Twistor geometry of rolling bodies
Pawel Nurowski (Center for Theoretical Physics of the Polish Academy of Sciences, Varsavia)
Aula Beltrami - Tuesday, June 21, 2016 h.15:00
Abstract. It is well known that the configuration space of two rigid bodies rolling on each other is a circle bundle S1->C->M over a 4-dimensional manifold M. If the motion of the two bodies is restricted by a non-holonomic constraint, which corresponds to rolling without slipping or twisting, then the bundle C is naturally equipped with a rank 2 distribution D. The distribution D, a rank 2-subbundle D of the tangent bundle TC, determines the space of possible velocities D_x compatible with the constraints at each point x of C. If the rolling bodies have a general shape the distribution D is generic, and belongs to a class of distributions considered by Elie Cartan in his famous 5-variables paper from 1910. Such distributions have local differential invariants, and most symmetric of them has split real form of the exceptional simple Lie group G2, as a group of local symmetries.
Circle bundles S1->T->M appear also in twistor theory, where M is a 4-dimensional pseudoriemannian manifold equipped with a metric of signature (2,2), and T is a bundle of real totally null planes over M.
In the talk I will discuss relations between the bundles C and T, and discuss the twistor interpretation of rolling without slipping or twisting. As an unexpected result I will give examples of shapes of the rolling bodies whose rolling distribution D has symmetry group isomorphic to the exceptional Lie group G2.
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