Projective geodesic extensions by conformal modifications in nonholonomic mechanics

2025/09/19 by Belrhazi, Malika, Mestdag, Tom
#37J06 #37J60 #53Z05 #70G45 #70G65 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.15863

Abstract

Projective geodesic extensions are reparametrizations of the trajectories of a nonholonomic mechanical system (with only a kinetic energy Lagrangian), in such a way that they can be interpreted as part of the geodesics of a Riemannian metric. We derive necessary and sufficient conditions for the existence of these extensions, in the case where the constrained Lagrangian remains preserved up to a conformal transformation. When the nonholonomic system has a symmetry group (a Chaplygin system), we clarify the relation between projective geodesic extensions and closely related concepts, such as ϕ-simplicity, invariant measures and Hamiltonization. Throughout the paper, new and relevant examples illustrate the key differences between all these concepts.

Citations

• Mechanical Hamiltonization of unreduced ϕ-simple Chaplygin systems
• Geodesic extensions of mechanical systems with nonholonomic constraints
• The geometry of nonholonomic Chaplygin systems revisited
• Reduction of nonholonomic systems in two stages and Hamiltonization

Related

• Geodesic extensions of mechanical systems with nonholonomic constraints
• Mechanical Hamiltonization of unreduced ϕ-simple Chaplygin systems
• The geodesics for Poincaré's half-plane: a nonstandard derivation
• An Underlying Geometrical Manifold for Hamiltonian Mechanics
• Force-Free Fields are Conformally Geodesic