General formulas for adiabatic invariants in nearly periodic Hamiltonian systems

@article{Burby2020GeneralFF,
  title={General formulas for adiabatic invariants in nearly periodic Hamiltonian systems},
  author={Joshua William Burby and Jonathan Squire},
  journal={Journal of Plasma Physics},
  year={2020},
  volume={86}
}
While it is well known that every nearly periodic Hamiltonian system possesses an adiabatic invariant, extant methods for computing terms in the adiabatic invariant series are inefficient. The most popular method involves the heavy intermediate calculation of a non-unique near-identity coordinate transformation, even though the adiabatic invariant itself is a uniquely defined scalar. A less well-known method, developed by S. Omohundro, avoids calculating intermediate sequences of coordinate… 
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8 Citations
Background Citations
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Methods Citations
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8 Citations

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References

SHOWING 1-10 OF 28 REFERENCES

Asymptotic Theory of Hamiltonian and other Systems with all Solutions Nearly Periodic

Consider a system of N ordinary first‐order differential equations in N dependent variables, and let the independent variable s not appear explicitly. Let the system depend on a small parameter e and…

Hamiltonian formulation of guiding center motion

A Hamiltonian theory of guiding center motion which uses rectangular coordinates in physical space and noncanonical coordinates in phase space is presented. The averaging methods preserve two…

Hamiltonian theory of guiding-center motion

Guiding-center theory provides the reduced dynamical equations for the motion of charged particles in slowly varying electromagnetic fields, when the fields have weak variations over a gyration…

Hamiltonian theory of guiding centre motion revisited

The classical problem of the motion of a charged particle in a slowly varying electromagnetic field is reconsidered in the framework of ‘pseudo-canonical transformations’ in a Hamiltonian formalism.…

Hamiltonian structure of the guiding center plasma model

The guiding center plasma model (also known as kinetic MHD) is a rigorous sub-cyclotron-frequency closure of the Vlasov-Maxwell system. While the model has been known for decades, and it plays a…

Adiabatic Invariance and Applications: From Molecular Dynamics to Numerical Weather Prediction

A wide class of Hamiltonian systems exhibit a mixture of slow motion with superimposed fast oscillations. Under the assumption of scale separation, these systems can be investigated using the…

Hamiltonian perturbation theory in noncanonical coordinates

The traditional methods of Hamiltonian perturbation theory in classical mechanics are first presented in a way which clearly displays their differential‐geometric foundations. These are then…

Guiding center dynamics as motion on a formal slow manifold in loop space

  • J. Burby
  • Physics
    Journal of Mathematical Physics
  • 2020
Since the late 1950's, the dynamics of a charged particle's ``guiding center" in a strong, inhomogeneous magnetic field have been understood in terms of near-identity coordinate transformations. The…

Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics.

One of the world’s leading mathematicians, Jurgen Moser developed theories in celestial mechanics and many other aspects of mathematics. He is most renowned for his work on the…

Magnetohydrodynamic motion of a two-fluid plasma

The two-fluid Maxwell system couples frictionless electron and ion fluids via Maxwell's equations. When the frequencies of light waves, Langmuir waves, and single-particle cyclotron motion are scaled…