# Pullback Transformations in Gyrokinetic Theory

## Abstract

The Pullback transformation of the distribution function is a key component of the gyrokinetic theory. In this paper, a systematic treatment of this subject is presented, and results from applications of the uniform framework developed are reviewed. The focus is on providing a clear exposition of the basic formalism which arises from the existence of three distinct coordinate systems in gyrokinetic theory. The familiar gyrocenter coordinate system, where the gyromotion is decoupled from the rest of particle's dynamics, is non-canonical and non-fabric. On the other hand, Maxwell's equations, which are needed to complete a kinetic system, are initially only defined in the fabric laboratory phase space coordinate system. The pullback transformations provide a rigorous connection between the distribution functions in gyrocenter coordinates and Maxwell's equations in laboratory phase space coordinates. This involves the generalization of the usual moment integrals originally defined on the cotangent fiber of the phase space to the moment integrals on a general 6D symplectic manifold, is shown to be an important step in the proper formulation of gyrokinetic theory. The resultant systematic treatment of the moment integrals enabled by the pullback transformation. Without this vital element, a number of prominent physics features, such as the presencemore »

- Authors:

- Publication Date:

- Research Org.:
- Princeton Plasma Physics Lab., NJ (US)

- Sponsoring Org.:
- USDOE Office of Science (SC) (US)

- OSTI Identifier:
- 811954

- Report Number(s):
- PPPL-3772

TRN: US0303279

- DOE Contract Number:
- AC02-76CH03073

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: PBD: 21 Jan 2003

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; DISTRIBUTION FUNCTIONS; FIBERS; KINETICS; PHASE SPACE; PHYSICS; TRANSFORMATIONS; GYROKINETIC EQUATIONS; KINETIC THEORY; MATHEMATICAL PHYSICS

### Citation Formats

```
H. Qin, and W.M. Tang.
```*Pullback Transformations in Gyrokinetic Theory*. United States: N. p., 2003.
Web. doi:10.2172/811954.

```
H. Qin, & W.M. Tang.
```*Pullback Transformations in Gyrokinetic Theory*. United States. doi:10.2172/811954.

```
H. Qin, and W.M. Tang. Tue .
"Pullback Transformations in Gyrokinetic Theory". United States. doi:10.2172/811954. https://www.osti.gov/servlets/purl/811954.
```

```
@article{osti_811954,
```

title = {Pullback Transformations in Gyrokinetic Theory},

author = {H. Qin and W.M. Tang},

abstractNote = {The Pullback transformation of the distribution function is a key component of the gyrokinetic theory. In this paper, a systematic treatment of this subject is presented, and results from applications of the uniform framework developed are reviewed. The focus is on providing a clear exposition of the basic formalism which arises from the existence of three distinct coordinate systems in gyrokinetic theory. The familiar gyrocenter coordinate system, where the gyromotion is decoupled from the rest of particle's dynamics, is non-canonical and non-fabric. On the other hand, Maxwell's equations, which are needed to complete a kinetic system, are initially only defined in the fabric laboratory phase space coordinate system. The pullback transformations provide a rigorous connection between the distribution functions in gyrocenter coordinates and Maxwell's equations in laboratory phase space coordinates. This involves the generalization of the usual moment integrals originally defined on the cotangent fiber of the phase space to the moment integrals on a general 6D symplectic manifold, is shown to be an important step in the proper formulation of gyrokinetic theory. The resultant systematic treatment of the moment integrals enabled by the pullback transformation. Without this vital element, a number of prominent physics features, such as the presence of the compressional Alfven wave and a proper description of the gyrokinetic equilibrium, cannot be readily recovered.},

doi = {10.2172/811954},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2003},

month = {1}

}