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.

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 $${{\mathcal {M}}}^{nh}_q$$ M q nh 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 $${{\mathcal {M}}}^{nh}_q$$ M q nh !

Refutation of the Quantum Theory Principles: Theorem

The theorem presented challenges the quantum mechanics and its relativistic theory generally posited as an ultimate unifying guideline of nature in fundamental and applied matters, refutes this theory, any bridges from it to the realm. We build the evidence on the rigorous statistical criteriaand arguments of compatibility at the interfaces not adduced previously against the theory. It calls in question the Born rule, particle-wave doublethink, probability sense of the quantum theory, any bridges from the theory to both Lagrangian and nonholonomic mechanics. The argumentation given to the matter of ambient noise impact at the interfaces by meaningful statistical methods paves the way towards the correct principles of causality, connectedness, robustness .