The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside. (English) Zbl 1308.70003
Summary: In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of a nonholonomic hinge. Equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler-Jacobi-Lie theorem, which is a new integration mechanism in nonholonomic mechanics. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found.
MSC:
| 70E18 | Motion of a rigid body in contact with a solid surface |
| 37J60 | Nonholonomic dynamical systems |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
References:
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