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Transactions of the American Mathematical Society

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Second order Lagrangian Twist systems: simple closed characteristics
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by J. B. Van den Berg and R. C. Vandervorst PDF
Trans. Amer. Math. Soc. 354 (2002), 1393-1420 Request permission


We consider a special class of Lagrangians that play a fundamental role in the theory of second order Lagrangian systems: Twist systems. This subclass of Lagrangian systems is defined via a convenient monotonicity property that such systems share. This monotonicity property (Twist property) allows a finite dimensional reduction of the variational principle for finding closed characteristics in fixed energy levels. This reduction has some similarities with the method of broken geodesics for the geodesic variational problem on Riemannian manifolds. On the other hand, the monotonicity property can be related to the existence of local Twist maps in the associated Hamiltonian flow. The finite dimensional reduction gives rise to a second order monotone recurrence relation. We study these recurrence relations to find simple closed characteristics for the Lagrangian system. More complicated closed characteristics will be dealt with in future work. Furthermore, we give conditions on the Lagrangian that guarantee the Twist property.
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Additional Information
  • J. B. Van den Berg
  • Affiliation: Division of Theoretical Mechanics, University of Nottingham, Nottingham NG7 2RD, United Kingdom
  • MR Author ID: 639255
  • Email:
  • R. C. Vandervorst
  • Affiliation: Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
  • Address at time of publication: Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, Georgia 30332-0190
  • Email:
  • Received by editor(s): January 18, 2000
  • Published electronically: November 8, 2001
  • Additional Notes: The first author was supported by grants TMR ERBFMRXCT980201 and NWO SIR13-4785
    The second author by grants ARO DAAH-0493G0199 and NIST G-06-605
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 1393-1420
  • MSC (1991): Primary 34C12, 49Jxx, 49S05, 70Hxx, 70Kxx
  • DOI:
  • MathSciNet review: 1873011