Wladimir Igorewitsch Arnold Books

Focusing on the interplay between topology, group theory, and geometry, this monograph presents a comprehensive analysis of ideal hydrodynamics and magneto-hydrodynamics. It explores complex problems within these fields, offering a unified perspective that integrates various mathematical approaches. First published in 1998, it serves as a valuable resource for understanding the intricate relationships between these disciplines in fluid dynamics.

This volume is a celebration of the state of mathematics at the end of the millennium. Produced under the auspices of the International Mathematical Union (IMU), the book was born as part of the activities of World Mathematical Year 2000. It consists of 28 articles written by influential mathematicians. Authors of 14 contributions were recognized in various years by the IMU as recipients of the Fields Medal, from K.F. Roth (Fields Medalist, 1958) to W.T. Gowers (Fields Medalist, 1998).

Focusing on the period from 1991 to 1995, this collection features 27 influential papers by renowned mathematician V.I. Arnold. His work explores a diverse range of topics, including Vassiliev's theory of invariants and knots, bifurcations of plane curves, and the combinatorial properties of Bernoulli, Euler, and Springer numbers. Additionally, Arnold delves into the geometry of wave fronts, the Berry phase, and the quantum Hall effect, showcasing his profound contributions to modern mathematics.

nen (die fast unverändert in moderne Lehrbücher der Analysis übernommen wurde) ermöglichten ihm nach seinen eigenen Worten, „in einer halben Vier telstunde“ die Flächen beliebiger Figuren zu vergleichen. Newton zeigte, daß die Koeffizienten seiner Reihen proportional zu den sukzessiven Ableitungen der Funktion sind, doch ging er darauf nicht weiter ein, da er zu Recht meinte, daß die Rechnungen in der Analysis bequemer auszuführen sind, wenn man nicht mit höheren Ableitungen arbeitet, sondern die ersten Glieder der Reihenentwicklung ausrechnet. Für Newton diente der Zusammenhang zwischen den Koeffizienten der Reihe und den Ableitungen eher dazu, die Ableitungen zu berechnen als die Reihe aufzustellen. Eine von Newtons wichtigsten Leistungen war seine Theorie des Sonnensy stems, die in den „Mathematischen Prinzipien der Naturlehre“ („Principia“) ohne Verwendung der mathematischen Analysis dargestellt ist. Allgemein wird angenommen, daß Newton das allgemeine Gravitationsgesetz mit Hilfe seiner Analysis entdeckt habe. Tatsächlich hat Newton (1680) lediglich be wiesen, daß die Bahnkurven in einem Anziehungsfeld Ellipsen sind, wenn die Anziehungskraft invers proportional zum Abstandsquadrat ist: Auf das Ge setz selbst wurde Newton von Hooke (1635-1703) hingewiesen (vgl. § 8) und es scheint, daß es noch von weiteren Forschern vermutet wurde.

Translated from the Russian by E. J. F. Primrose „Remarkable little book.“ -SIAM REVIEW V. I. Arnold, who is renowned for his lively style, retraces the beginnings of mathematical analysis and theoretical physics in the works (and the intrigues!) of the great scientists of the 17th century. Some of Huygens' and Newton's ideas. several centuries ahead of their time, were developed only recently. The author follows the link between their inception and the breakthroughs in contemporary mathematics and physics. The book provides present-day generalizations of Newton's theorems on the elliptical shape of orbits and on the transcendence of abelian integrals; it offers a brief review of the theory of regular and chaotic movement in celestial mechanics, including the problem of ports in the distribution of smaller planets and a discussion of the structure of planetary rings.