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Second order Lagrangian Twist systems: simple closed characteristics

Authors: J. B. Van den Berg and R. C. Vandervorst
Journal: Trans. Amer. Math. Soc. 354 (2002), 1393-1420
MSC (1991): Primary 34C12, 49Jxx, 49S05, 70Hxx, 70Kxx
Published electronically: November 8, 2001
MathSciNet review: 1873011
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Abstract: 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

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

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
Article copyright: © Copyright 2001 American Mathematical Society