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**Variations on a theme by Kepler.**
*(English)*
Zbl 0724.70006

The theme of this book is the role of the symmetry Lie groups and Lie algebras in the explanation of the laws of classical and quantum physics, illustrated on the concrete situation of the classical Kepler motion, and of its quantum counterpart, the hydrogen atom. The first sections are devoted to the study of the O(4)-symmetry, which explains the classical inverse square law, and the spectrum of the hydrogen atom. Then, the Kepler manifold \(T^+S^ 3=T^*S^ 3\setminus (the\) zero section) is studied under various aspects: as a coadjoint orbit of SO(2,4) or SU(2,2), as a symplectic Marsden-Weinstein reduction of SI(2,\({\mathbb{R}})\), and as the space of forward null geodesics on the conformal completion of Minkowski space. The study also includes the quantization of this manifold and the computation of the hydrogen spectrum, the derivation of the Kepler Hamiltonian from a geodesic flow in five dimension, etc. The intervention of the Minkowski space led to the final chapter of the book which studies homogeneous models in general relativity, and ends with a theory of Kostant, where electromagnetism and gravitation are unified in a Kaluza-Klein type theory in six dimensions.

Reviewer: I.Vaisman (Haifa)

### MSC:

70F15 | Celestial mechanics |

81R99 | Groups and algebras in quantum theory |

83C20 | Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory |

53C80 | Applications of global differential geometry to the sciences |

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |