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The affine motion of two-dimensional (2d) incompressible fluids surrounded byvacuum can be reduced to a completely integrable and globally solvableHamiltonian system of ordinary differential equations for the deformationgradient in ${\rm SL}(2,{\mathbb R})$. In the case of perfect fluids, themotion is given by geodesic flow in ${\rm SL}(2,{\mathbb R})$ with theEuclidean metric, while for magnetically conducting fluids (MHD), the motion isgoverned by a harmonic oscillator in ${\rm SL}(2,{\mathbb R})$. A completeclassification of the dynamics is given including rigid motions, rotatingeddies with stable and unstable manifolds, and solutions with vanishingpressure. For perfect fluids, the displacement generically becomes unbounded,as $t\to\pm\infty$. For MHD, solutions are bounded and genericallyquasi-periodic and recurrent.

Affine motion of 2d incompressible fluids surrounded by vacuum and flows in ${\rm SL}(2,{\mathbb R})$