Variational Principles for Relaxation of Toroidal Plasmas with Constraints*

A substantial fraction of Bob Dewar's work in theoretical plasma physics addressed the problem of relaxation of toroidal plasmas with constraints. He was inspired by the classic papers of M. D. Kruskal, R. M. Kulsrud and W. Newcomb on ideal MHD as well as by the highly infuential paper by J. B. Taylor on relaxation in non-ideal plasmas in which the denumerably infinite number of topological constraints of ideal MHD are broken. Beginning in the 1980's, when I had the privilege to be his first PhD thesis student, he led an original and deeply thoughtful research program on this difficult problem, which produced some extremely interesting analytical and computational results. Bob had a singular ability to hold a difficult problem in his head for decades, and chipped away at it relentlessly. I will review in this talk Bob's elegant formulations of variational principles for toroidal plasmas with constraints over nearly three decades. The applications of his ideas are numerous---a unified theory of relaxation of toroidal plasmas unstable to tearing modes in reversed-field pinches and tokamkas, the calculation of three-dimensional toroidal MHD equilibria in stellarators with anti-diffusive barriers and magnetic islands, and methods of regularization of 3D MHD equilibria with dynamical constraints. The fruits of his efforts are inspiring, and one of his significant legacies.