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A formalism for the construction of energy principles for dissipative systems is presented. It is shown that dissipative systems satisfy a conservation law for the bilinear Hamiltonian provided the Lagrangian is time invariant. The energy on the other hand, differs from the Hamiltonian by being quadratic and by having a negative definite time derivative (positive power dissipation). The energy is a Lyapunov functional whose definiteness yields necessary and sufficient stability criteria. The stability problem of resistive magnetohydrodynamic (MHD) is addressed: the energy principle for ideal MHD is generalized and the stability criterion by Tasso [Phys. Lett. 147, 28 (1990)] is shown to be necessary in addition to sufficient for real growth rates. An energy principle is found for the inner layer equations that yields the resistive stability criterion DR<O in the incompressible limit, whereas the tearing mode criterion delta< 0 is shown to result from the conservation law of the bilinear concomitant in the resistive layer.

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Authors: PLETZER A, Centre de Recherches en Physique des Plasmas, EURATOM Association-Confederation Suisse, Ecole Polytechnique Federale de Lausanne (CH)
Bibliographic Reference: An article published in: Physics of Plasmas, vol.4. No.9, pp.3141-3151
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