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From the history of physics


Topological phase in classical mechanics

 a,  b
a Federal Research Center A.V. Gaponov-Grekhov Institute of Applied Physics of the Russian Academy of Sciences, ul. Ulyanova 46, Nizhny Novgorod, 603000, Russian Federation
b V. I. Kuznetsov Federal State Unitary Enterprise Research Institute of Applied Mechanics, ul. Aviamotornaya 55, Moscow, 111123, Russian Federation

The historical development of the concept of the topological phase in classical mechanics from the mid-19th century to the present is discussed. There are three stages to be recognized in this period. The first, the mid-19th century stage, is concerned with studying the kinematics of rigid body rotation and includes such milestone developments as the Euler theorem on finite rotation of rigids, Gauss formula for the sum excess of the angles of a spherical polygon, Rodrigues’s proof of the noncommutativity property of two finite rotations, and, finally, Hamilton’s Lectures on Quaternions where the solid angle theorem is formulated and proved. The experimental discovery of the nonholonomic error of gyroscopes and its exhaustive explanation by A Yu Ishlinskii represent the second stage. The third stage, which started in the 1980s, has witnessed the rediscovery of the nonholonomic effect in the framework of Hamiltonian formalism and is dominated by the study of how the topological phase — or an additional angle — forms in a mechanical system being treated in action-angle variables.

Fulltext pdf (317 KB)
Fulltext is also available at DOI: 10.1070/PU2003v046n09ABEH001635
PACS: 01.65.+g, 03.65.Vf, 45.40.Cc (all)
DOI: 10.1070/PU2003v046n09ABEH001635
URL: https://ufn.ru/en/articles/2003/9/d/
000188171400004
Citation: Malykin G B, Kharlamov S A "Topological phase in classical mechanics" Phys. Usp. 46 957–965 (2003)
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Оригинал: Малыкин Г Б, Харламов С А «Топологическая фаза в классической механике» УФН 173 985–994 (2003); DOI: 10.3367/UFNr.0173.200309e.0985

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