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Normal Modes, Symmetries and Stability

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Complex Hamiltonian Dynamics

Part of the book series: Springer Series in Synergetics ((SSSYN,volume 10))

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Abstract

The present Chapter studies nonlinear normal modes (NNMs) of coupled oscillators from an altogether different perspective. Focusing entirely on periodic boundary conditions and using the Fermi Pasta Ulam β (FPU − β) and FPU − α models as examples, we demonstrate the importance of discrete symmetries in locating and analyzing exactly a class of NNMs called one-dimensional “bushes”, depending on a single periodic function \(\hat{q}(t)\). Using group theoretical arguments one can similarly identify n-dimensional bushes described by \(\hat{{q}}_{1}(t),\ldots,\hat{{q}}_{n}(t)\), which represent quasiperiodic orbits characterized by n incommensurate frequencies. Expressing these solutions as linear combinations of single bushes, it is possible to simplify the linearized equations about them and study their stability analytically. We emphasize that these results are not limited to monoatomic particle chains, but can apply to more complicated molecular structures in two and three spatial dimensions, of interest to solid state physics.

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  1. Department of Mathematics, University of Patras, Patras, Greece

    Tassos Bountis

  2. MPI for the Physics of Complex Systems, Dresden, Germany

    Haris Skokos

  3. Physics Department, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Haris Skokos

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Bountis, T., Skokos, H. (2012). Normal Modes, Symmetries and Stability. In: Complex Hamiltonian Dynamics. Springer Series in Synergetics, vol 10. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27305-6_4

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