Jacobi fields in nonholonomic mechanics

In this paper, we define Jacobi fields for nonholonomic mechanics using a similar characterization than in Riemannian geometry. We give explicit conditions to find Jacobi fields and finally we find the nonholonomic Jacobi fields in two equivalent ways: the first one, using an appropriate complete lift of the nonholonomic system and, in the second one, using the curvature and torsion of the associated nonholonomic connection.

Invariant Submanifolds of Genus 5 and a Cantor Staircase in the Nonholonomic Model of a Snakeboard

In this paper, we address the free (uncontrolled) dynamics of a snakeboard consisting of two wheel pairs fastened to a platform. The snakeboard is one of the well-known sports vehicles on which the sportsman executes necessary body movements. From the theoretical point of view, this system is a direct generalization of the classical nonholonomic system of the Chaplygin sleigh. We carry out a topological and qualitative analysis of trajectories of this dynamical system. An important feature of the problem is that the common level set of first integrals is a compact two-dimensional surface of genus 5. We specify conditions under which the reaction forces infinitely increase during motion and the so-called phenomenon of nonholonomic jamming is observed. In this case, the nonholonomic model ceases to work and it is necessary to use more complex mechanical models incorporating sliding, elasticity, etc.