This publication is predicated at the Colloquium Lectures awarded by way of Shlomo Sternberg in 1990. The authors delve into the mysterious position that teams, specifically Lie teams, play in revealing the legislation of nature by means of targeting the typical instance of Kepler movement: the movement of a planet lower than the charm of the sunlight based on Kepler's legislation. Newton learned that Kepler's moment law--that equivalent parts are swept out in equivalent times--has to do with the truth that the strength is directed radially to the sunlight. Kepler's moment legislations is de facto the statement of the conservation of angular momentum, reflecting the rotational symmetry of the approach in regards to the beginning of the strength. In latest language, we'd say that the crowd $O(3)$ (the orthogonal staff in 3 dimensions) is answerable for Kepler's moment legislation. through the top of the 19th century, the inverse sq. legislations of appeal was once noticeable to have $O(4)$ symmetry (where $O(4)$ acts on a component of the six-dimensional part house of the planet). Even higher teams have because been came upon to be fascinated by Kepler movement. In quantum mechanics, the instance of Kepler movement manifests itself because the hydrogen atom. Exploring this circle of principles, the 1st a part of the booklet was once 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 show $O(4)$ 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 significant position during this publication. The final part of the ebook stories a number of the cosmological versions during this comparable conformal classification (and having various isometry teams) from the perspective of projective geometry. The computation of the hydrogen spectrum presents an indication of the main that enlarging the section area can simplify the equations of movement within the classical surroundings and reduction within the quantization challenge within the quantum surroundings. The authors supply a quick precis of the homological quantization of constraints and an inventory of modern purposes to many fascinating finite-dimensional settings. The publication closes with an overview of Kostant's concept, 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 concept in six dimensions.
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