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.^[1][2][3] In hydrodynamics, they serve the purpose of describing the motion of inviscid, incompressible fluids.^[4] The formal definition requires advanced knowledge of analysis of PDEs.^[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