Geometric calculus for second-order differential equations and generalizations of the inverse problem of Lagrangian mechanics

Abstract

Abstract. We review the main features of the geometric calculus which has been introduced over the past 15 years in the study of second-order ordinary differential equations and then explain how a recently introduced generalization of the inverse problem of Lagrangian mechanics can be very concisely dealt with by this calculus in an intrinsic way. This paper is an account of the lecture with the same title presented at

Jacobi Fields and Conjugate Points for a Projective Class of Sprays

“…Equivalenty, Jacobi fields can be characterized as solutions of a system of second-order ordinary linear differential equations, the so-called generalized Jacobi equations. In order to introduce them we first recall two important operators that one can associate to a sode (but see [14] for a short review on this material).…”

Lagrangian descriptions of dissipative systems: a review

“…Another elegant differential form approach is the one of Henneaux [46], which also works in the same form in field theories [47]. For recent reviews of differential geometric approach see [48, 49].…”

The Inverse Problem of the Calculus of Variations and the Stabilization of Controlled Lagrangian Systems

“…We prefer to use in this paper this geometric approach to the Helmholtz conditions, over the more analytical style of Douglas' paper, for the reason that it can be conveniently applied (in the next section) to a (non-coordinate) frame of eigenvectors of Φ. More details on this calculus may be found in the review paper [14].…”

“…It is well-known that the sode Γ defines a non-linear connection which ensures that every vector field Z on J 1 π can be split in a horizontal and a vertical part (see e.g. [8,11,14]). This observation leads to the definition of two operators.…”

“…We wish to find sufficient conditions for that equilibrium to be stable. Note first that the Jacobian of the system (14) in the equilibrium has a zero eigenvalue, and that, as a consequence, the equilibrium can never be linearly stable. Second, for systems of second-order differential equations, one may also consider a second, more geometric, linearization process, where the linearized equations are given by the matrix that corresponds to the Jacobi endomorphism Φ (see e.g.…”