General formulas for adiabatic invariants in nearly periodic Hamiltonian systems

  title={General formulas for adiabatic invariants in nearly periodic Hamiltonian systems},
  author={Joshua William Burby and Jonathan Squire},
  journal={Journal of Plasma Physics},
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… 

Nearly-periodic maps and geometric integration of noncanonical Hamiltonian systems

M. Kruskal showed that each continuous-time nearly-periodic dynamical system admits a formal U(1) symmetry, generated by the so-called roto-rate. When the nearlyperiodic system is also Hamiltonian,

Normal stability of slow manifolds in nearly periodic Hamiltonian systems

M. Kruskal showed that each nearly-periodic dynamical system admits a formal U(1) symmetry, generated by the so-called roto-rate. We prove that such systems also admit nearly-invariant manifolds of

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. A continuous-time dynamical system with parameter ε is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as ε approaches 0. Nearly-periodic maps are

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In a strong, inhomogeneous magnetic field, charged particle dynamics may be studied in the guiding-centre approximation, which is known to be Hamiltonian. When the magnetic field is quasisymmetric,

On the derivation of guiding center dynamics without coordinate dependence

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pl as mp h ] 1 8 O ct 2 02 0 Approximate symmetries of guiding-centre motion

Quasisymmetry builds a third invariant for charged-particle motion besides energy and magnetic moment. We address quasisymmetry at the level of approximate symmetries of first-order guiding-centre

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In a magnetic field, transitions between classes of guiding-centre motion can lead to cross-field diffusion and escape. We say a magnetic field is isodrastic if guiding centres make no transitions

Minimizing Separatrix Crossings through Isoprominence

A simple property of magnetic fields that minimizes bouncing to passing type transitions of guiding center orbits is defined and discussed. This property, called isoprominence, is explored through the



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

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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