Euler–Arnold equation

In mathematical physics and differential geometry, the Euler–Arnold equations are a class of partial differential equations (PDEs) that describe the geodesic flow on infinite-dimensional Lie groups equipped with right-invariant metrics. These equations generalize classical mechanical systems, such as rigid body motion and ideal fluid flow (Euler equation), by interpreting their evolution as geodesic flow on a group of transformations. Introduced by Vladimir Arnold in 1966, this perspective unifies diverse PDEs arising in fluid dynamics, elasticity, and other areas, in a common geometrical framework. In hydrodynamics, they serve the purpose of describing the motion of inviscid, incompressible fluids. 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.