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.^[1][2][3] In hydrodynamics, they serve the purpose of describing the motion of inviscid, incompressible fluids.^[4][5] 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).^[6]

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.^[7]

Context

In 1966, Arnold published the paper "Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits" ('On the differential geometry of infinite-dimensional Lie groups and its applications to the hydrodynamics of perfect fluids'), in which he presented a common geometric interpretation for both the Euler's equations for rotating rigid bodies and the Euler's equations of fluid dynamics, this effectively linked topics previously thought to be unrelated, and enabled mathematical solutions to many questions related to fluid flows and their turbulence.^[8][9][10]

Further reading

• Terence Tao's blog: https://terrytao.wordpress.com/2010/06/07/the-euler-arnold-equation/
• The original source: Arnold, V. "Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits." Annales de l'Institut Fourier (Grenoble) 16 (1966), 319–361. (in French) DOI: 10.5802/aif.233
• Jae Min Lee (2018), Geometry and Analysis of some Euler-Arnold Equations, PhD thesis, City University of New York