立命館大学数理科学科

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タイトル： A geometric approach to stochastic extensions of nonholonomic constraints

講演者： François Gay-Balmaz（CNRS – LMD – Ecole Normale Supérieure）

アブストラクト：

We propose several stochastic extensions of nonholonomic constraints for mechanical systems and study the effects on the dynamics and on the conservation laws. Our approach relies on a stochastic extension of the Lagrange-d’Alembert framework. The mechanical system we focus on is the example of a Routh sphere, i.e., a rolling unbalanced ball on the plane. We interpret the noise in the constraint as either a stochastic motion of the plane, random slip or roughness of the surface. Without the noise, this system possesses three integrals of motion: energy, Jellet and Routh. Depending on the nature of noise in the constraint, we show that either energy, or Jellet, or both integrals can be conserved, with probability 1. We also present some exact solutions for particular types of motion in terms of stochastic integrals. Inspired by this example, we then consider two different ways of including stochasticity in nonholonomic systems. We show that when the noise preserves the linearity of the constraints, then energy is.  preserved. For other types of noise in the constraint, e.g., in the case of an affine noise, the energy is not conserved. This approach is illustrated with a class of Lagrangian mechanical systems on Lie groups, with constraints of “rolling ball type”. We conclude with numerical simulations illustrating our theories, and some pedagogical examples of noise in constraints for other nonholonomic systems popular in the literature, such as the nonholonomic particle, the rolling disk and the Chaplygin sleigh.

日時：2022年1月24日（月） 15:00～16:00

日時：2021年12月20日（月） 16:30～18:00

<<立命館大学幾何学セミナー>>

<<立命館大学幾何学セミナー>>

<<立命館大学幾何学セミナー>>

<<立命館大学幾何学セミナー>>

<<立命館大学幾何学セミナー>>

We discuss central limit theorems (CLTs) for non-symmetric random walks on covering graphs of finite graphs whose covering transformation groups are groups of polynomial volume growth. By realizing the covering graph into a certain nilpotent Lie group through a discrete harmonic map, we show that the limiting semigroup is generated by the sub-Laplacian with a non-trivial linear drift on the nilpotent Lie group equipped with the Albanese metric. Furthermore, under the centered condition, we establish a functional CLT (i.e., Donsker-type invariance principle) in a Hölder space over the nilpotent Lie group by employing the theory of rough paths. As a refinement of the CLT, we also establish the Edgeworth expansion of the random walk even if the random walk is non-symmetric. A part of this talk is based on joint work with Satoshi Ishiwata (Yamagata) and Hiroshi Kawabi (Keio).