The affine motion of two-dimensional (2d) incompressible fluids surrounded by vacuum can be reduced to a completely integrable and globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in $$\mathrm{SL}(2,\mathbb {R})$$. In the case of perfect fluids, the motion is given by geodesic flow in $$\mathrm{SL}(2,\mathbb {R})$$ with the Euclidean metric, while for magnetically conducting fluids (MHD), the motion is governed by a harmonic oscillator in $$\mathrm{SL}(2,\mathbb {R})$$. A complete classification of the dynamics is given including rigid motions, rotating eddies with stable and unstable manifolds, and solutions with vanishing pressure. For perfect fluids, the displacement generically becomes unbounded, as $$t\rightarrow \pm \infty $$. For MHD, solutions are bounded and generically quasi-periodic and recurrent.

中文翻译：

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$$\mathrm{SL}(2,\mathbb {R})$$SL(2,R)中真空和流动包围的二维不可压缩流体的仿射运动

对于 $$\mathrm{SL}(2,\mathbb {R })$$。在完美流体的情况下，运动由具有欧几里德度量的 $$\mathrm{SL}(2,\mathbb {R})$$ 中的测地线流给出，而对于导磁流体 (MHD)，运动是由 $$\mathrm{SL}(2,\mathbb {R})$$ 中的谐振子控制。给出了动力学的完整分类，包括刚性运动、具有稳定和不稳定流形的旋转涡流以及具有消失压力的解决方案。对于完美流体，位移一般是无界的，如 $$t\rightarrow\pm\infty $$。对于 MHD，