# Topological methods in hydrodynamics

@inproceedings{Arnold1998TopologicalMI, title={Topological methods in hydrodynamics}, author={Vladimir I. Arnold and Boris A. Khesin}, year={1998} }

A group theoretical approach to hydrodynamics considers hydrodynamics to be the differential geometry of diffeomorphism groups. The principle of least action implies that the motion of a fluid is described by the geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy. Investigation of the geometry and structure of such groups turns out to be useful for describing the global behavior of fluids for large time intervals.

## 1,413 Citations

### Geometric hydrodynamics and infinite-dimensional Newton’s equations

- MathematicsBulletin of the American Mathematical Society
- 2021

We revisit the geodesic approach to ideal hydrodynamics and present a related geometric framework for Newton’s equations on groups of diffeomorphisms and spaces of probability densities. The latter…

### Geodesics on extensions of Lie groups and stability: the superconductivity equation

- Mathematics, Physics
- 2001

### The Geometry of Barotropic Flow

- Mathematics
- 2013

In this paper we construct a new noninvariant Riemannian metric on the semidirect product of the diffeomorphism group of a manifold and the space of positive functions on that manifold, which has the…

### Geometric Hydrodynamics of Compressible Fluids

- Mathematics
- 2020

We develop a geometric framework for Newton's equations on infinite-dimensional configuration spaces to describe numerous fluid dynamical equations. According to V. Arnold, the Euler equations of an…

### Lagrangian description, symplectization, and Eulerian dynamics of incompressible fluids

- Mathematics, Physics
- 2016

Eulerian dynamical equations in a three-dimensional domain are used to construct a formal symplectic structure on time-extended space. Symmetries, invariants, and conservation laws are related to…

### The Motion of Solid Bodies in Potential Flow With Circulation: A Geometric Outlook

- Physics
- 2008

The motion of a circular body in 2D potential flow is studied using symplectic reduction. The equations of motion are obtained starting front a kinetic-energy type system on a space of embeddings and…

### Navier-Stokes Equation and Diffusions on the Group of Homeomorphisms of the Torus

- Mathematics
- 2007

A stochastic variational principle for the (two dimensional) Navier-Stokes equation is established. The velocity field can be considered as a generalized velocity of a diffusion process with values…

### Geodesic Equations on Diffeomorphism Groups

- Mathematics
- 2008

We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant L 2 or H 1…

### A note on the stationary Euler equations of hydrodynamics

- MathematicsErgodic Theory and Dynamical Systems
- 2015

This note concerns stationary solutions of the Euler equations for an ideal fluid on a closed 3-manifold. We prove that if the velocity field of such a solution has no zeroes and real analytic…

## References

SHOWING 1-10 OF 563 REFERENCES

### Groups of diffeomorphisms and the motion of an incompressible fluid

- Mathematics
- 1970

In this paper we are concerned with the manifold structure of certain groups of diffeomorphisms, and with the use of this structure to obtain sharp existence and uniqueness theorems for the classical…

### Geodesics and curvature of a group of diffeomorphisms and motion of an ideal fluid

- Mathematics
- 1992

Motion of an ideal fluid is represented as geodesics on the group of all volume-preserving diffeomorphisms. An explicit form of the geodesic equation is presented for the fluid flow on a three-torus…

### Riemannian curvature on the group of area-preserving diffeomorphisms (motions of fluid) of 2-sphere

- Mathematics
- 1997

### Finite-mode analogs of 2D ideal hydrodynamics: coadjoint orbits and local canonical structure

- Mathematics, Physics
- 1991

### ON THE GEOMETRY OF THE GROUP OF DIFFEOMORPHISMS AND THE DYNAMICS OF AN IDEAL INCOMPRESSIBLE FLUID

- Mathematics
- 1987

The author studies the geometric properties of the group of volume-preserving diffeomorphisms of a region. This group is the configuration space of an ideal incompressible fluid, the trajectories of…

### Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations

- Mathematics
- 1999

The three-dimensional motion of an incompressible inviscid fluid is classically described by the Euler equations but can also be seen, following Arnold [1], as a geodesic on a group of…

### Two-dimensional ideal magnetohydrodynamics and differential geometry

- Mathematics, Physics
- 1993

It is shown that equations of two-dimensional ideal magnetohydrodynamics may be regarded as geodesic equations on appropriate infinite dimensional Lie group. The physical interpretation of such a…