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 Shaggy Steed of Physics.- The Heavens and the Elements.- The Law of Motion.- Classical Mechanics: The Heavens.- Quantum Mechanics: The Elements.- The Hidden Unity of Space and Time.- The Manifold Universe.

The Shaggy Steed is an unassuming figure from Irish folklore who reveals himself as an inspiring teacher of the forces hidden in the universe. This book celebrates an unassuming bit of physics that also turns out to be an inspiring teacher. The two-body problem-the motion of two bodies bound by the inverse-square force of gravity and electricity-is the Shaggy Steed of Physics, guiding the reader to an understanding of both the forces and the mathematical beauty hidden in the physical world. The book begins with an exposition of the action principle and its revelation of invariants created by the symmetries of nature. It then turns to the two-body problem and the grand unifying themes of symmetry and topology in physics, both classical and quantum. On the scale of the solar system this motion generates the Kepler ellipse-the fundamental orbit of celestial mechanics. On the microscopic scale of the quantum the same motion generates the hydrogen atom-the primal element. This remarkable unity of the heavens and the elements rests upon hidden symmetry-the symmetry of rotation in four dimensions and the three-sphere upon which they are topologically imaged. This richly endowed symmetry is a paradigm for the unitary symmetries of the elementary particles and pointed the way to them. This new edition preserves the spirit of the original while deepening key topics. The remarkable relations between the generators of symmetry (such as rotation generators) and the dynamical invariants of the symmetry (such as the hidden invariant of Kepler motion) are now elaborated. Readers are led through the states of the hydrogen atom by a master quantum number orchestrating the symmetries of the three-sphere. And KAM Theory-the greatest achievement in celestial mechanics of the twentieth century-is given its place in the story.

From the reviews of the second edition: "In the present book, the shaggy steed of physics is the two-body problem, a problem that guides the reader through an interesting tour of the mathematical beauty hidden in the physical world. Actually, the author takes an approach which is opposite to that usually taken by the physicist who often marvels at the physical beauty hidden in the mathematical world. This reverse approach makes the reading of this book both interesting and illuminating for the professional physicist or chemist." (Fernande Grandjean and Gary J. Long, Physicalia Vol. 57 (3), 2005) "The book celebrates an unassuming bit of physics that also turns out to be an inspiring teacher. The two-body problem - the motion of two bodies bound by the inverse-square force of gravity and electricity - is the Shaggy Steed of Physics, guiding the reader to an understanding of both the forces and the mathematical beauty hidden in the physical world. … The second edition preserves the spirit of the original while deepening key topics." (Zentralblatt für Didaktik der Mathematik, January, 2005) "This is the second edition of a very interesting little book first published in 1994. The author writes with considerable conviction, passion, and clarity on the topic of mechanics as the one major unifying theme … . Overall, I regard this to be an excellent book, especially important for those in the community who would like to understand the big picture, the connections and the similarities between the major physical theories which describe our universe … ." (J M Hill, The Physicist, Vol. 41 (5), September/October, 2004) "Oliver’s new edition … of his intellectual history of mechanics and the transition from classical mechanics to quantum mechanics enhances the first edition in several areas. … Oliver writes elegantly, almost to the point of being flowery. … One delightful feature not often found in works of this type isthe inclusion of an appendix that contains brief technical biographies and portraits of the major contributors to the topic … . Summing up: Highly recommended. Upper-division undergraduate and graduate students; professionals." (A. Spero, CHOICE, June, 2004)