Morse families and Dirac systems

Abstract

Dirac structures and Morse families are used to obtain a geometric formalism that unifies most of the scenarios in mechanics (constrained calculus, nonholonomic systems, optimal control theory, higher-order mechanics, etc.), as the examples in the paper show. This approach generalizes the previous results on Dirac structures associated with Lagrangian submanifolds. An integrability algorithm in the sense of Mendela, Marmo and Tulczyjew is described for the generalized Dirac dynamical systems under study to determine the set where the implicit differential equations have solutions. the presymplectic structure by the dynamics under the assumption of the integrability of the Dirac structure. The paper ends with a section devoted to future work. The final appendix contains details about the relation between the Dirac structures of interest to us and some natural maps in mechanics.Acknowledgements.

Generalized port-Hamiltonian DAE systems

“…Following [1], and inspired by [2], we have given a geometric definition of generalized port-Hamiltonian DAE systems, defined by pairs of Dirac structures and Lagrangian subspaces. For physical models, the Dirac structure corresponds to the interconnection structure of the system, while the Lagrangian subspace corresponds to the definition of their energy.…”

“…Recently, and from different points of view [2,1], it was noted that a second type of algebraic constraints can be formulated by generalizing the Hamiltonian function H(x) = 1 2 x T Qx to a Lagrangian subspace of X × X * . This latter notion is defined as follows, resembling 2 the previous definition of a Dirac structure.…”

“…On the other hand it was recently observed in [2], see also the subsequent work [8,7,3,9], that by generalizing the definition of linear port-Hamiltonian systems algebraic constraints may arise in different ways as well. At the same time in [1], motivated primarily by considerations in the geometric formulation of Lagrangian systems, systems with kinematic constraints, and optimal control, the definition of port-Hamiltonian systems was generalized by replacing the gradient of the Hamiltonian function in the port-Hamiltonian dynamics by a Lagrangian submanifold which is not necessarily the graph of the gradient of a Hamiltonian. This leads to algebraic constraints in the state variables which are of a different nature than those originating from effort constraints corresponding to Dirac structures.…”

“…For simplicity of exposition we will concentrate on linear time-invariant finite-dimensional systems, and moreover on the lossless case (no energy-dissipation) without external variables (inputs/outputs). For developments concerning time-varying linear generalized pH DAE systems or infinite-dimensional linear pH DAE systems we refer to [8,2], and for the nonlinear case to [1].…”

“…Conceptually, the current paper is closest to [1] by emphasizing the geometric definition of generalized port-Hamiltonian systems as pairs of a Dirac structure and a Lagrangian subspace, while some constructions (as well as the emphasis on the linear case) are inspired by [2]. The paper is structured as follows.…”