Cartan Forms and Second Variation for Constrained Variational Problems

Abstract

Using the Cartan form of first order constrained variational prob lems introduced earlier we define the second variation. This definition coin cides in the unconstrained case with the usual one in terms of the double Lie derivative of the Lagrangian density, an expression, that in the constrained case does not work. The Hessian metric and other associated concepts intro duced in this way are compared with those obtained through the Lagrange multiplier rule. The theory is illustrated with an example of isoperimetrie problem.

A historical review on nonholomic mechanics

“…Vakonomic dynamics will not be discussed here in detail, but we refer to [4,[8][9][10]59,[54][55][56]106,107] for more details. For the relation to Sub-Riemannian geometry we refer to [110].…”

Variational integrators in discrete vakonomic mechanics

“…A doctrine of renewed interest in the last years is, certainly, the nonholonomic mechanics, either in its properly speaking nonholonomic version or in its vakonomic formulation [3,4,8,11,13,17,20,21]. With roots in the famous d'Alambert principle and in the well-known Lagrange's variational problem, respectively, its actual interest has been addressed very specially to the field theory (see, e.g.…”

“…With roots in the famous d'Alambert principle and in the well-known Lagrange's variational problem, respectively, its actual interest has been addressed very specially to the field theory (see, e.g. [8,9,23]) and to the discrete mechanics (see, e.g., [7,15,16]). Being the starting data identical for both formulations-a Lagrangian density and a constraint submanifold on the configuration 1-jet bundle of a mechanical system-they differ, in fact, in how to obtain the equations of the movement, that is: either by applying the d'Alambert principle (generalized in 1932 by Chetaev) in the first case, or by finding the extremals of the Lagrange problem defined by such data under the modern version developed in the 80's by the russian school (Arnold, Kozlov, Nejshtadt,…) with the name of vakonomic mechanics: "variational axiomatic kind approximation to the mechanics".…”