mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...

, the Euler–Arnold equations are a class of

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

s (PDEs) that describe the evolution of a velocity field when the Lagrangian flow is a

geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

in a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

of smooth transformations (see

groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...

). It connects

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

of infinite-dimensional Lie groups ('infinite-dimensional differential geometry') ideas to PDEs theory ideas. They are named after

Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

and

Vladimir Arnold Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to s ...

. In

hydrodynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in ...

, they serve the purpose of describing the motion of inviscid, incompressible

fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...

s. 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

ODE An ode (from ) is a type of lyric poetry, with its origins in Ancient Greece. Odes are elaborately structured poems praising or glorifying an event or individual, describing nature intellectually as well as emotionally. A classic ode is structu ...

s 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 group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

Burgers' equation Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and ...

, Korteweg–De Vries equation, Camassa–Holm equation, Hunter–Saxton equation, and many more.

Context

In 1966, Arnold published the paper "" ('On the

of infinite-dimensional Lie groups and its applications to the

of perfect fluids'), in which he presented a common

geometric Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

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 In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between ...

.IAMP News Bulletin, July 2010, pp. 25–26
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Context

Further reading

References

See also