Morse theory on spaces of braids and Lagrangian dynamics

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dc.creatorGhrist, R. W.
dc.creatorBerg, J. B. Van den
dc.creatorVandervorst, R. C.
dc.date2001-05-10
dc.date2002-10-03
dc.date.accessioned2026-07-07T04:41:38Z
dc.date.available2026-07-07T04:41:38Z
dc.descriptionIn 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.
dc.descriptionRevised version: numerous changes in exposition. Slight modification of two proofs and one definition; 55 pages, 20 figures
dc.identifierhttps://arxiv.org/abs/math/0105082
dc.identifierhttp://arxiv.org/abs/math/0105082
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/61444
dc.subjectDynamical Systems
dc.subjectMathematical Physics
dc.subjectGeometric Topology
dc.subjectSymplectic Geometry
dc.subject37J45; 37B30; 57M25
dc.titleMorse theory on spaces of braids and Lagrangian dynamics
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