Integrators For Nonholonomic Systems Research Articles

Moving energies as first integrals of nonholonomic systems with affine constraints

In nonholonomic mechanical systems with constraints that are affine (linear nonhomogeneous) functions of the velocities, the energy is typically not a first integral. It was shown in Fassò and Sansonetto (2016 J. Nonlinear Sci. 26 519–44) that, nevertheless, there exist modifications of the energy, called there moving energies, which under suitable conditions are first integrals. The first goal of this paper is to study the properties of these functions and the conditions that lead to their conservation. In particular, we enlarge the class of moving energies considered in Fassò and Sansonetto (2016 J. Nonlinear Sci. 26 519–44). The second goal of the paper is to demonstrate the relevance of moving energies in nonholonomic mechanics. We show that certain first integrals of some well known systems (the affine Veselova and LR systems), which had been detected on a case-by-case way, are instances of moving energies. Moreover, we determine conserved moving energies for a class of affine systems on Lie groups that include the LR systems, for a heavy convex rigid body that rolls without slipping on a uniformly rotating plane, and for an n-dimensional generalization of the Chaplygin sphere problem to a uniformly rotating hyperplane.

Linear integrals of non-holonomic systems with non-linear constraints

The problem of the existence of linear integrals of the equation of motion of a mechanical system, subjected to non-linear constraints, is considered. Existing results for holonomic systems, and also for non-holonomic systems with linear constraints, are extended to systems with non-linear non-holonomic constraints. Examples are given.

The problem of first integral of nonholonomic systems

In this paper, the problem of first integrals of a nonholonomic system is discussed. The aim of this work is concentra.ted on finding the condition for existence of first integrals. The obtained results are applied for the construction of linear and quadratic integrals of nonholonomic systems. It obtains two important affirmations; they are: Any first integral could be treated as a particular nonholonomic constraint and contrarily, any nonholonomic constraint could be regarded as a first integral of the nonholonomic system.

First integrals of non-holonomic systems and their generators

We discuss various aspects of mechanical systems with general (nonlinear) non-holonomic constraints from the perspective of presymplectic geometry. We begin by introducing a 2-form on the evolution space of a system having the property, among others, of modelling the unconstrained dynamics. Using this 2-form we then characterize a unique second-order dynamics on the constraint submanifold through a simple geometrical implementation of Chetaev's concept of virtual work. We also give necessary and sufficient conditions in order for the reduced dynamics to admit a non-holonomic Lagrangian formulation. Finally, we study the structure of a set of vector fields on the constraint submanifold which generates all first integrals of a constrained system. The relationships with a previously proposed set of vector fields in non-conservative holonomic mechanics and with known generalizations of Noether's theorem for non-holonomic systems are analysed.