Arnold's geometric formulation of the Euler equations

Remarks

Here $g(t)$ maps the initial position of each fluid particle to its position at time $t$. In Arnold's 1966 paper the correspondence is formal, since $G$ is not a Lie group in the finite-dimensional sense. Ebin and Marsden made it rigorous in 1970 on groups of Sobolev class diffeomorphisms.

Example: for $M$ the circle $\R / \Z$ with the standard metric, $\operatorname{div} v = v'$, so the only divergence-free fields are constants and $G$ is the group of rotations. The geodesics are the uniform rotations $g(t)(x) = x + c t$, whose Eulerian velocity is the constant $c$, and constant $u$ with constant $p$ solves the Euler equations on the circle.

Non-example: the path of rotations $g(t)(x) = x + t^2$ in the same group is not a geodesic. Its Eulerian velocity $u = 2t$ has $\partial_t u = 2$, while $-\nabla p = -p'$ has mean zero on the circle, so it cannot equal $2$.

The same scheme with other groups and metrics gives other equations. The rotation group $SO(3)$ gives the Euler equations of a rigid body. The circle diffeomorphism group with the $L^2$ metric gives $\partial_t u + 3 u \, \partial_x u = 0$, a rescaled inviscid Burgers equation (see diffeomorphism group).