# Toroidal regularization of the guiding center Lagrangian

## Abstract

In the Lagrangian theory of guiding center motion, an effective magnetic field B* = B+ (m/e)v _{∥}∇ x b appears prominently in the equations of motion. Because the parallel component of this field can vanish, there is a range of parallel velocities where the Lagrangian guiding center equations of motion are either ill-defined or very badly behaved. Moreover, the velocity dependence of B* greatly complicates the identification of canonical variables and therefore the formulation of symplectic integrators for guiding center dynamics. Here, this letter introduces a simple coordinate transformation that alleviates both these problems simultaneously. In the new coordinates, the Liouville volume element is equal to the toroidal contravariant component of the magnetic field. Consequently, the large-velocity singularity is completely eliminated. Moreover, passing from the new coordinate system to canonical coordinates is extremely simple, even if the magnetic field is devoid of flux surfaces. We demonstrate the utility of this approach in regularizing the guiding center Lagrangian by presenting a new and stable one-step variational integrator for guiding centers moving in arbitrary time-dependent electromagnetic fields.

- Authors:

- Courant Inst. of Mathematical Sciences, New York, NY (United States)
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Fusion Energy Sciences (FES) (SC-24)

- OSTI Identifier:
- 1424084

- Alternate Identifier(s):
- OSTI ID: 1409875

- Report Number(s):
- LLNL-JRNL-737871

Journal ID: ISSN 1070-664X; TRN: US1801902

- Grant/Contract Number:
- AC52-07NA27344; FG02-86ER53223; AC05-06OR23100; AC52-07NA2734

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 24; Journal Issue: 11; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; Hamiltonian mechanics; Energy; Probability theory; General physics; Plasma gyrokinetics; Stochastic processes; Mathematical physics; Nuclear fusion power; Plasma confinement; Tokamaks; Plasma Physics and Thermonuclear Processes

### Citation Formats

```
Burby, J. W., and Ellison, C. L..
```*Toroidal regularization of the guiding center Lagrangian*. United States: N. p., 2017.
Web. doi:10.1063/1.5004429.

```
Burby, J. W., & Ellison, C. L..
```*Toroidal regularization of the guiding center Lagrangian*. United States. doi:10.1063/1.5004429.

```
Burby, J. W., and Ellison, C. L.. Wed .
"Toroidal regularization of the guiding center Lagrangian". United States.
doi:10.1063/1.5004429.
```

```
@article{osti_1424084,
```

title = {Toroidal regularization of the guiding center Lagrangian},

author = {Burby, J. W. and Ellison, C. L.},

abstractNote = {In the Lagrangian theory of guiding center motion, an effective magnetic field B* = B+ (m/e)v∥∇ x b appears prominently in the equations of motion. Because the parallel component of this field can vanish, there is a range of parallel velocities where the Lagrangian guiding center equations of motion are either ill-defined or very badly behaved. Moreover, the velocity dependence of B* greatly complicates the identification of canonical variables and therefore the formulation of symplectic integrators for guiding center dynamics. Here, this letter introduces a simple coordinate transformation that alleviates both these problems simultaneously. In the new coordinates, the Liouville volume element is equal to the toroidal contravariant component of the magnetic field. Consequently, the large-velocity singularity is completely eliminated. Moreover, passing from the new coordinate system to canonical coordinates is extremely simple, even if the magnetic field is devoid of flux surfaces. We demonstrate the utility of this approach in regularizing the guiding center Lagrangian by presenting a new and stable one-step variational integrator for guiding centers moving in arbitrary time-dependent electromagnetic fields.},

doi = {10.1063/1.5004429},

journal = {Physics of Plasmas},

number = 11,

volume = 24,

place = {United States},

year = {Wed Nov 22 00:00:00 EST 2017},

month = {Wed Nov 22 00:00:00 EST 2017}

}

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