arXiv subjects

J. W. Burby

Publications and source records attributed to J. W. Burby.

47 records · Page 3Linked to original sources

arXiv2015

Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory

The dynamics of charged particles in general linear focusing lattices with quadrupole, skew-quadrupole, dipole, and solenoidal components, as well as torsion of the fiducial orbit and variation of beam energy is parameterized using a generalized Courant-Snyder (CS) theory, which extends the original CS theory for one degree of freedom to higher dimensions. The envelope function is generalized into an envelope matrix, and the phase advance is generalized into a 4D symplectic rotation, or an U(2) element. The 1D envelope equation, also known as the Ermakov-Milne-Pinney equation in quantum mechanics, is generalized to an envelope matrix equation in higher dimensions. Other components of the original CS theory, such as the transfer matrix, Twiss functions, and CS invariant (also known as the Lewis invariant) all have their counterparts, with remarkably similar expressions, in the generalized theory. The gauge group structure of the generalized theory is analyzed. By fixing the gauge freedom with a desired symmetry, the generalized CS parameterization assumes the form of the modified Iwasawa decomposition, whose importance in phase space optics and phase space quantum mechanics has been recently realized. This gauge fixing also symmetrizes the generalized envelope equation and express the theory using only the generalized Twiss function β.The generalized phase advance completely determines the spectral and structural stability properties of a general focusing lattice. For structural stability, the generalized CS theory enables application of the Krein-Moser theory to greatly simplify the stability analysis. The generalized CS theory provides an effective tool to study coupled dynamics and to discover more optimized lattice design in the larger parameter space of general focusing lattices.

physics.acc-ph↗

arXiv2015

Equivalence of two independent calculations of the higher order guiding center Lagrangian

The difference between the guiding center phase-space Lagrangians derived in [J.W. Burby, J. Squire, and H. Qin, Phys. Plasmas {\bf 20}, 072105 (2013)] and [F.I. Parra, and I. Calvo, Plasma Phys. Control. Fusion {\bf 53}, 045001 (2011)] is due to a different definition of the guiding center coordinates. In this brief communication the difference between the guiding center coordinates is calculated explicitly.

physics.plasm-ph↗

arXiv2014

Hamiltonian formulation of the gyrokinetic Vlasov-Maxwell equations

The gyrokinetic Vlasov-Maxwell equations are cast as an infinite-dimensional Hamiltonian system. The gyrokinetic Poisson bracket is remarkably simple and similar to the Morrison-Marsden-Weinstein bracket for the Vlasov-Maxwell equations. By identifying many of the bracket's Casimirs, this work enables (i) the derivation of gyrokinetic equilibrium variational principles and (ii) the application of the energy-Casimir method and the method of dynamically-accessible variations to study stability properties of gyrokinetic equilibria.

physics.plasm-ph↗

arXiv2014

Variational integration for ideal magnetohydrodynamics with built-in advection equations

Newcomb's Lagrangian for ideal magnetohydrodynamics (MHD) in Lagrangian labeling is discretized using discrete exterior calculus. Variational integrators for ideal MHD are derived thereafter. Besides being symplectic and momentum-preserving, the schemes inherit built-in advection equations from Newcomb's formulation, and therefore avoid solving them and the accompanying error and dissipation. We implement the method in 2D and show that numerical reconnection does not take place when singular current sheets are present. We then apply it to studying the dynamics of the ideal coalescence instability with multiple islands. The relaxed equilibrium state with embedded current sheets is obtained numerically.

physics.plasm-ph↗

arXiv2014

Hamiltonian mechanics of generalized eikonal waves

In accordance with the Keller-Maslov global WKB theory, a semiclassical scalar wave field is best encoded as a triple consisting of (i) a Lagrangian submanifold $Λ$ in the ray phase space, (ii) a density $μ$ on $Λ$, and (iii) an overall phase factor $ϕ$. We present the Hamiltonian structure of the Cauchy problem for such a "geometric semiclassical state" in the special case where the wave operator is Hermetian. Variational, symplectic, and Poisson formulations of the time evolution equations for $(Λ,μ,ϕ)$ are identitfied. Because we work in terms of the Keller-Maslov global WKB ansatz, as opposed to the more restrictive $ψ=a \exp(i S/ε)$, all of our results are insensitive to the presence of caustics. In particular, because the variational principle is insensitive to caustics, the latter may be used to construct structure-perserving numerical integrators for scalar wave equations.

math-ph↗

arXiv2014

Variational integrators for perturbed non-canonical Hamiltonian systems

Finite-dimensional non-canonical Hamiltonian systems arise naturally from Hamilton's principle in phase space. We present a method for deriving variational integrators that can be applied to perturbed non-canonical Hamiltonian systems on manifolds based on discretizing this phase-space variational principle. Relative to the perturbation parameter $ε$, this type of integrator can take $O(1)$ time steps with arbitrary accuracy in $ε$ by leveraging the unperturbed dynamics. Moreover, these integrators are coordinate independent in the sense that their time-advance rules transform correctly when passing from one phase space coordinate system to another.

math-ph↗

arXiv2014

Initializing and stabilizing variational multistep algorithms for modeling dynamical systems

Backward error initialization and parasitic mode control are well-suited for use in algorithms that arise from a discrete variational principle on phase-space dynamics. Dynamical systems described by degenerate Lagrangians, such as those occurring in phase-space action principles, lead to variational multistep algorithms for the integration of first-order differential equations. As multistep algorithms, an initialization procedure must be chosen and the stability of parasitic modes assessed. The conventional selection of initial conditions using accurate one-step methods does not yield the best numerical performance for smoothness and stability. Instead, backward error initialization identifies a set of initial conditions that minimize the amplitude of undesirable parasitic modes. This issue is especially important in the context of structure-preserving multistep algorithms where numerical damping of the parasitic modes would violate the conservation properties. In the presence of growing parasitic modes, the algorithm may also be periodically re-initialized to prevent the undesired mode from reaching large amplitude. Numerical examples of variational multistep algorithms are presented in which the backward error initialized trajectories outperform those initialized using highly accurate approximations of the true solution.

physics.comp-ph↗

arXiv2013

Gyrosymmetry: Global Considerations

In the guiding center theory, smooth unit vectors perpendicular to the magnetic field are required to define the gyrophase. The question of global existence of these vectors is addressed using a general result from the theory of characteristic classes. It is found that there is, in certain cases, an obstruction to global existence. In these cases, the gyrophase cannot be defined globally. The implications of this fact on the basic structure of the guiding center theory are discussed. In particular it is demonstrated that the guiding center asymptotic expansion of the equations of motion can still be performed in a globally consistent manner when a single global convention for measuring gyrophase is unavailable. The latter fact is demonstrated directly by deriving a new expression for the guiding-center Poincaré-Cartan form exhibiting no dependence on the choice of perpendicular unit vectors.

math-ph↗

arXiv2013

Automation of The Guiding Center Expansion

We report on the use of the recently-developed Mathematica package \emph{VEST} (Vector Einstein Summation Tools) to automatically derive the guiding center transformation. Our Mathematica code employs a recursive procedure to derive the transformation order-by-order. This procedure has several novel features. (1) It is designed to allow the user to easily explore the guiding center transformation's numerous non-unique forms or representations. (2) The procedure proceeds entirely in cartesian position and velocity coordinates, thereby producing manifestly gyrogauge invariant results; the commonly-used perpendicular unit vector fields $e_1,e_2$ are never even introduced. (3) It is easy to apply in the derivation of higher-order contributions to the guiding center transformation without fear of human error. Our code therefore stands as a useful tool for exploring subtle issues related to the physics of toroidal momentum conservation in tokamaks.

physics.plasm-ph↗

arXiv2013

The Hamiltonian Mechanics of Stochastic Acceleration

We show how to find the physical Langevin equation describing the trajectories of particles undergoing collisionless stochastic acceleration. These stochastic differential equations retain not only one-, but two-particle statistics, and inherit the Hamiltonian nature of the underlying microscopic equations. This opens the door to using stochastic variational integrators to perform simulations of stochastic interactions such as Fermi acceleration. We illustrate the theory by applying it to two example problems.

math-ph↗

arXiv2013

A Lagrangian perspective on the stability of ideal MHD equilibria with flow

We take a careful look at two approaches to deriving stability criteria for ideal MHD equilibria. One is based on a tedious analysis of the linearized equations of motion, while the other examines the second variation of the MHD Hamiltonian computed with proper variational constraints. For equilibria without flow, the two approaches are known to be fully consistent. However, for equilibria with flow, the stability criterion obtained from the constrained variation approach was claimed to be stronger than that derived using the linearized equations of motion. We show this claim is incorrect by deriving and comparing both criteria within the same framework. It turns out that the criterion obtained from the constrained variation approach has stricter requirements on the initial perturbations than the other. Such requirements naturally emerge in our new treatment of the constrained variation approach using the Euler-Poincaré structure of ideal MHD, which is more direct and simple than the previous derivation from the Poisson perspective.

physics.plasm-ph↗