Lagrange, Euler, and Kovalevskaya tops

rdf:langString Lagrange, Euler, and Kovalevskaya tops

rdf:langString Leonhard Euler

rdf:langString Joseph-Louis Lagrange

rdf:langString Sofia Vasilyevna Kovalevskaya

rdf:langString Leonhard Euler, Joseph-Louis Lagrange, and Sofia Vasilyevna Kovalevskaya

rdf:langString Lagrange portrait.jpg

rdf:langString Leonhard Euler.jpg

rdf:langString Sofja Wassiljewna Kowalewskaja 1.jpg

rdf:langString In classical mechanics, the precession of a rigid body such as a spinning top under the influence of gravity is not, in general, an integrable problem. There are however three (or four) famous cases that are integrable, the Euler, the Lagrange, and the Kovalevskaya top. In addition to the energy, each of these tops involves three additional constants of motion that give rise to the integrability. The Euler top describes a free top without any particular symmetry, moving in the absence of any external torque in which the fixed point is the center of gravity. The Lagrange top is a symmetric top, in which two moments of inertia are the same and the center of gravity lies on the symmetry axis. The Kovalevskaya top is a special symmetric top with a unique ratio of the moments of inertia which satisfy the relation That is, two moments of inertia are equal, the third is half as large, and the center of gravity is located in the plane perpendicular to the symmetry axis (parallel to the plane of the two equal points). The nonholonomic Goryachev–Chaplygin top (introduced by D. Goryachev in 1900 and integrated by Sergey Chaplygin in 1948) is also integrable. Its center of gravity lies in the equatorial plane. It has been proven that no other holonomic integrable tops exist.