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Extracting multidimensional phase space topology from periodic orbits.

Stephan Gekle1, Jörg Main, Thomas Bartsch

  • 1Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany.

Physical Review Letters
|October 10, 2006
PubMed
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We found a way to order periodic orbits in complex Hamiltonian systems. This allows reconstructing phase space structures using only these orbits, as shown for the hydrogen atom.

Area of Science:

  • Physics
  • Quantum Mechanics
  • Classical Mechanics

Background:

  • Hamiltonian systems describe the dynamics of many physical systems.
  • Understanding the behavior of periodic orbits is crucial for characterizing phase space.
  • The hydrogen atom in external fields presents a complex dynamical system.

Purpose of the Study:

  • To establish a hierarchical ordering of periodic orbits in strongly coupled multidimensional Hamiltonian systems.
  • To demonstrate that phase space structures can be quantitatively reconstructed from periodic orbits alone.
  • To apply and illustrate these findings using the hydrogen atom in crossed electric and magnetic fields.

Main Methods:

  • Developing a method for hierarchical ordering of periodic orbits.

Related Experiment Videos

  • Utilizing the identified periodic orbits to reconstruct phase space.
  • Applying the methodology to the specific case of the hydrogen atom.

    • Main Results:

    • A robust hierarchical ordering of periodic orbits was established for the studied system.
    • Quantitative reconstruction of phase space structures was achieved solely from periodic orbits.
    • The findings were successfully illustrated for the hydrogen atom in crossed electric and magnetic fields.

    Conclusions:

    • Periodic orbits provide a fundamental basis for understanding complex Hamiltonian dynamics.
    • The developed method offers a powerful tool for phase space reconstruction.
    • This approach is applicable to experimentally relevant systems like the hydrogen atom in external fields.