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.
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.
“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.”
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 If the Riemannian manifold is flat, then the geodesic equation becomes the familiar -derivative. 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 system in -body 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 under the influence of a quadratic potential energy. Here, the configuration space is the unit sphere -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 diffeomorphisms, while the fiber over a volume form -preserving 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 .
One can see that for a flat manifold this is a flat metric on as it is the , on diffeomorphisms -metric regarded as vector functions This metric is right-invariant for the . (but not the -action -action): for since the change of coordinates leads to the factor , in the integrand.
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 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