Abstract
From Hamilton's principle and a factorization Ansatz we derive a class of exact solutions for three-dimensional motion in ideal, compressible MHD. These exact nonlinear solutions are motions generated by time-dependent affine transformations, under which the fluid rotates, circulates and deforms. They reduce to three-dimensional self-similar solutions when rotation is absent. Continuous symmetries of Hamilton's principle for the affine MHD motions generate various constants of motion. Discrete symmetries establish duality relations among classes of solutions. In a special case, rotational and circulatory MHD motion is expressed as classical mechanical motion upon its own symmetry group, the Lie group O(4), in the well-known Arnold-Lax-Euler commutator form, M ̊ = [ω, M] .
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