Radial kinetic nonholonomic trajectories are Riemannian geodesics!

Nonholonomic mechanics describes the motion of systems constrained by nonintegrable constraints. One of its most remarkable properties is that the derivation of the nonholonomic equations is not variational in nature. However, in this paper, we prove (Theorem 1.1) that for kinetic nonholonomic systems, the solutions starting from a fixed point q are true geodesics for a family of Riemannian metrics on the image submanifold Mnh q of the nonholonomic exponential map. This implies a surprising result: the kinetic nonholonomic trajectories with starting point q, for sufficiently small times, minimize length in Mnhq!
D. Martín de Diego and A. Anahory Simoes acknowledge financial support from the Spanish Ministry of Science and Innovation, under grants PID2019-106715GB-C21 and ?Severo OchoaProgramme for Centres of Excellence in R&D? (CEX2019-000904-S). A. Anahory Simoes is supported by the FCT (Portugal) research fellowship SFRH/BD/129882/2017 partially funded by the European Union (ESF). J.C. Marrero acknowledges the partial support by European Union (Feder) grant PGC2018-098265-B-C32.