Hydromagnetic Equilibrium Ii. Stability in the Variational Formulation.

In a few recent papers (Woltjer," 2 Chandrasekhar3) a beginning has been made with a variational formulation of the problems of hydromagnetic equilibrium. A similar approach was followed by Kruskal and Kulsrud4 who examined the static equilibrium in a toroid from the variational point of view, making largely use of topological arguments. It seems somewhat doubtful whether this approach can be generalized to nonstatic equilibria in a more general geometry. The basic idea behind the variational formulation is the following. The hydromagnetic equations (in the absence or dissipative terms) admit certain integrals of motion. A hydromagnetic system cannot change into an arbitrary different system, but only into systems characterized by the same integrals. Thus the integrals may be considered to be the constraints of the system. If now the energy has a minimum value compatible with the constraints, the system must be in stable equilibrium. For a change of configuration would mean an increase in energy, which is not possible since the energy is conserved. An alternative formulation of the variational principle is that if a system is able to dissipate energy, without changing the other integrals of motion, it will move toward a state of minimum energy, compatible with the constraints. The advantages of a variational formulation are twofold. The relation between the various possible equilibria becomes more clear. Also a variational principle is useful in computing configurations that cannot be dealt with analytically. Two points in our previous papers need some clarification. Firstly, it is clear that our set of integrals is not complete. As shown by Kruskal and Kulsrud, the variational principle should give all possible equilibrium solutions, if a complete set of integrals is used as constraints. The fact that we did not obtain all solutions demonstrates that our set of integrals is not complete.* Although the relevance of various integrals may depend on special symmetries in the geometry of the problem, there is no guarantee that no more important solutions exist. This remark applies equally well to our discussion2 as to the one by Chandrasekhar3 who discussed axisymmetric incompressible media. In fact for this case a complete set of integrals can be obtained" and it can be shown explicitly that the variational formulation and the direct solution of the equilibrium equations lead to identical results. The second point which needs some amplification and which was stated incor-