# Geometric hydrodynamics and infinite-dimensional Newton’s equations

To the memory of Vladimir Arnold and Jerry Marsden, pioneers of geometric hydrodynamics, who left in 2010, ten years ago.

## Abstract

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 setting is sufficiently general to include equations of compressible and incompressible fluid dynamics, magnetohydrodynamics, shallow water systems and equations of relativistic fluids. We illustrate this with a survey of selected examples, as well as with new results, using the tools of infinite-dimensional information geometry, optimal transport, the Madelung transform, and the formalism of symplectic and Poisson reduction.

The only way to get rid of dragons is to have one of your own.

## 1. Introduction

The Euler equations of hydrodynamics describe motions of an incompressible and inviscid fluid occupying a fixed domain (with or without boundary). In the 1960s V. I. Arnold discovered that these equations are precisely the geodesic equations of a right-invariant metric on the group of diffeomorphisms preserving the volume element of the domain Reference 5. This beautiful observation, combining the early work of Hadamard on geodesic flows on surfaces with the dynamical systems ideas of Poincaré and Kolmogorov and using analogies with classical mechanics of rigid bodies, inspired many researchers—one of the first was J. E. Marsden. Their combined efforts led to remarkable developments, such as formulation of new stability criteria for fluid motions Reference 4Reference 7Reference 23Reference 24, explicit calculation of the associated Hamiltonian structures and first integrals Reference 8Reference 53Reference 54, development of symplectic reduction methods Reference 52Reference 53, introduction of Riemannian geometric techniques to the study of diffeomorphism groups including explicit computations of curvatures, conjugate points, diameters Reference 5Reference 56Reference 57Reference 70, detailed studies of regularity properties of the solution maps of the fluid equations in Lagrangian and Eulerian coordinates Reference 20Reference 21, construction of similar configuration spaces for other partial differential equations of hydrodynamic origin Reference 8Reference 33, etc.

In this paper, based on the research pioneered and developed by Arnold, Marsden and many others, we present a broad geometric framework which includes an infinite-dimensional generalization of the classical Newton’s equations of motion to the setting of diffeomorphism groups and spaces of probability densities. This approach has a wide range of applicability and covers a large class of important equations of mathematical physics. Our goal is twofold. We start by presenting a concise survey of various geodesic and Newton’s equations, thus introducing the reader to the rapidly expanding field of geometric hydrodynamics, and revisiting a few standard examples from the point of view advocated here. We then also include a number of selected new results to illustrate the flexibility and utility of this approach.

We focus primarily on the geometric aspects and emphasize formal procedures leaving until the end analytic issues which in most cases can be resolved using standard methods once an appropriate functional-analytic setting (e.g., Fréchet, Hölder, or Sobolev) is adopted. The corresponding tame Fréchet framework is described in more detail in Appendix B. Our main tools include the Wasserstein metric of optimal transport, the infinite-dimensional analogue of the Fisher–Rao information metric, the Madelung transform, and the formalism of symplectic and Poisson reduction, all of which are defined in the paper. The early sections should be accessible to mathematicians with only general background in geometry. In later sections some acquaintance with the basic material found, for example, in the monographs Reference 7Reference 8Reference 51 will be helpful.

Needless to say, it is not possible to give a comprehensive survey of such a vast area of geometric hydrodynamics in such a limited space, therefore our emphasis on certain topics and the choice of examples are admittedly subjective. (The epigraph to the paper is our take on the Laws of Nature, on the tamed structures discussed below, as well as a counterpoint to the beautiful epigraph in the monograph Reference 14, quoted here in the footnote.Footnote^{1}) We nevertheless hope that this paper provides a flavour of some of the results in this beautiful area pioneered by V. Arnold and J. Marsden.

^{1}

“There once lived a man

who learned how to slay dragons

and gave all he possessed

to mastering the art.

After three years

he was fully prepared but,

alas, he found no opportunity

to practise his skills.”

—Dschuang Dsi

**Backlinks**: Reference 1, Reference 2.

### 1.1. Geodesics and Newton’s equations: finite-dimensional examples

A curve is a geodesic in a Riemannian manifold if it satisfies the *equation of geodesics*, namely

where

From the point of view of classical mechanics, the geodesic equation Equation 1.1 describes motions *Newton’s equations*

One of the classical examples of Newton’s equations is the

which becomes infinite on the diagonals

Another classical example is provided by the C. Neumann problem Reference 63 describing the motion of a single particle on an

The C. Neumann system is related to the geodesic flow on the ellipsoid defined by the equation

where

### 1.2. Three motivating examples from hydrodynamics

We now make a leap from finite to infinite dimensions. Our aim is to show that many well-known PDEs of hydrodynamical pedigree can be cast as Newton’s equations on infinite-dimensional manifolds. Indeed, groups of smooth diffeomorphisms arise naturally as configuration spaces of compressible and incompressible fluids. We begin with three famous examples. Consider a connected compact Riemannian manifold

When the

Here

As we shall see below, both of the above equations turn out to be examples of equations of geodesics on diffeomorphism groups with Lagrangians given by the corresponding kinetic energy. However, the Lagrangian in our next example will include also a potential energy. Consider the equations of a compressible (barotropic) fluid describing the evolution of a velocity field

These equations can be interpreted as Newton’s equations on the full diffeomorphism group of

### 1.3. Riemannian metrics and their geodesics on spaces of diffeomorphisms and densities

Let us next see how differential geometry of diffeomorphism groups manifests itself in the above equations. Given a Riemannian manifold

Namely, assume that the Riemannian volume form

One can see that for a flat manifold

### 1.4. First examples of Newton’s equations on diffeomorphism groups

### 1.5. Other related equations

Newton’s equations for fluids discussed in the present paper are assumed to be conservative systems with a potential force. However, the subject concerning Newton’s equations is broader, and we mention briefly two topics related to nonconservative Newton’s equations for compressible and incompressible fluids that are beyond the scope of this paper.

First, observe that the dissipative term

can be viewed as a (linear) friction force while the equation itself can be seen as Newton’s equation on

can be seen as Newton’s equations on