Rubber rolling over a sphere |
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Authors: | J Koiller K Ehlers |
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Institution: | 1.Funda??o Getulio Vargas,Rio de Janeiro,Brazil;2.Truckee Meadows Community College,Reno,USA |
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Abstract: | “Rubber” coated bodies rolling over a surface satisfy a no-twist condition in addition to the no slip condition satisfied
by “marble” coated bodies 1]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures
of the curves on the surfaces at corresponding points are equal. The associated distribution in the 5 dimensional configuration
space has 2–3–5 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for
rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptional group G
2). The 2–3–5 nonholonomic geometries are classified in a companion paper 2] via Cartan’s equivalence method 3]. Rubber rolling
of a convex body over a sphere defines a generalized Chaplygin system 4–8] with SO(3) symmetry group, total space Q = SO(3) × S
2 and base S
2, that can be reduced to an almost Hamiltonian system in T*S
2 with a non-closed 2-form ωNH. In this paper we present some basic results on the sphere-sphere problem: a dynamically asymmetric but balanced sphere of
radius b (unequal moments of inertia I
j but with center of gravity at the geometric center), rubber rolling over another sphere of radius a. In this example ωNH is conformally symplectic 9]: the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular
there is an invariant measure, whose density is the determinant of the reduced Legendre transform, to the power p = 1/2(b/a − 1). Using sphero-conical coordinates we verify the result by Borisov and Mamaev 10] that the system is integrable for
p = −1/2 (ball over a plane). They have found another integrable case 11] corresponding to p = −3/2 (rolling ball with twice the radius of a fixed internal ball). Strikingly, a different set of sphero-conical coordinates
separates the Hamiltonian in this case. No other integrable cases with different I
j are known.
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Keywords: | nonholonomic mechanics reduction Chaplygin systems |
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