Geometric hydrodynamics and infinite-dimensional Newton’s equations

By Boris Khesin, Gerard Misiołek, Klas Modin

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.

Evgeny Schwartz, The Dragon

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.⁠Footnote1) 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 stands for the covariant derivative on and the dot denotes the -derivative. If the Riemannian manifold is flat, then the geodesic equation becomes the familiar in any local Euclidean coordinates on .

From the point of view of classical mechanics, the geodesic equation Equation 1.1 describes motions of a system driven only by its kinetic energy. More general systems may depend also on a potential energy. Indeed, if is a configuration space of some physical system (a Riemannian manifold) and represents its potential energy (a differentiable function), then satisfies Newton’s equations

One of the classical examples of Newton’s equations is the -body system in . Introducing coordinates one can regard as the configuration space of the system. If the bodies have masses , then their kinetic energy is and hence corresponds to a Riemannian metric on of the form . If denotes the gravitational constant, then the potential energy is given by the expression

which becomes infinite on the diagonals . The corresponding Lagrangian function is , while the total energy of the system (its Hamiltonian) is . We shall revisit this system in a fluid dynamical context below.

Another classical example is provided by the C. Neumann problem Reference 63 describing the motion of a single particle on an -sphere under the influence of a quadratic potential energy. Here, the configuration space is the unit sphere in while the phase space is the tangent bundle of the sphere. The potential energy of the system is given by , where and is a positive-definite symmetric matrix. As before, the Lagrangian function is the difference of the kinetic and the potential energies

The C. Neumann system is related to the geodesic flow on the ellipsoid defined by the equation , see, e.g., Moser Reference 62, Sec. 3. The corresponding Hamiltonian system on the cotangent bundle is integrable and, if the eigenvalues of are all different, then the first integrals, expressed in canonical coordinates and , are explicitly given by

where and are the components of and with respect to the eigenbasis of . We will see that an infinite-dimensional analogue of the C. Neumann problem naturally arises in the context of information geometry, while its integrability in infinite dimensions remains an intriguing open problem.

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 of dimension (for our purposes may be a domain in ) and assume that it is filled with an inviscid fluid (either a gas or a liquid). When the group of diffeomorphisms of is equipped with an metric (essentially, the metric corresponding to the fluid’s kinetic energy, as we shall discuss later) its geodesics describe the motions of noninteracting particles in whose velocity field satisfies the inviscid Burgers equation

When the metric is restricted to the subgroup of diffeomorphisms of that preserve the Riemannian volume form , then its geodesics describe the motions of an ideal (that is, inviscid and incompressible) fluid in whose velocity field satisfies the incompressible Euler equations

Here is the pressure function whose gradient is defined uniquely by the divergence-free condition on the velocity field and can be viewed as a constraining force. (If has a nonempty boundary, then is also required to be tangent to ).

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 and a density function , namely

These equations can be interpreted as Newton’s equations on the full diffeomorphism group of . In this case the pressure is a prescribed function of density and this dependence, called the equation of state, determines the fluid’s potential energy. In sections below we shall also consider general equations with the term replaced by the gradient , where denotes an arbitrary thermodynamical work function; cf. section 4.3.

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 , we equip the group of all diffeomorphisms of with a (weak) Riemannian metric and a natural fibration.

Namely, assume that the Riemannian volume form has the unit total volume (or total mass) and regard it as a reference density on . Now consider the projection of diffeomorphisms onto the space of (normalized) smooth densities on . The diffeomorphism group is fibered over by means of this projection as follows: the fiber over is the subgroup of -preserving diffeomorphisms, while the fiber over a volume form consists of all diffeomorphisms that push to , or, equivalently, . (Note that diffeomorphisms from act transitively on smooth normalized densities, according to Moser’s theorem.) In other words, two diffeomorphisms and belong to the same fiber if and only if for some diffeomorphism .

Remark 1.1.

It is worth comparing “the functional dimensions” of the fiber and the base . The space of densities can be thought of as the space of functions of variables, where . On the other hand, the group consists of

i)

isometries in dimension (e.g., for it is ),

ii)

symplectic diffeomorphisms in dimension (e.g., for these are Hamiltonian diffeomorphisms, locally described by a function of 2 variables), and

iii)

in dimensions these diffeomorphisms are subject to the only constraint on the Jacobian: (i.e., one equation on functions of variables).

Therefore, in the fibration the fiber is small compared to the base in dimension , the fiber and the base are about the same size in dimension , and the fiber becomes much bigger than the base starting with dimension .

Definition 1.2.

Now define an -metric on by the formula

where is a tangent vector at the point , i.e., a map such that for each , while stands for the pointwise Riemannian product at the point .

One can see that for a flat manifold this is a flat metric on , as it is the -metric on diffeomorphisms regarded as vector functions . This metric is right-invariant for the -action (but not the -action): for , since the change of coordinates leads to the factor in the integrand.

Remark 1.3.

Consider the following optimal mass transport problem: Find a map that pushes the measure forward to another measure of the same total volume and attains the minimum of the -cost functional among all such maps ( denotes here the Riemannian distance function on ). The minimal cost of transport defines the following Kantorovich–Wasserstein distance on the space of densities :

The mass transport problem admits a unique solution for Borel maps and densities (defined up to measure-zero sets), called the optimal map , see, e.g., Reference 12Reference 55Reference 77. In the smooth setting the Kantorovich–Wasserstein distance is generated by a (weak) Riemannian metric on the space of smooth densities Reference 9Reference 64, which we call the Wasserstein–Otto metric and describe in detail in section 2.2. Thus both and can be regarded as infinite-dimensional Riemannian manifolds for the and Wasserstein–Otto metrics, respectively.

Remark 1.4.

Later we will see (following Reference 64) that the corresponding projection is a Riemannian submersion from the diffeomorphism group onto the density space , i.e., the map respecting the above metrics. Recall that for two Riemannian manifolds and a submersion is a smooth map which has a surjective differential and preserves lengths of horizontal tangent vectors to . For a bundle this means that on there is a distribution of horizontal spaces orthogonal to fibers and projecting isometrically to the tangent spaces to . Geodesics on can be lifted to horizontal geodesics in , and the lift is unique for a given initial point in .

Note also that horizontal (i.e., normal to fibers) spaces in the bundle consist of right-translated gradient fields. In short, this follows from the Hodge decomposition for vector fields on : any vector field decomposes uniquely into the sum of a divergence-free field and a gradient field , which are -orthogonal to each other, . The vertical tangent space at the identity coincides with , while the horizontal space is . The vertical space (tangent to a fiber) at a point consists of , divergence-free vector fields right-translated by the diffeomorphism , while the horizontal space is given by the right-translated gradient fields, . The -type metric on horizontal spaces for different points of the same fiber projects isometrically to one and the same metric on the base, due to the -invariance of the metric. Now the Riemannian submersion property follows from the observation that the Wasserstein–Otto metric is Riemannian and is generated by the metric on gradients; see Reference 9.

Example 1.5.

Geodesics in the full diffeomorphism group with respect to the above -metric have a particularly simple description for a flat manifold ; cf. Reference 11Reference 20. In that case the group is locally (a dense subset of) the -space of vector-functions , and hence is flat, while its geodesics are straight lines. If is the velocity field of the flow in defined by , then the geodesic equation becomes , which in turn is equivalent to the inviscid Burgers equation

Furthermore, from the viewpoint of exterior geometry, the Euler equation can be regarded as an equation with a constraining force acting orthogonally to the submanifold of volume-preserving diffeomorphisms and keeping the geodesics confined to that submanifold.

Remark 1.6.

Analytical studies of the differential geometry of the incompressible Euler equations began with the paper of Ebin and Marsden Reference 20 and continued with Reference 21Reference 57Reference 70 and others. The approach via generalized flows was proposed by Brenier Reference 11. Many aspects of this approach to the group of all diffeomorphisms and their relation to the Kantorovich–Wasserstein space of densities and problems of optimal mass transport are discussed in Reference 47Reference 77Reference 82. There is also a finite-dimensional matrix version of the submersion framework and decomposition of diffeomorphsims; see Reference 12. In the finite-dimensional optimal mass transport on discussed in Reference 59 the probability distributions are multivariate Gaussians and the transport maps are linear transformations. The corresponding dynamics turned out to be closely related to many finite-dimensional flows studied in the literature: Toda-lattice, isospectral flows, and an entropy gradient interpretation of the Brockett flow for matrix diagonalization. A sub-Riemannian version of the exterior geometry of with vector fields tangent to a bracket generating distribution in , as well as a nonholonomic version of Moser lemma, is described in Reference 1Reference 36. For a symplectic reduction formulation to the above Riemannian submersion see section 3.2.

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

Example 1.7 (Shallow water equation as a Newton’s equation).

We next proceed to describe Newton’s equations on the diffeomorphism group . To this end we consider the case of a potential on which depends only on the density carried by a diffeomorphism , i.e., the potential for is a pullback for the projection , where as we take a simple quadratic function

on the space of densities. It turns out that with this potential we obtain shallow water equations. There are several equivalent formulations, depending on the functional setting.

Proposition 1.8.

Newton’s equations with respect to the -metric Equation 2.1 and the potential Equation 1.5 take the following forms:

on

where ;

the shallow water equations on

where is the horizontal velocity field and is the water depth;

for the gradient velocity they assume the Hamilton–Jacobi form

Remark 1.9.

The latter form can be regarded as an equation on . Since is a quadratic function, equations Equation 1.7 can be interpreted as a Hamiltonian form of an infinite-dimensional harmonic oscillator with respect to the Wasserstein–Otto metric Equation 2.8. We will prove this theorem in a more general setting of a barotropic fluid (cf. equation Equation 1.4) with an arbitrary potential in section 4.3; here .

Example 1.10 (The -body problem as a Newton’s equation).

Newton’s law of gravitation states that for a body with mass distribution , the associated potential is , where is the gravitational constant. Following the above framework, the potential function on is given by

where is a (suitably defined) inverse Laplacian with appropriate boundary conditions.

The corresponding fluid system is described by

Thus, we have arrived at a fluid dynamics formulation of a continuous Newton mass system under the influence of gravity: a “fluid particle” positioned at experiences a gravitational pull corresponding to the potential . In particular, if we obtain the well-known Green’s function for the Laplacian

We now wish to study weak solutions to these equations where the mass distribution is replaced by an atomic measure

for point masses positioned at . The differential-geometric setting is as follows. We have a Riemannian metric on (the Wasserstein—Otto metric) and a potential function on (the Newton potential). The group acts on the (finite-dimensional) manifold of atomic measures with particles. Clearly, we have . The isotropy subgroup for this action on is

Although the horizontal distribution is not defined rigorously, it is formally given by vector fields with support on . With this notion of horizontality, the projection , given by , is a Riemannian submersion with respect to the weighted Riemannian structure on , given by . For , substituting the atomic measure into the formula Equation 1.8 and using the Green’s function for gives

The resulting finite-dimensional Riemannian structure together with this potential function defines the kinetic and potential energies giving rise to the -body problem.

Remark 1.11.

In the wake of Arnold’s work, various approaches to infinite-dimensional generalizations of Newton’s equations Equation 1.2 have been considered in special settings. Those of perhaps most interest from our point of view were proposed by Smolentsev Reference 72Reference 73, who used diffeomorphism groups to describe the motions of a barotropic fluid, and by Ebin Reference 19, who used a similar framework to study, among others, the incompressible limit of slightly compressible fluids. In the early 1980s, Doebner, Goldin, and Sharp Reference 17Reference 27 began to develop links between representations of diffeomorphism groups, ideal fluids, and nonlinear quantum systems, revisiting in the process the classical transform of Madelung Reference 48Reference 49. More recently, motivated by the problems of optimal transport, von Renesse Reference 79 used it to relate the Schrödinger equations with a variant of Newton’s equations defined on the space of probability measures (see Section 9 below for details). A similar objective, but driven partly by motivation from information geometry and statistics, can be found in a recent paper of Molitor Reference 60.

In what follows we will systematically describe how one can conveniently study various equations of mathematical physics, including all the examples listed in Table 1, from a unified point of view as certain Newton’s equations. Our goal is to present a rigorous infinite-dimensional geometric framework that unifies Arnold’s approach to incompressible and inviscid hydrodynamics and its relatives with various generalizations of Newton’s equations Equation 1.2 such as those mentioned above, to provide a very general setting for systems of hydrodynamical origin on diffeomorphism groups and spaces of probability densities. We will also survey the setting of the Hamiltonian reduction, which establishes a correspondence between various representations of these equations.

Remark 1.12.

More precisely, given a compact -dimensional manifold , we will equip the group of diffeomorphisms and the space of nonvanishing probability densities with the structures of smooth infinite-dimensional manifolds (see Appendix B for details) and study Newton’s equations on these manifolds viewed as the associated configuration spaces.

As a brief preview of what follows, let be the subgroup of diffeomorphisms preserving the Riemannian volume form of . Consider the fibration of the group of all diffeomorphisms over the space of densities

discussed by Moser Reference 61, whose cotangent bundles and are related by a symplectic reduction; cf. Section 3 below. Moser’s construction can be used to introduce two different algebraic objects: the first is obtained by identifying with the left cosets

and the second by identifying it with the right cosets

In this paper we will make use of both identifications.

In order to define Newton’s equations on and , and to investigate their mutual relations, we will choose Riemannian metrics on both spaces so that the natural projections corresponding to Equation 1.9 or Equation 1.10 become (infinite-dimensional) Riemannian submersions. We will consider two such pairs of metrics. In Section 2, using left cosets, we will study a noninvariant -metric on together with the Wasserstein–Otto metric on . In Section 7, using right cosets, we will focus on a right-invariant metric on and the Fisher–Rao information metric on . Extending the results of von Renesse Reference 79, we will then derive in Section 9 various geometric properties of the Madelung transform. This will allow us to represent Newton’s equations on as Schrödinger-type equations for wave functions.

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 in the viscous Burgers equation

can be viewed as a (linear) friction force while the equation itself can be seen as Newton’s equation on with a nonpotential force. Similarly, observe that the Navier–Stokes equations of a viscous incompressible fluid

can be seen as Newton’s equations on