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 \({{\mathcal {M}}}^{nh}_q\) 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 \({{\mathcal {M}}}^{nh}_q\)!

非完整力学描述了受不可积分约束约束的系统的运动。其最显着的特性之一是非完整方程的推导本质上不是变分的。然而，在本文中，我们证明（定理 1.1）对于动力学非完整系统，从不动点q开始的解是图像子流形上的一系列黎曼度量的真实测地线\({{\mathcal {M}}} ^{nh}_q\)的非完整指数映射。这意味着一个令人惊讶的结果：以q为起点的动力学非完整轨迹，对于足够小的时间，最小化\({{\mathcal {M}}}^{nh}_q\) 中的长度！