Nonlinear constraints in nonholonomic mechanics

Abstract

In this paper we have obtained some dynamics equations, in the presence of nonlinear nonholonomic constraints and according to a lagrangian and some Chetaev-like conditions. Using some natural regular conditions, a simple form of these equations is given. In the particular cases of linear and affine constraints, one recovers the classical equations in the forms known previously, for example, by Bloch and all [3,4]. The case of time-dependent constraints is also considered. Examples of linear constraints, time independent and time depenndent nonlinear constraints are considered, as well as their dynamics given by suitable lagrangians. All examples are based on classical ones, such as those given by Appell's machine.

On a cotangent form of a nonholonomic lagrangian dynamics

“…Proposition 3.4 [20,Proposition 4]If C : NF → T M is a nonlinear constraint and S ∈ X(NF ) is a C-semispray, then the local formula…”

“…Proposition 3.5 [20,Proposition 6]If the lagrangian L is C-regular, then the integral curves, solutions of equations of motion of the extended nonholonomic system, are the integral curves of a C-semispray S.…”

“…The result stated in Theorem 3.1 gives a synthetic form of linear and regular-nonlinear cases. This result can be adapted to other situations; for example, in the case of time dependent constraints, or of a time independent lagrangian (as studied in [16] and the examples of the Appell-Hammel dynamic system in an elevator and the riemannian flow (see [20]).…”

“…See, for example, monograph [6] and the citations therein. But nonlinear constraints are also involved in nonholonomic mechanics; see, for example [20] and the citations therein.…”

“…We notice that the lagrangian setting is based on the source [20]. We use basic constructions and results on vector bundles and foliations from [7,19].…”