Lagrangian systems with higher order constraints

Abstract

A class of mechanical systems subject to higher order constraints (i.e., constraints involving higher order derivatives of the position of the system) are studied. We call them higher order constrained systems (HOCSs). They include simplified models of elastic rolling bodies, and also the so-called generalized nonholonomic systems (GNHSs), whose constraints only involve the velocities of the system (i.e., first order derivatives in the position of the system). One of the features of this kind of systems is that D’Alembert’s principle (or its nonlinear higher order generalization, the Chetaev’s principle) is not necessarily satisfied. We present here, as another interesting example of HOCS, systems subjected to friction forces, showing that those forces can be encoded in a second order kinematic constraint. The main aim of the paper is to show that every HOCS is equivalent to a GNHS with linear constraints, in a canonical way. That is to say, systems with higher order constraints can be described in terms of one with linear constraints in velocities. We illustrate this fact with a system with friction and with Rocard’s model [Dynamique Générale des Vibrations (1949), Chap. XV, p. 246 and L’instabilité en Mécanique; Automobiles, Avions, Ponts Suspendus (1954)] of a pneumatic tire. As a by-product, we introduce some applications on higher order tangent bundles, which we expect to be useful for the study of intrinsic aspects of the geometry of such bundles.