2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61444In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index and provides a gloablization of `lap number' techniques used in scalar parabolic PDEs. In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits.Revised version: numerous changes in exposition. Slight modification of two proofs and one definition; 55 pages, 20 figuresDynamical SystemsMathematical PhysicsGeometric TopologySymplectic Geometry37J45; 37B30; 57M25text