Euler–Arnold equation

In mathematical physics, the Euler–Arnold equations are a class of partial differential equations (PDEs) that describe the evolution of a velocity field when the Lagrangian flow is a geodesic in a group of smooth transformations (see groupoid). It connects differential geometry of infinite-dimensional Lie groups ('infinite-dimensional differential geometry') ideas to PDEs theory ideas. They are named after Leonhard Euler and Vladimir Arnold. In hydrodynamics, they serve the purpose of describing the motion of inviscid, incompressible fluids. The formal definition requires advanced knowledge of analysis of PDEs.

A great number of results related to this are included in now called Euler–Arnold theory, whose main idea is to geometrically interpret ODEs on infinite-dimensional manifolds as PDEs (and vice-versa).

Many PDEs from fluid dynamics are just special cases of the Euler–Arnold equation when viewed from suitable Lie groups: Burgers' equation, Korteweg–De Vries equation, Camassa–Holm equation, Hunter–Saxton equation, and many more.

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