Symmetries, integrability, and magnetic axis coordinates [1 ETCS, MATH-02/B]

On the one hand, the relationship between symmetry fields and first integrals plays a fundamental role in the integrability of vector fields. On the other hand, integrable or near-integrable magnetic fields are prominent in the design of plasma confinement devices. In this minicourse, the relationship between symmetry fields and first integrals of divergence-free vector fields is explored in three dimensions in light of its relevance to plasma physics and magnetic confinement fusion. The existence of flux coordinates for magnetic fields in high generality is given, even without needing to assume a symmetry, as in the cases of magnetohydrostatics or quasi-symmetry.

- David Perrella, Nathan Duignan, David Pfefferlé; Existence of global symmetries of divergence-free fields with first integrals. J. Math. Phys. 1 May 2023; 64 (5): 052705.
- J. W. Burby, N. Duignan, J. D. Meiss; Integrability, normal forms, and magnetic axis coordinates. J. Math. Phys. 1 December 2021; 62 (12): 122901.