Inventiones mathematicae

, Volume 152, Issue 2, pp 369–432

Morse theory on spaces of braids and Lagrangian dynamics

  • R.W. Ghrist
  • J.B. Van den Berg
  • R.C. Vandervorst

Abstract.

In 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.¶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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • R.W. Ghrist
    • 1
  • J.B. Van den Berg
    • 2
  • R.C. Vandervorst
    • 3
  1. 1.Department of Mathematics, University of Illinois, Urbana, IL 61801, USAUS
  2. 2.Department of Applied Mathematics, University of Nottingham, UKGB
  3. 3.Department of Mathematics, Free University Amsterdam, De Boelelaan 1081, Amsterdam, NetherlandsNL
  4. 4.CDSNS, Georgia Institute of Technology, Atlanta, GA 30332-0160, USAUS

Personalised recommendations