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On the integration theory of equations of nonholonomic mechanics. (English) Zbl 1006.37040
The paper deals with the problem of integration of a motion in nonholonomic systems. By means of the well-known theory of differential equations with an invariant measure, new integrable systems are discovered. Among them there are generalizations of Chaplygin’s problem of rolling a nonsymmetric ball on the plane and of the Suslov problem on rotation of a rigid body with a fixed point. The structure of the dynamics of systems on the invariant manifold in the integrable problems is suggested. Some new ideas in the theory of integration of the equations in nonholonomic mechanics are given. The first of them consists in using known integrals as the constraints. The second is the use of resolvable groups of symmetries in nonholonomic systems. Existence conditions of invariant measures with analytical density for differential equations of nonholonomic systems are given.
This article is a reprint of Adv. Mech. USSR 8, No. 3, 85-107 (1985).

MSC:
37J60 Nonholonomic dynamical systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70E40 Integrable cases of motion in rigid body dynamics
01A75 Collected or selected works; reprintings or translations of classics
70F25 Nonholonomic systems related to the dynamics of a system of particles
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