Higher-order constrained systems on fibered manifolds: An exterior differential systems approach

TLDR: In this article, the concept of higher-order Lagrangean systems as a Lepagean two-form defined on a certain jet prolongation of a fibered manifold over a one-dimensional base is recalled.

Abstract: Some recent results on higher-order Lagrangean systems are presented. The concept of higher-order Lagrangean system as a Lepagean two-form defined on a certain jet prolongation of a fibered manifold over a one-dimensional base is recalled. The dynamics then can be defined by a distribution (the Euler-Lagrange distribution) which generally is of non-constant rank. This approach leads to a natural geometric definition of regularity and a geometric classification of constrained systems. Since a Lagrangean system is understood as a class of equivalent Lagrangians (which can be of different orders), the theory, including a Hamilton formulation, is independent on the choice of a particular Lagrangian for the Lagrangean system under consideration. Relations to the symplectic, presymplectic, cosymplectic and precosymplectic geometry are discussed.