A geometrical framework for the study of non-holonomic Lagrangian systems

Abstract

A Lagrangian system subject to linear non-holonomic constraints may be represented in several different geometrical frameworks. We describe one such framework, involving the addition of an extra term to the Cartan 2-form Ω of the unconstrained system, which is dual to the traditional approach of adding a reaction force to the unconstrained dynamical vector field Γ. We show how this framework is closely related to the method of constructing a 2-form Ω M when the constraints are given by a connection on an auxiliary bundle, as described in our earlier work.

General framework for nonholonomic mechanics: Nonholonomic systems on Lie affgebroids

“…For instance, introducing modified Ereshmann connections (nonholonomic connection) [3], using a symplectic distribution on the constraint submanifold [1,35] and distinguishing the different casuistic depending on the 'position' of the Lie group of symmetries acting on the system and the nonholonomic distribution or in a Poisson context [22] (see [6] and references therein). Moreover, using jet bundle techniques, many authors studied the geometry of nonholonomic systems admitting an extension to explicitly time-dependent nonholonomic systems [4,14,16,30,33,34] and ready for extension to nonholonomic field theories [37]. Today we find an scenario with a very rich theory but with an important lack: a general framework unifying the different casuistic (unreduced and reduced equations, systems subjected to linear or affine constraints, time-dependent or time-independent systems...).…”

Non-holonomic mechanics: A geometrical treatment of general coupled rolling motion

“…3.3, i.e. (15), (16), (17), (18). Comparing the obtained Newton equations with the system of deformed equations of motion (26) we can see that…”

“…Conditions (15) and (16) represent the fact that the cylinder W rolls without slipping inside the ring V, conditions (17) and (18) The sense of second two conditions is immediately clear from two facts: the velocity V = (Ẋ, 0,Ż) is parallel with the tangent plane to the terrain profile, and for rolling without slipping it holds V = Rβ,β being the angular speed of V. Let us perform a derivation of constraints (15) and (16) (see figure (4)). The components of the relative velocity vector v 1 are…”

A Lie algebroid framework for non-holonomic systems

“…Indeed, the biggest difference between our approach and many others is that the fundamental form δθ L and the dynamics are not a differential form or a vector field on W, but a form and a section of the prolongation bundle T i W → W. A framework that seems to be closely related to ours is [21,22] (although their set-up is more general since also time-dependent systems were included). In those papers, two fundamental 2-forms on T W have been considered.…”

“…Similar to before, the, not unrelated, bundles (τ * ) ρ , (τ * ) λ and (µ * ) ρ will also come into the picture. In the special example of systems with symmetry, we should be able to relate the first equation in (21), in a Hamiltonian formulation, to the momentum equation (for a recent survey see [2]). …”