Variations on a Theme by Kepler (Colloquium Publications) by Victor W. Guillemin and Shlomo Sternberg

By Victor W. Guillemin and Shlomo Sternberg

This e-book relies at the Colloquium Lectures awarded via Shlomo Sternberg in 1990. The authors delve into the mysterious position that teams, specially Lie teams, play in revealing the legislation of nature by way of concentrating on the primary instance of Kepler movement: the movement of a planet lower than the allure of the solar in keeping with Kepler's legislation. Newton discovered that Kepler's moment law--that equivalent components are swept out in equivalent times--has to do with the truth that the strength is directed radially to the solar. Kepler's moment legislations is absolutely the statement of the conservation of angular momentum, reflecting the rotational symmetry of the procedure concerning the foundation of the strength. In state-of-the-art language, we'd say that the crowd $O(3)$ (the orthogonal crew in 3 dimensions) is chargeable for Kepler's moment legislations. through the top of the 19th century, the inverse sq. legislations of allure was once noticeable to have $O(4)$ symmetry (where $O(4)$ acts on a section of the six-dimensional part house of the planet). Even better teams have considering that been discovered to be excited by Kepler movement. In quantum mechanics, the instance of Kepler movement manifests itself because the hydrogen atom. Exploring this circle of rules, the 1st a part of the ebook used to be written with the overall mathematical reader in brain. the rest of the ebook is aimed toward experts. It starts with an indication that the Kepler challenge and the hydrogen atom express $O(4)$ symmetry and that the shape of this symmetry determines the inverse sq. legislations in classical mechanics and the spectrum of the hydrogen atom in quantum mechanics. the distance of regularized elliptical motions of the Kepler challenge (also referred to as the Kepler manifold) performs a important function during this e-book. The final element of the ebook reviews some of the cosmological versions during this similar conformal category (and having various isometry teams) from the perspective of projective geometry. The computation of the hydrogen spectrum offers a demonstration of the primary that enlarging the part house can simplify the equations of movement within the classical atmosphere and relief within the quantization challenge within the quantum environment. The authors supply a quick precis of the homological quantization of constraints and an inventory of contemporary functions to many attention-grabbing finite-dimensional settings. The booklet closes with an summary of Kostant's conception, during which a unitary illustration is linked to the minimum nilpotent orbit of $SO(4,4)$ and during which electromagnetism and gravitation are unified in a Kaluza-Klein-type conception in six dimensions.