Lagrangian Systems with Higher Order Constraints

CENDRA, H.; GRILLO, S.D.

JOURNAL OF MATHEMATICAL PHYSICS

Amer. Inst. Phys.

Lugar: Huntington Quadrangle Melville NY; Año: 2007 vol. 48 p. 1 - 35

0022-2488

A class of mechanical systems subject to higher order constraints i.e., constraintsinvolving higher order derivatives of the position of the system are studied. Wecall them higher order constrained systems ,HOCSs. They include simplified modelsof 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 kindof 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 anotherinteresting example of HOCS, systems subjected to friction forces, showingthat those forces can be encoded in a second order kinematic constraint. The mainaim of the paper is to show that every HOCS is equivalent to a GNHS with linearconstraints, in a canonical way. That is to say, systems with higher order constraintscan 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 Généraledes Vibrations 1949, Chap. XV, p. 246 and L’instabilité en Mécanique; Automobiles,Avions, Ponts Suspendus 1954 of a pneumatic tire. As a by-product, weintroduce some applications on higher order tangent bundles, which we expect tobe useful for the study of intrinsic aspects of the geometry of such bundles.i.e., constraintsinvolving higher order derivatives of the position of the system are studied. Wecall them higher order constrained systems ,HOCSs. They include simplified modelsof 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 kindof 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 anotherinteresting example of HOCS, systems subjected to friction forces, showingthat those forces can be encoded in a second order kinematic constraint. The mainaim of the paper is to show that every HOCS is equivalent to a GNHS with linearconstraints, in a canonical way. That is to say, systems with higher order constraintscan 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 Généraledes Vibrations 1949, Chap. XV, p. 246 and L’instabilité en Mécanique; Automobiles,Avions, Ponts Suspendus 1954 of a pneumatic tire. As a by-product, weintroduce some applications on higher order tangent bundles, which we expect tobe useful for the study of intrinsic aspects of the geometry of such bundles.