Geometry of constrained connection systems

Abstract Inspired by work by de Diego et al (Simoes et al 2021 Anal. Math. Phys. 11 1–28; Simoes et al 2022 J. Phys. A: Math. Theor. 55 1–53), we introduce constrained connection systems which are a generalization of kinetic nonholonomic mechanical systems. For a given point <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> , we prove that there exists a family of Riemannian metrics such that the geodesics for the constrained connection system starting from q are minimizers for the Riemannian metrics. We define Jacobi fields for constrained connection systems and derive Jacobi equations. Then two examples are investigated: a constrained mechanical system and a kinetic nonholonomic mechanical system. For the constrained mechanical system, we provide a characterization of the global form of the geodesic vector field; we also provide remarkable examples of Jacobi fields.