delight others, particularly
those just coming of age in physics and mathematics and caught up in
its fascination. It is to you that I especially offer this book as a spirited
companion; for I wrote the book I would have loved to have placed in my
hands as a young student.
My tale of mechanics is not an alternative to rigorous study and the
exerciseofproblem-solvingrequiredtomasterphysicalprinciples. Norisit
intended as a diversion from masterful texts such as Arnold Sommerfeld’s
Mechanics, the ebullient works of V. I. Arnol’d in mechanics, or the Me-
chanics and the Quantum Mechanics of L. D. Landau and E. M. Lifshitz.
I offer my telling of this traditional story as a soulful companion to the
textbooks.
Experienced old-timers know thes stories intimately. I would be de-
lighted if they find a thing or two to catch their fancy in my rendering
of the tale. (Stories, especially the most traditional, are never retold in
precisely the same way.)
It is my hope that those whose main interests lie outside physics may
also be drawn into this story of the mathematical splendor of the world;
and I welcome you. Mathematics is a central mode of expression in this
kind of story-telling. Mathematics is part of our language, an important
meansofperceptionanddiscernment. Isuggestthatmathematicalphysics
is not only a language but a visual art whose structure, symmetries, and
iconographyreflectthesymmetryandbeautyoftheworldthatitdescribes.
Basic undergraduate mathematics for science, engineering, and mathe-
matics students provides a sufficient level with which one may enter into
the story. For those readers with deeper interest and greater mathematical
preparation I offer a set of notes with more extensive (and in some cases