# Magnetic field lines, Hamiltonian dynamics, and nontwist systems

## Abstract

Magnetic field lines typically do not behave as described in the symmetrical situations treated in conventional physics textbooks. Instead, they behave in a chaotic manner; in fact, magnetic field lines are trajectories of Hamiltonian systems. Consequently the quest for fusion energy has interwoven, for 50 years, the study of magnetic field configurations and Hamiltonian systems theory. The manner in which invariant tori breakup in symplectic twist maps, maps that embody one and a half degree-of-freedom Hamiltonian systems in general and describe magnetic field lines in tokamaks in particular, will be reviewed, including symmetry methods for finding periodic orbits and Greene's residue criterion. In nontwist maps, which describe, e.g., reverse shear tokamaks and zonal flows in geophysical fluid dynamics, a new theory is required for describing tori breakup. The new theory is discussed and comments about renormalization are made. (c) 2000 American Institute of Physics.

- Authors:

- Department of Physics and Institute for Fusion Studies, University of Texas at Austin, Austin, Texas 78712 (United States)

- Publication Date:

- OSTI Identifier:
- 20216536

- Resource Type:
- Journal Article

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 7; Journal Issue: 6; Other Information: PBD: Jun 2000; Journal ID: ISSN 1070-664X

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; HAMILTONIANS; MAGNETIC FIELDS; PLASMA SIMULATION; TRAJECTORIES; PLASMA CONFINEMENT; MAGNETOHYDRODYNAMICS; THEORETICAL DATA

### Citation Formats

```
Morrison, P. J.
```*Magnetic field lines, Hamiltonian dynamics, and nontwist systems*. United States: N. p., 2000.
Web. doi:10.1063/1.874062.

```
Morrison, P. J.
```*Magnetic field lines, Hamiltonian dynamics, and nontwist systems*. United States. doi:10.1063/1.874062.

```
Morrison, P. J. Thu .
"Magnetic field lines, Hamiltonian dynamics, and nontwist systems". United States. doi:10.1063/1.874062.
```

```
@article{osti_20216536,
```

title = {Magnetic field lines, Hamiltonian dynamics, and nontwist systems},

author = {Morrison, P. J.},

abstractNote = {Magnetic field lines typically do not behave as described in the symmetrical situations treated in conventional physics textbooks. Instead, they behave in a chaotic manner; in fact, magnetic field lines are trajectories of Hamiltonian systems. Consequently the quest for fusion energy has interwoven, for 50 years, the study of magnetic field configurations and Hamiltonian systems theory. The manner in which invariant tori breakup in symplectic twist maps, maps that embody one and a half degree-of-freedom Hamiltonian systems in general and describe magnetic field lines in tokamaks in particular, will be reviewed, including symmetry methods for finding periodic orbits and Greene's residue criterion. In nontwist maps, which describe, e.g., reverse shear tokamaks and zonal flows in geophysical fluid dynamics, a new theory is required for describing tori breakup. The new theory is discussed and comments about renormalization are made. (c) 2000 American Institute of Physics.},

doi = {10.1063/1.874062},

journal = {Physics of Plasmas},

issn = {1070-664X},

number = 6,

volume = 7,

place = {United States},

year = {2000},

month = {6}

}