TY - JOUR
AU - Khesin, B A
AB -  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. We begin with a survey of conservation laws for incompressible and barotropic fluid flows and superconductivity. These laws are determined by the infinitesimal structure of the corresponding diffeomorphism groups (i.e. the structure of their Lie algebras and coalgebras). For example, the equations of an inviscid incompressible fluid are Hamiltonian on the coadjoint orbits of the group of volume-preserving diffeomorphisms (Arnold 1966, 1969a, 1989). It is well known that for a two-dimensional flow there is an infinite number of enstrophy-type integrals [ Sf(curl v) d2x], and in a three-dimensional case there is the total helicity integral [S (curl v, v) d3x]. It turns out that these ideal hydrodynamics equations (as well as barotropic fluid and superconductivity equations) have an infinite 145 0066-4189/92/0115-0145$02.00 ARNOLD & KHESIN number of invariants 
TI - Topological Methods in Hydrodynamics
JF - Annual Review of Fluid Mechanics
DO - 10.1146/annurev.fl.24.010192.001045
DA - 1992-01-01
UR - https://www.deepdyve.com/lp/annual-reviews/topological-methods-in-hydrodynamics-CNZPVnJUtZ
SP - 145
EP - 166
VL - 24
IS - 1
DP - DeepDyve
ER -