|Affiliations:||Mathematical Institute of the Serbian Academy of Sciences and Arts||Title:||Connections and Time Reperametrizations in Nonholonomic Mechanics||Journal:||Advances in Nonlinear Science||First page:||58||Last page:||59||Conference:||International Conference “Scientific Heritage of Sergey A. Chaplygin: nonholonomic mechanics, vortex structures and hydrodynamics, Cheboksary, 2-6 June 2019||Issue Date:||2019||Rank:||M30||URL:||http://umu.chuvsu.ru/chaplygin2019/docs/TezisChap.pdf||Abstract:||
We consider nonholonomic system (M,L,D) on configuration space M given with Lagrangian Land nonintegrable distribution D defined by linear nonholonomic constraints. The equations of motion are obtained from the Lagrange-d’Alembert principle. In classical works of Synge , Vranceanu , Shouten , Wagner [15,16] the problem of motion of nonholonomic systems from the geometric point of view is considered. The equations can be rewritten in terms of suitable vector bundle connection∇P over configuration space M:
∇P ̇q ̇q=−gradDV.
In the case when the potential V vanishes, the solutions becomes the geodesic lines of ∇P. We recall on the extensions of the vector-bundle connection to the linear connection on TM considered in [3,17] and , as well as on so called partial connection (see ). We compare various approaches in geometrical formulation of nonolonomic systems by using affine connections, including the Chaplygin reduction performed by Bakša . Although mentioned objects are very well studied, some natural relationships between them are pointed out. In addition, we consider the Newton type equations on a Riemannian manifold (M,g) and look for a conformal metric g∗=f2g such that solutions of the Newton equations, after a time reparametrization, become the geodesic lines of g. This is a generalization of the Chaplygin multiplier method for Hamiltonization of G-Chaplygin systems [4,5]. Also, we obtain variants of the Maupertuis principle in nonholonomic mechanics as they are given in [1,11].
|Publisher:||Institute for Computer Science, Moscow - Izhevsk||Project:||Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems|
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