Partial Dirac Structures and Dynamical Systems

TL;DR

This paper extends the concept of Dirac structures to infinite-dimensional convenient manifolds, exploring their geometric properties, variational principles, and applications to geodesics in Banach manifolds.

Contribution

It introduces partial Dirac structures in infinite dimensions and adapts finite-dimensional geometric results to this setting, including variational methods and geodesic characterizations.

Findings

Extended Dirac structures to convenient manifolds.

Adapted variational principles for infinite-dimensional constraints.

Characterized normal geodesics for conical Finsler metrics.

Abstract

In a previous paper (PeCa24), the notion of Dirac structure in finite dimension was extended to the convenient setting. In particular, we introduce the notion of \emph{partial Dirac structure on a convenient manifold} and look for which all geometrical results in finite dimension which are still true in this infinite dimensional framework. Note that this context is justified by many mechanical infinite dimensional examples which recover all the classical ones, such as Hilbert, Banach, Fr\'{e}chet context or direct limits of Banach spaces. Another reason is that if we want to extend the variational technics, as in YoMa06II, to the infinite dimensional setting, the category of convenient vector spaces is cartesian closed, which is not the case for the category of locally convex vector spaces and so this variational approach does not work in this last setting. In this second part, first, we look for an adaptation of the results obtained in YoMa06II for a variational approach of constraint Lagrangian on a subbundle of a Banach manifold. Then we study the same type of problem but for constraint Lagrangians on a \textit{singular} distribution. Theses results are finally applied to the characterization of normal geodesics for a conical Finsler metric on a Banach manifold.