Necessity of the energy principles for magnetostatic stability

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Abstract
Three energy principles for magnetostatic stability are known that are supposed to give necessary and sufficient conditions. For this reason their minimization has been the subject of a lot of work in plasma physics. Indeed one can easily justify the sufficiency of the conditions of stability devised from these energy principles. But to the demonstrate their necessity one usually assumes that the operators of the perturbed linearized motions have a complete spectrum of eigenfunctions in the space of square integrable functions. We show the weakness of that assumption and propose two new demonstrations of the necessity of energy principles for stability that require less stringent assumptions. The first demonstration involves some properties of the Laplace transform. In the second one we use the integral invariants of the linearized motion. The conclusions in both cases are identical: if one finds a trial function which makes the potential energy negative, the equilibrium is unstable. We give lower and upper bounds for the growth rate of the unstable perturbation.

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