The Godbillon-Vey Invariant as a Restricted Casimir of Three-dimensional Ideal Fluids

We show the Godbillon-Vey invariant arises as a `restricted Casimir' invariant for three-dimensional ideal fluids associated to a foliation. We compare to a finite-dimensional system, the rattleback, where analogous phenomena occur.


Introduction
The topological aspect of ideal fluids has its origins in the transport of vorticity. A consequence is the conservation of helicity, which measures the average linking of vortex lines [1,2,3]. In the Hamiltonian formulation of ideal fluids as an infinite-dimensional Lie-Poisson system, helicity appears as a Casimir invariant, a degeneracy in the Lie-Poisson bracket [4].
The state of an ideal fluid on a homology 3-sphere is specified by the vorticity, a divergence-free vector field. A Casimir in an ideal fluid is invariant under all volume-preserving diffeomorphisms of the domain, so can be said to measure a topological property of the vorticity. For a generic vorticity field, helicity is the only topological invariant [5,6,7]. However, higher order invariants can be defined in special cases. Here we study the Godbillon-Vey invariant, GV , which can be associated to a vorticity field tangent to a codimension-1 foliation [8,9,10,11]. GV originates in the theory of foliations [12,13]; in ideal fluids it measures topological helical compression of vortex lines [8]. The goal of this paper is to show how GV fits naturally into the Lie-Poisson Hamiltonian formulation of ideal fluids [4] as a 'restricted Casimir' invariant. In particular, we consider a set S of ideal fluids where the Lie-Poisson bracket has an additional degeneracy associated to the Lie subalgebra of volume-preserving vector fields tangent to a foliation, which may vary within S. On S we construct a modified Lie-Poisson type bracket, in terms of which the Godbillon-Vey invariant appears as a Casimir.
Imposing this degeneracy also forces the helicity to vanish and in this sense GV is hierarchical, in a manner analogous to that suggested by Arnold and Khesin [14]. Recent work [15,16] has studied similar hierarchical structures in Hamiltonian systems, where a singular region in phase space with a Poisson operator of decreased rank can itself be considered as a Poisson submanifold, on which new Casimir invariants appear. What we describe can be considered an example of this phenomenon. We note also that a foliation on M can be defined through an exact sequence of vector bundles where N = T F/T M is the normal bundle of the foliation. This sequence passes to the volume-preserving case, and we note the connection to the classification of Casimir invariants coming from Lie algebra extensions [17].
There is a finite-dimensional example in the Lie-Poisson formulation of the 'rattleback' spinning top [18], where corresponding phenomena occur: there is a submanifold of phase space where the Poisson operator has an additional degeneracy associated to a Lie subalgebra; on this submanifold the primary Casimir vanishes and a new restricted Casimir appears.
In the finite-dimensional rattleback case, perturbation of the system around the singular manifold leads to interesting dynamical properties [18]. Our own analysis of the Godbillon-Vey invariant elsewhere also suggests a strong connection to dynamics; GV provides a global and local obstruction to steady flow and can be used to estimate the rate of change of vorticity [8]. With that in mind, we suggest that flows with GV = 0 (or perturbations thereof) may prove particularly interesting from a dynamical perspective. Finally, we note our light touch regarding rigour.

Lie-Poisson Systems
See e.g. [17] for a description. Let g be a Lie algebra associated to a group G, with g * its dual. Given an element α ∈ g * and two elements U, V ∈ g we form the bracket where ·, · : g * × g → R is the natural pairing between the Lie algebra and its dual, and [·, ·] is the Lie bracket of g. This is then used to define the Lie-Poisson bracket where the (functional) derivative δF/δα is identified with an element of g by the relation The sign in (3) depends on whether we consider right-invariant or leftinvariant function(al)s on g * with respect to the coadjoint representation of G, but is irrelevant for our purposes. Coupled with a Hamiltonian function on g * , this specifies the system. The noncanonical nature of the Lie-Poisson bracket allows for the existence of Casimir invariants, C, given by the property {F, C} = 0 for any function F . We define the coadjoint bracket [·, ·] † : g × g * → g * as This allows us to give the condition for C to be a Casimir as In this paper we will be interested in sets of points α ∈ S ⊂ g * where there is a non-generic degeneracy associated to a subalgebra h α ⊂ g, such that (7) α, U = 0, for U ∈ h α . For a given α ∈ g * , let β = ad * g α, g ∈ G. Then β is orthogonal to the subalgebra h β = ad g h α , so that S will, in general, be a set of coadjoint orbits in g * . The precise specification of admissible subalgebras and the subset S in a general formulation is left intentionally vague.

Finite Dimensional Example: the Rattleback
An idealised description of the chiral dynamics of the rattleback spinning top [18] can be formulated as a Lie-Poisson system based on the threedimensional Lie algebra with Bianchi classification VI h<−1 , spanned by three elements, P , R, S with Lie bracket (8) [ Physically P , R, and S are associated to pitching, rolling and spinning motions respectively, and h is a geometric parameter related to the aspect ratio of the top. The dynamical variable is an element of the dual space g * which we write as a lowercase tuple (p, r, s), in terms of which the dynamics are [18,19] The Hamiltonian of this system is given by H = (p 2 + r 2 + s 2 )/2. At a generic point in g * the Lie-Poisson bracket has a one-dimensional kernel, associated to the Casimir which one can check is conserved by the dynamics (9). There is a two-dimensional Abelian subalgebra h ⊂ VI h<−1 , spanned by P, R. The set of points M ⊂ g * orthogonal to h is the singular line (0, 0, s), so that on M the Casimir C = 0. On M the Lie-Poisson bracket is trivial, so the dynamics are trivial (one can see this by setting p = r = 0 in (9)). It follows that s is a constant of the motion on M only. Finally, note that M can be thought of as a one-dimensional Poisson manifold with trivial Poisson bracket, and with respect to this bracket s is a Casimir invariant (as is any function of s), so that s is a restricted Casimir invariant of the rattleback system. Physically it corresponds to simple spinning motion of the top.

The Godbillon-Vey Invariant as a Restricted Casimir in Three-Dimensional Ideal Fluids
Now we see how the same pattern of phenomena is found in three-dimensional ideal fluids on a manifold M . We assume throughout that M is a homology 3-sphere (one can take M = S 3 ).

Ideal Hydrodynamics and Helicity.
In the Lie-Poisson formulation of ideal fluids [4,14], g is the Lie algebra of volume-preserving vector fields on M with respect to a volume form µ, so that L U µ = 0 for U ∈ g and the 2-form ι U µ is closed. The dynamical variable is given by an element of the dual space g * , the smooth part of which can be identified as Ω 1 (M )/dΩ 0 (M ), the space of differential 1-forms modulo exact forms, and each element is given by a coset [α], with specific representative α. We will suppress the coset notation []. The pairing ·, · : g * × g → R is given by (11) α which does not depend on the representative 1-form α. In this case the Lie-Poisson bracket takes the form The coadjoint bracket is given as Where the vorticity field W ∈ g is given by dα = ι W µ. Helicity is defined as

4.2.
Foliations and g * . We now consider a codimension-1 foliation F α of M such that α ∈ g * satisfies (17) α, X = 0, for X in the subalgebra of volume-preserving vector fields h α ⊂ g that are tangent to the leaves of F α . Let β α be a defining form for F α and consider the family of closed 2-forms d(hβ α ), h a function, then the vector field Y defined by ι Y µ = d(hβ α ) is an element of h α . By assumption on α, As h is arbitrary, β α ∧ dα = 0. Recall the vorticity field W defined by dα = ι W µ, it follows that As an immediate consequence, the helicity, α, W , vanishes. Now let γ be a closed loop tangent to F. The quantity (20) is invariant under leafwise homotopies of γ, so that α defines a class [α] F in the foliated cohomology group H 1 (F α ). In fact Consider a family of smooth vector fields G λ ∈ g with support in a tubular neighbourhood of γ of diameter ∼ λ (with respect to a metric), tending to the singular vector field with support γ and constant flux φ as λ → 0, so that D ι G λ µ → φ as λ → 0, where D is a disk pierced by γ. Then α, G λ → φI g as λ → 0. But | α, G λ | < λC for some constant C, so I g = 0.
As γ was arbitary, [α] F = 0 ∈ H 1 (F α ). As an immediate consequence, any representative of the coset [α] ∈ Ω 1 (M )/dΩ 0 (M ) can be written as f β α + dg for functions f, g. There is then a canonical representative form α c = f β α , which we write as α in subsequent sections. With this choice of representative the helicity density vanishes, α c ∧ ι W µ = 0.

4.3
. Ω 1 I (M ) and the Functional Derivative. We may write α = f β α . For simplicity we will assume that f = 0, and we choose β α so that α = β α is a defining form for F α . Then, with this canonical choice, the subset of g * we are considering can be identified with the space of nonvanishing integrable 1-forms on M , which we write as Ω 1 I (M ) (this space is not connected, we consider a single arbitrary connected component). We note that Ω 1 I (M ) is no longer a vector space.
We would like to define functional derivatives on Ω 1 I (M ). For a oneparameter family α t ∈ Ω 1 I (M ), t ≥ 0 with time derivativeα t , we writė α =α 0 and α = α 0 . Now for a functional F on Ω 1 I (M ) the functional derivative is defined as and we identify δF /δα with a vector field as in Section 4.1. Because we have a canonical choice of α, we no longer require invariance under gauge transformations and so are not restricted to volume-preserving vector fields. We suppose instead δF /δα ∈ X(M )/Ξ α where X(M ) is the space of smooth vector fields on M and Ξ α ⊂ X(M ) is an α dependent subset satisfying α, U = 0 for U ∈ Ξ α . Our characterisation of Ξ α below is not complete, but is sufficient for our purposes. First, we will show that it is non-empty. As α t is integrable we have (23) α t ∧ dα t = 0.
In particular this gives for any function f , so that fields V satisfying are elements of Ξ α . Now we give two properties of general elements of Ξ α . Firstly we note that any field in Ξ α must be tangent to F α . We can choose α t = exp(gt)α, so thatα = gα for an arbitrary function g. Now suppose U is not tangent to F α , then by an appropriate choice of g we can force gα, U = 0, so U / ∈ Ξ α . Secondly we note that any element V of Ξ α must satisfy d(ι V µ) = η ∧ (ι V µ), where η is a 1-form defined by the relation dα = α ∧ η. We can choose α t to be generated by a family of diffeomorphisms, so thatα = L U α for U ∈ X(M ). Then, writing ν = α ∧ σ = ι V µ, V ∈ Ξ α we require and since ι U α is arbitrary we find dα ∧ σ + d(α ∧ σ) = 0, or Any element of Ξ α must then be tangent to F α and satisfy (27). This is not a complete characterisation, there are vector fields satisfying these two conditions which are not elements of Ξ α . This is demonstrated by example in section 4.6. We speculate that vector fields of the form (25) fully characterise Ξ α . where now α ∈ Ω 1 I (M ) and the functional derivatives are cosets in X(M )/Ξ α . The bracket must not depend on the choice of representative vector field for the functional derivative. Consider a vector field A on M such that ι A α = 0 and d(ι A µ) = η ∧ ι A µ. From the previous section we know all elements of Ξ α have these properties. Then Now ι A µ = α ∧ σ for some 1-form σ. Then we have it follows that the bracket {F, G} I does not depend on the choice of representative vector field for the functional derivatives and becomes a Poisson bracket on Ω 1 I (M ). Finally, we note that if F is the restriction to Ω 1 I (M ) of a functional on g * , then its functional derivative is still an element of g, all elements of which are representative vector fields of a coset in X(M )/Ξ α , and the bracket (28) reproduces the Lie-Poisson bracket of the original ideal fluid formulation. In particular, we can recover Euler's equations by choosing the appropriate Hamiltonian.
4.5. The Godbillon-Vey Invariant. For a codimension-1 foliation F on a closed manifold M , the Godbillon-Vey class [12,13] is an element GV ∈ H 3 (M ; R), if M is a closed 3-manifold H 3 (M ; R) = R and GV ∈ R is a diffeomorphism invariant of the foliation. Let β be a defining 1-form for F, then the integrability condition β ∧ dβ = 0 implies there is a 1-form η such that The 3-form η ∧ dη is closed and GV is defined as β is only defined up to multiplication by a non-zero function, and η is only defined up to addition of a multiple of β, but under these transformations η ∧ dη changes by an exact 3-form, so GV is well-defined. By construction GV is a diffeomorphism invariant of F. Where we use the fact that dη = α ∧ γ. Using dα = α ∧ η, we find dα = α ∧ η + α ∧η, so that Now we consider the 2-form χ = 2(η ∧γ −dγ), there is a freedom in χ arising from freedom in η and γ. We may make the transformations (36) η → η + f α, γ → γ + f η − df + gα, for functions f, g. Under these transformations one finds (37) χ → χ − 2(gdα + d(gα)), which does not affect the value of dGV /dt as per (24). Now observe that α ∧ χ = 0 and dχ = η ∧ χ. Writing χ = ι T µ, we find T is tangent to F α and satisfies (27). Using (29) we find (38) {F, GV } I = 0 for any functional F on Ω 1 I (M ) so that GV is a restricted Casimir of threedimensional ideal fluids. I am extremely grateful to PJ Morrison for a hugely enlightening discussion. I would also like to acknowledge many useful conversations with JH Hannay.