On the notion of Jacobi fields in constrained calculus of variations

Massa, Enrico ; Pagani, Enrico

Communications in Mathematics, Tome 24 (2016) no. 2, pp. 91-113.

Résumé

In variational calculus, the minimality of a given functional under arbitrary deformations with fixed end-points is established through an analysis of the so called \emph {second variation}. In this paper, the argument is examined in the context of constrained variational calculus, assuming piecewise differentiable extremals, commonly referred to as \emph {extremaloids}. The approach relies on the existence of a fully covariant representation of the second variation of the action functional, based on a family of \emph {local} gauge transformations of the original Lagrangian and on a set of scalar attributes of the extremaloid, called the corners' \emph {strengths} \cite {mlp}. In discussing the positivity of the second variation, a relevant role is played by the \emph {Jacobi fields}, defined as infinitesimal generators of $1$-parameter groups of diffeomorphisms preserving the extremaloids. Along a piecewise differentiable extremal, these fields are generally discontinuous across the corners. A thorough analysis of this point is presented. An alternative characterization of the Jacobi fields as solutions of a suitable \emph {accessory variational problem} is established.

Détail

Classification : 49J--, 53B05, 58F05, 65K10, 70D10, 70Q05
Keywords: constrained variational calculus; second variation; Jacobi fields.