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Symplectic maps, variational principles, and transport. (English) Zbl 1160.37302

This is a well-written survey article on the recent developments concerning symplectic maps which are the discrete-time analog of Hamiltonian motion. They arise in many applications including accelerator, chemical, condensed-matter, plasma, and fluid physics. Twist maps correspond to Hamiltonians for which the velocity is a monotonic function of the canonical momentum. Twist maps have a Lagrangian variational formulation. One-parameter families of twist maps typically exhibit the full range of possible dynamics-from simple or integrable motion to complex or chaotic motion. One class of orbits, the minimizing orbits, can be found throughout this transition; the properties of the minimizing orbits are discussed in detail. Among these orbits are the periodic and quasiperiodic orbits, which form a scaffold in the phase space and constrain the motion of the remaining orbits. The theory of transport deals with the motion of ensembles of trajectories. The variational principle provides an efficient technique for computing the flux escaping from regions bounded by partial barriers formed from minimizing orbits. Unsolved problems in the theory of transport include the explanation for algebraic tails in correlation functions, and its extension to maps of more than two dimensions. (From the summary)
A large number of references is also provided.

MSC:

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37E40 Dynamical aspects of twist maps
70H05 Hamilton’s equations
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