Ideal hydrodynamics on Lie groups

Physica D 221 (2006) 84–91 www.elsevier.com/locate/physd

Ideal hydrodynamics on Lie groups Mikhail V. Deryabin Mads Clausen Instituttet, Syddansk Universitet, Sønderborg, Denmark Received 18 January 2006; received in revised form 18 July 2006; accepted 18 July 2006 Available online 17 August 2006 Communicated by B. Sandstede

Abstract In this paper we consider the reduction to a Lie group of geodesic flows of left- or right-invariant metrics: for a fixed value of momentum we describe the reduced vector field on the Lie group, as well as its (right or left-invariant) symmetry fields. The reduced vector field has an important hydrodynamic interpretation: it is a stationary flow of an ideal fluid with a constant pressure. We use the explicit expressions for the reduced vector field and its symmetry fields to define “secondary hydrodynamics”, i.e., we study the reduction for the Euler equations of an ideal fluid, that describe the geodesics of a right-invariant metric on a Lie group SDiff(M) of the volume-preserving diffeomorphisms of a Riemannian manifold M, to the group SDiff(M). For a “typical” coadjoint orbit we find all symmetry fields of a reduced flow, and, as a corollary, we get a simple proof for nonexistence of new invariants of coadjoint orbits, which are the integrals of local densities over the flow domain. c 2006 Elsevier B.V. All rights reserved.

Keywords: Ideal hydrodynamics; Lie groups; Invariants of coadjoint orbits

1. Introduction In the 30’s, E.T. Whittaker suggested the following reduction procedure for the Euler top: by fixing values of the Noether integrals, the Hamiltonian vector field on the cotangent bundle T ∗ S O(3) can be uniquely projected onto the group S O(3) [1]. Thus we get a dynamical system (i.e., a vector field) on the group S O(3) itself. The Whittaker reduction is valid for any Hamiltonian system on a cotangent bundle T ∗ G to a Lie group G, provided the Hamiltonian is invariant under the left (or right) shifts on the group G. An important example of such a Hamiltonian system is a geodesic flow of a left-(right-)invariant metric on a Lie group. The Whittaker reduction to a Lie group can be regarded as a part of a general well-known construction, known as the Marsden–Weinstein reduction of Hamiltonian systems with symmetries [2]: for the above case, if we reduce the Hamiltonian system to the Lie group G, and then factorize the reduced vector field by the orbits of its symmetry fields, then, by the Marsden–Weinstein theorem, we get the same Hamiltonian system on a coadjoint orbit on the dual algebra g∗ , as if we

E-mail address: [email protected]. c 2006 Elsevier B.V. All rights reserved. 0167-2789/$ - see front matter doi:10.1016/j.physd.2006.07.019

first reduced the system to the dual algebra g∗ , and then to the coadjoint orbit (see also [3], Appendix 5). In contrast to the Marsden–Weinstein reduction, not much attention has been paid to the Whittaker reduction alone. However, it is itself worth studying. It turns out that a vector field, reduced to a Lie group G has a remarkable hydrodynamic interpretation: it is a velocity field for a stationary flow of an ideal fluid, that flows on the group G (viewed as a Riemannian manifold), and is incompressible with respect to some left- (or right-)invariant measure on G, see [4,5] and [6] for details. The reduction to a Lie group is also useful for a series of applications, which include stability theory, noncommutative integration of Hamiltonian systems, control theory and discretization, see, e.g., [6–9]. As we show below, the reduced vector field defines a stationary flow with constant pressure, and this may play an important role in studying the differential geometry of diffeomorphism groups, cf. [10–13]. Notice, however, that most of these results have been essentially finite-dimensional. In [6], Appendix 1, it was suggested to study “secondary hydrodynamics” — the Whittaker reduction procedure for flows an ideal fluid. The dynamics of an ideal incompressible fluid is defined by a rightinvariant Hamiltonian on the Lie group of volume-preserving diffeomorphisms of a Riemannian manifold, and thus the above

85

M.V. Deryabin / Physica D 221 (2006) 84–91

reduction to the group is possible. The “vector field”, reduced to the group of the volume-preserving diffeomorphisms has its own “secondary” hydrodynamic interpretation, being a “stationary flow” field on the diffeomorphism group. In this article, we first review the Whittaker reduction and its hydrodynamic essence, and provide an explicit expression for the reduction of a geodesic flow of a left- or right-invariant metric onto a Lie group. For any Lie group (including those that are not Banach manifolds) we find both the reduced vector field and its “symmetry fields”, i.e., left- or right-invariant fields on the group that commute with our reduced vector field. These fields have also a hydrodynamical meaning: these are the vortex vector fields for our stationary flow (i.e., they annihilate the vorticity 2-form), cf. [6]. Then, we use these results to define the “secondary hydrodynamics”, i.e., we study the reduction for the Euler equations of an ideal fluid, that describe the geodesics of a right-invariant metric on a Lie group SDiff(M) of the volumepreserving diffeomorphisms of a Riemannian manifold M, to the group SDiff(M). To get a reduced vector field, we fix a coadjoint orbit. In Ref. [14], certain special coadjoint orbits were identified with point-vortex dynamics. This identification can also be thought of as a reduction to the group for these special coadjoint orbits (the particle velocity is explicitly defined for the point vortices system). Notice that in this case, the “reduced” system (i.e., the system of point vortices in 2D) is a finite-dimensional system, while the Whittaker reduction on the Lie group SDiff(M) in the general case is an infinitedimensional system. We will pay most attention to the symmetry fields of the reduced flow. For a “typical” coadjoint orbit we find all symmetry fields of a reduced flow, which are the variational derivatives of the invariants of coadjoint orbits, found in [15]. The term “typical” means that a certain vector field does not admit any symmetries. Coadjoint orbits with no extra symmetries were also considered in [16] in relation to the topology of steady even-dimensional flows. Now, as an invariant of a coadjoint orbit generates a symmetry field for the reduced flow, we conclude readily that there are no new invariants of coadjoint orbits for ideal fluid flows, linearly independent of the classical ones on these “typical” orbits (these invariants should be functionals on the dual algebra, such that the variational derivative of such a functional is a smooth vector field from the Lie algebra SVect(M) of divergence-free vector fields on M). If we can take for granted the fact that the “typical” cases are everywhere dense in the space of all 1-forms on M in some appropriate topology, then we readily conclude that there are no invariants of coadjoint orbits, which are integrals of local densities over the flow domain, which are linearly independent of the classical ones. Thus, there are no new invariants of coadjoint orbits for ideal fluid flows in the above class of functionals. The results on the absence of new integral invariants complement the theorem on local invariants, i.e., on the local description of isovorticed fields existing at a generic point in ideal hydrodynamics [17]. This theorem states that at a generic point, in the even-dimensional case, there is

only one local invariant, which is the vorticity function, while in the odd-dimensional case, there are no local invariants at all. Some of the results on the reduction of geodesic flows to Lie groups were obtained in [8]. 2. Reduction of a geodesic flow to a Lie group First we recall some basic facts on the Euler equations (see, e.g., [17]). Let G be an arbitrary Lie group, g be its Lie algebra, and g∗ be the corresponding dual algebra. Here, the group G can be infinite-dimensional, and not necessarily a Banach manifold, but we assume that the exponential map exp : g → G exists (it exists for the group of volumepreserving diffeomorphisms SDiff(M)). Any vector g˙ ∈ Tg G and any covector m ∈ Tg∗ G can be translated to the group unity by left or right shifts. As a result we obtain the vectors ωc , ωs ∈ g and the momenta m c , m s ∈ g∗ : ωc = L g−1 ∗ g, ˙

ωs = Rg−1 ∗ g, ˙

m c = L ∗g m,

m s = Rg∗ m.

In the sequel the following relation plays the central role: m c = Adg∗ m s ,

(2.1)

Adg∗ : g∗ → g∗ being the group coadjoint operator. If we fix the “momentum in space” m s , then relation (2.1) defines a coadjoint orbit. The Casimir functions are the functions of m c , that are invariants of coadjoint orbits. For example, for the Euler top, the Casimir function is the length of the kinetic momentum. Let A : g → g∗ be a positive definite symmetric operator (inertia operator) defining a scalar product on the Lie algebra. This operator defines a left- or right-invariant inertia operator A G (and thus a left- or right-invariant metric) on the group G. For example, in the left-invariant case, A G = L ∗−1 · A · L g−1 ∗ . g If the metric is left-invariant, then the geodesics of this metric are described by the Euler equations m˙c = ad ∗A−1 m m c . c

Here : → g∗ is the coadjoint representation of ξ ∈ g. The Euler equations follow from the fact that “the momentum in space” m s is constant, whereas “the momentum in the body” m c is obtained from m s by (2.1), see [17]. Given a solution of the Euler equations ωc = A−1 m c , the trajectory on the group is determined by the relation L g−1 ∗ g˙ = ωc . adξ∗

g∗

Remark. Strictly speaking, in the infinite dimensional case the operator A is invertible only on a regular part of the dual algebra g∗ . In our case this means, that some natural restriction on values of m s (or m c ) have to be imposed (see [17]). In the case of a right-invariant metric m c is constant, the Euler equations read m˙s = −ad ∗A−1 m m s , and the trajectory s on the group is determined by the equation Rg−1 ∗ g˙ = ωs . The result of the reduction onto the group is a vector field v(g) ∈ T G such that the trajectory on the group is defined by the equation g˙ = v(g). The field v(g) will be referred to as reduced.

86

M.V. Deryabin / Physica D 221 (2006) 84–91

Proposition 2.1 (The Whittaker Reduction). For the case of the left-invariant or the right-invariant metric, the vector field v(g) has the form v(g) = L g∗ A−1 Adg∗ m s

(2.2)

Proof of Theorem 2.3. It is sufficient to show that

and, respectively v(g) = Rg∗ A−1 Adg∗−1 m c .

(2.3)

Notice that in Proposition 2.1, to find the reduced vector field we do not need the Hamiltonian equations on T ∗ G and the explicit expression for the Noether integrals. We only need the Lie group structure and the inertial operator. This is important for generalizations to the infinite-dimensional case. Unlike for the Marsden–Weinstein reduction, we do not have to assume nondegeneracy conditions on the momenta m c or m s . Proof. We consider only the case of the left-invariant metric; for the right-invariant case the proof is similar. Relation (2.1) determines the function m c = m c (m s , g) on the group G depending on m s as a parameter. From the equality ωc = A−1 m c and L g−1 ∗ g˙ = ωc follows that for any g ∈ G, L g−1 ∗ g˙ = A−1 m c (m s , g) 

The reduced vector fields (2.2) and (2.3) are in general neither left- nor right-invariant themselves. An important exception is when the inertia operator defines a Killing metric on the Lie algebra. However, the reduced covector fields are always right- or left-invariant. Let us assume that the metric is left-invariant. The case of a right-invariant metric is treated in a similar way. Proposition 2.2. If the metric is left-invariant, then the reduced covector field m(g) = A G v(g) is right-invariant. Proof. Indeed, ∗−1 −1 ∗ ∗−1 m(g) = L ∗−1 g AL g −1 ∗ v(g) = L g A A L g R g m s

= Rg∗−1 m s .  Let w(g) ∈ T G be a right-invariant vector field on the group G, which is defined by a vector ξ ∈ g: w(g) = Rg∗ ξ . We fix a momentum m s . Theorem 2.3. For the momentum m s fixed, the vector field w(g) on G is a symmetry field of the reduced system v(g) if and only if the vector ξ satisfies the condition adξ∗ m s = 0.

v((exp τ ξ )g) = L (exp τ ξ )∗ v(g) if and only if the condition of the theorem is fulfilled. Indeed,

Here m s , respectively m c , is constant.

which implies (2.2).

family of the left shifts on the group G: g → (exp τ ξ )g, see, for example, [18], as we have assumed that the exponential map exists. Notice also that, in view of Proposition 2.2, under the assumption of Theorem 2.3, the Lie derivative L w(g) m(g) = 0.

(2.4)

In the finite-dimensional case this means that the flows of the vector fields v(g), w(g) on the group commute. In the infinitedimensional case one should be more accurate: the equation g˙ = v(g) is a partial integral-differential equation, rather than an ordinary differential equation, hence, strictly speaking, it is not clear if it has a solution. On the other hand, equation g˙ = Rg∗ ξ always has a solution, which is a one-parametric

∗ v((exp τ ξ )g) = L (exp τ ξ )g∗ A−1 Ad(exp τ ξ )g m s ∗ = L exp τ ξ ∗ L g∗ A−1 Adg∗ (Adexp τ ξ m s ).

The last term equals L (exp τ ξ )∗ v(g) for any g ∈ G if and only if ∗ Adexp τ ξ ms = ms

for all values of the parameter τ . Differentiating the last relation by τ we arrive at the statement of the theorem.  Vectors ξ ∈ g that satisfy condition (2.4) are called the isotropy vectors. For any functional F : g∗ → R one can define its variational derivative δ F/δm: d F(m + sw) , m, w ∈ g∗ (w, δ F/δm) = ds s=0 see, e.g., [17]. In the finite-dimensional situation, a variational derivative (or a differential) of a function on the dual algebra g∗ is always a vector on the Lie algebra g. In the infinitedimensional case one has to be more careful, as not any linear operator on the dual algebra g∗ can be represented by a smooth vector field from the Lie algebra g. We consider an important example in Section 6. However, for such functions (or Lie groups), when this is the case, Theorem 2.3 describes all the invariants of coadjoint orbits, due to the following observation. Consider a subspace F of functions on the dual algebra g∗ , such that the variational derivatives of these functions belong to the Lie algebra g. Proposition 2.4. A function F(m) ∈ F is constant on a coadjoint orbit Adg∗ m s only if its differential generates (by the right shifts) a symmetry field to the reduced flow with the momentum m s . Proof. Let F be constant on the coadjoint orbit Adg∗ m s : F(Adg∗ m s ) = const for any g ∈ G. Then,   δ F(m s ) ∗ , ada m s = 0 = (a, adδ∗F(m s )/δm m s ) δm s) is for any a ∈ g, thus adδ∗F(m s )/δm m s = 0 and Rg∗ δ F(m δm a symmetry field to the reduced flow with the momentum ms . 

This proposition is an important tool in finding the invariants of coadjoint orbits. Suppose that the differentials of the known invariants constitute all the symmetry fields. Then this automatically leads to nonexistence of additional invariants of

M.V. Deryabin / Physica D 221 (2006) 84–91

87

coadjoint orbits among the functionals from F. In the finitedimensional case, finding invariants of coadjoint orbits reduces to an analysis of some algebraic equations. In the infinitedimensional case, it reduces to studying a system of ordinary differential equations.

Recall now that the reduced covector field is right-invariant (Proposition 2.2). Thus, the condition adη∗ m s = 0 is equivalent to L η(g) m(g) = 0, where m(g) is the right-invariant 1-form (being equal to m s at g = id), and η(g) = R∗g η is the rightinvariant symmetry field. By the homotopy formula,

3. Stationary flows and asymptotic directions

0 = L η(g) m(g) = i η(g) dm(g) + d(η(g), m(g)) = i η(g) dm(g),

We now formulate some results on the hydrodynamic character of the reduced vector fields from the previous section. Consider first the Euler equations for an ideal incompressible fluid, that flows on a Riemannian manifold M:

as (η(g), m(g)) = (η, m s ) = const for all g (both vector and covector fields are right-invariant). We now define a vortex vector field, as an annihilator of the vorticity 2-form. Then the condition i η(g) dm(g) = 0 is exactly the definition of a vortex field. Thus, we have proved.

∂v + ∇v v = −∇ p, div v = 0, ∂t where ∇v v is the covariant derivative of the fluid velocity vector v by itself with respect to the Riemannian connection and p is a pressure function. Consider a geodesic vector field u on the manifold M. Locally it always exists, but it may not be defined globally on M — take a two-sphere as a simple example. Then u is a stationary flow of the ideal fluid with a constant pressure. Indeed, as u is a geodesic vector field, its derivative along itself is zero: ∇u u = 0. Remark. The converse is of course not true: there are stationary flows that are not geodesics of the Riemannian metric. Stationary flows with constant pressure play an important role in studying the differential geometry of diffeomorphism groups, see [10–12]: they define asymptotic directions on the subgroup of the volume-preserving diffeomorphisms of the group of all diffeomorphisms. The stationary flows with constant pressure form a background for hydrodynamics of Euler equations on Lie groups. Consider a Hamiltonian system on a finite-dimensional Lie group G, with a left-invariant Hamiltonian, which is quadratic in the momenta. This Hamiltonian defines a leftinvariant metric on the Lie group G. As we reduce this system to the group, the reduced vector field is globally defined on G, and is a geodesic vector field of the Riemannian metric, defined by the left-invariant Hamiltonian, and it defines a stationary flow of an ideal fluid on G. Thus, the reduced vector field (2.2) (and (2.3)) is a stationary flow on the Lie group G with left- (right-)invariant metric. An immediate corollary of Proposition 2.2 is Proposition 3.1. There is an isomorphism between the stationary flows with constant pressure, defined by a left-invariant metric on a finite-dimensional Lie group G, and the space of rightinvariant covector fields on this group. Remark. This result is a generalization of [13], where it was shown that every left-invariant vector field on a compact Lie group equipped with a bi-invariant metric is asymptotic: if a Hamiltonian defines the bi-invariant metric on the Lie algebra, then the reduced vector field (2.2) is itself left-invariant. Moreover, its flows (which are right shifts on the Lie group G) are isometries of this metric (see, e.g., [19]).

Proposition 3.2. Any symmetry field to the reduced vector field is a vortex vector field. Consider now ideal hydrodynamics on a Riemannian manifold M. The Euler equations of an ideal fluid are the geodesics of a right-invariant metric on a Lie group SDiff(M) of volume-preserving diffeomorphisms of the manifold M. Thus, we can apply the above procedure, and reduce the geodesic flow to the group SDiff(M). The reduced “vector field” can be thought of as a stationary flow of an “ideal fluid”, which flows on the Lie group SDiff(M). This justifies the term “secondary hydrodynamics” in the infinite-dimensional case (cf. [6]). This reduction to the group is described in the next section. 4. Reduction of the Euler equations for ideal incompressible fluid Let M be a Riemannian manifold of dimension n. Consider a Lie group SDiff(M) of diffeomorphisms of this manifold, which preserve volume µ on M and, if the boundary ∂ M is nonempty, transform the boundary to itself. The corresponding Lie algebra g = SVect(M) is the space of zero divergence vector fields on M that are tangent to the boundary ∂ M. The dual algebra g∗ is the quotient Ω 1 /dΩ 0 of differential 1-forms on M modulo exact 1-forms. The action of the coadjoint operator on g∗ coincides with the standard action of diffeomorphisms on cosets: Adg∗ [u] = g ∗ [u], [u] ∈ Ω 1 /dΩ 0 , and the operator of the coadjoint representation is the Lie derivative: adξ∗ [u] = L ξ [u] [17]. The energy quadratic form Z 1 E= (v, v)µ, (4.1) 2 M defines the right-invariant metric on the group SDiff(M). Here (v, v) is the scalar product determined by the Riemannian metric on M and µ is the volume form. The inertia operator A : g → g∗ is specified by the condition Z hAv, wi = (v, w)µ M

v, w ∈ g being zero divergence vector fields on M that are tangent to the boundary.

88

M.V. Deryabin / Physica D 221 (2006) 84–91

According to the principle of stationary action, motions of the ideal fluid on the manifold M are geodesics of the rightinvariant metric (4.1). The latter are described by the Euler equations on the dual algebra g∗ , ∂[u] = −ad ∗A−1 [u] [u] = −L v [u], ∂t

(4.2)

Let w(g) be a left-invariant field on the group SDiff(M), which is defined by a vector ξ ∈ SVect(M): w(g) = L g∗ ξ . Theorem 4.1. The phase flow of the vector field w(g) on the group SDiff(M) is a symmetry of the reduced field v(g) if and only if the vector ξ satisfies the condition

where the “momentum” [u] ∈ Ω 1 /∂Ω 0 , Av = [u]. For a particular choice of a 1-form u ∈ [u] we get the classical equation

i ξ du 0 = d f,

∂u = −L v u + d f, ∂t

Proof. This theorem is a corollary of Theorem 2.3 and the homotopy formula L v ω = i v dω + di v ω. Indeed, the condition adξ∗ [u 0 ] = 0 means that L ξ [u 0 ] = [L ξ u 0 ] = 0. Hence,  L ξ u 0 = dF. On the other hand, L ξ u 0 = i ξ du 0 + di ξ u 0 .

see [17]. Now we fix the “body momentum” u c = u 0 and apply our general results on the reduction of a geodesic flow on a group. According to Proposition 2.1, the reduced field v(g) has the form (2.3), which can be interpreted in the following way. At each point g we act on the coset [u 0 ] by diffeomorphisms g −1 , then we apply the inverse operator A−1 . As a result, we obtain a zero divergence vector field on M. The vector v(g) represents a vector field on M which is obtained from the zero divergence field A−1 Adg∗−1 [u 0 ] by applying a differential of the right shift, i.e., by “particle relabeling”. Let a diffeomorphism g ∈ SDiff(M) map initial particles’ positions x0 to x(t, x0 ). As g is invertible, one can express x0 = x0 (x, t). The “space momentum” 1-form u s = u s (x, t) is expressed through the “body momentum” u c (x0 ) as u s (x, t) = Adg∗−1 u c (x0 ) =

∂ x0i u ci (x0 (x, t))dx j . ∂x j

Then, vs = A−1 u s . Suppose for simplicity, that the metric on M is Euclidean. Let function f be the solution to the equation 1 f = δu s , such that on the boundary ∂ M of the domain M the following relation holds: ∗(u s − d f )|∂ M = 0. Here δ = ∗d∗, where ∗ is the Hodge operator. This solution exists, and one can write it down explicitly, which, for example, can be done if one finds the Green’s function. The explicit computation of the reduced field thus relies on the possibility of finding the Green’s function. Now, vs = u s − grad f. Indeed, div vs = div u s − div (grad f ) = 0 and v||∂ M, as ∗(u s − d f ) = i v µ (µ is the Euclidean volume form). The equation on the group can be written for an inverse mapping g −1 as   d −1 ∂ x0 ∂ x0 ∂ x0 (g ) = −L g−1 ∗ Rg−1 ∗ g; ˙ =− vs x 0 , . dt ∂t ∂x ∂x

(4.3)

where d f is the differential of some function f on M.

Remark. It is interesting to note that since the isotropy vectors form a subalgebra, their distribution is integrable. Indeed, at any point the commutator (that is minus Poisson bracket of the vector fields) of any two vectors belongs to the same distribution. The structure of the symmetry fields in even and odd dimensions is the following. Let dim M = 2n. For ideal fluid flows on an even-dimensional manifold there always exists the vorticity function λ = du n /µ, where [u] ∈ g∗ [17]. Proposition 4.2. Let ξ ∈ SVect(M) satisfy the condition of Theorem 4.1. Then ξ commutes with the vector field η, defined by i η µ = dλ ∧ (du 0 )(n−1) . Proof. As the vector field ξ satisfies the condition of Theorem 4.1, 0 = L ξ (du 0 )n = L ξ λµ = (L ξ λ)µ + λL ξ µ. In the above relation, L ξ µ = 0, as ξ ∈ SVect(M) and thus is divergence-free. Thus, L ξ λ = i ξ dλ = 0. Now, take i [ξ,η] µ = L ξ i η µ − i η L ξ µ = L ξ i η µ = di ξ (dλ ∧ (du 0 )(n−1) ) = 0, as i ξ dλ = 0 by the above, and i ξ du 0 = d f by Theorem 4.1. 

There are many such vector fields ξ (at least if the boundary of M is empty): for any function f (λ), a field ξ , defined by i ξ µ = d f (λ) ∧ (du 0 )n−1 is divergence-free and commutes with η. Notice that all of them are linearly dependent. Let now the dimension of the manifold M be odd: dim M = 2n + 1. Then, the vorticity lines are integral curves of the vorticity field η, which is set by the following condition [17]: i η µ = (du)n . Proposition 4.3. Let ξ ∈ SVect(M) satisfy the condition of Theorem 4.1. Then ξ commutes with the vorticity field η : i η µ = (du 0 )n .

89

M.V. Deryabin / Physica D 221 (2006) 84–91

If the manifold M has no boundary, then the vorticity field η itself is a symmetry field (as in this case it belongs to the Lie algebra SVect(M)).

as the boundary of the manifold M is empty. Here Z w ∧ iη µ ([w], η) = M

Proof. We use following identity:

is the pairing of the dual algebra element [w] and the vector η ∈ SVect(M). 

i [ξ,η] µ = L ξ i η µ − i η L ξ µ = 0 : indeed, L ξ µ = 0, as ξ is divergence-free, and L ξ (du 0 )n = 0 by the condition of Theorem 4.1.  Thus, the question of symmetry fields for the reduced vector field v(g), being defined on the Lie group SDiff(M), is reduced to the question of existence of symmetry fields for a usual vector field on M. Let the boundary of M be empty. Then Propositions 4.2 and 4.3 describe all the symmetry fields in the “typical” situation: in the even-dimensional case, all such fields are defined by i ξ µ = d f (λ)∧(du 0 )n−1 , while in the odddimensional case there is essentially one such field, defined by i ξ µ = (du 0 )n . The term “typical” means that the above vector fields do not admit extra symmetries. 5. Symmetry fields and invariants of coadjoint orbits For the Lie group SDiff(M), we get from Propositions 4.2 and 4.3, that in the “typical” case there are no other symmetries than described above. We will show below that those symmetry fields are the differentials of the Casimir functions, found in [15]. Thus, by Proposition 2.4, from the absence of new symmetry fields we readily conclude the nonexistence of new invariants of coadjoint orbits from the class of functionals on the dual algebra, such that their variational derivatives are smooth vector fields from the Lie algebra SVect(M). Consider again the Lie group SDiff(M) of volumepreserving diffeomorphisms of a manifold M. First, assume that M has no boundary. Then the integrals Z Z I ([u]) = u ∧ (du)n , I f ([u]) = f (λ)µ, M

M

λ = (du)n /µ,

(5.1)

are Casimir functions for the Euler equations in the odd- and even-dimensional case respectively [15,17]. As above, here [u] is the coset of a 1-form u, µ is an invariant volume form, and the function f : R → R is arbitrary. Proposition 5.1. Let dim(M) = 2n + 1. Then the variational derivative of the Casimir function I is a vorticity vector (n + 1)η, where i η µ = (du)n . Proof. Since both du and dw are 2-forms, the wedge product du ∧ dw = dw ∧ du is commutative and the derivative d (du + sdw)n = ndw ∧ (du)n−1 . ds s=0 Thus Z d (u + sw) ∧ (du + sdw)n ds M s=0 Z n = w ∧ (du) + u ∧ ndw ∧ (du)n−1 M Z = (n + 1) w ∧ (du)n = ([w], (n + 1)η), M

Proposition 5.2. Let dim(M) = 2n. Then the variational derivative of the Casimir function I f equals ξ f ∈ SVect(M), which is a divergence-free vector field on M, linearly dependent with the vorticity vector ξ , defined by the condition i ξ du = dλ. The proof is similar to that of Proposition 5.1. We now will need the following assumption: we will here assume that there exist a Green’s function, i.e., the inertia operator A−1 can be written as   Z n −1 −1 ˜ G(x, y)δu(y)d y , A [u](x) = A u(x) + d M

where A˜ −1 is the standard “lifting of indices”, and δ = ∗d∗. With this assumption, from Proposition 2.4, together with Propositions 5.1 and 5.2 we readily get the following result. Theorem 5.3. Let the flow domain M have no boundary, and let F be an invariant of coadjoint orbits, which is an integral over flow domain M of a smooth function of the fluid velocity v and a finite number of its derivatives. Then, on a “typical” coadjoint orbit, the variational derivative δ F/δ[u] equals the variational derivative of the Casimir functions (5.1), up to a constant multiplier in the odd-dimensional case. Remark. This is a stronger fact than linear dependence of vector fields on M. Proof. In Section 2 we defined a subspace F, which in our case are functions on the dual algebra with the Lie group SDiff(M), such that the variational derivatives of these functions belong to SVect(M). To prove the theorem, we only have to show that the functions on the dual algebra to SVect(M), that are integral invariants of local densities, i.e., the integrals over M of smooth functions of v and a finite number of its derivatives, belong to F. Indeed, if this is the case, then by Propositions 5.1 and 5.2, we get that for a “typical” coadjoint orbit, there are no other symmetries than the above ones. By Proposition 2.4, these are differentials of the Casimirs (5.1). We show now that the variational derivative of any integral invariant of local densities is a smooth divergence-free vector field on M. Let the flow domain M have no boundary. Consider an integral Z Z F(v)dn x = F(v([u]))dn x, M

M

where v([u]) = A−1 [u] — it should be defined as a functional on cosets [u]. To calculate its variational derivative, one should replace u by u +w, and differentiate with respect to  at  = 0. One can see that the result is a sum of the following terms: Z ([i]) F˜i j w j dn x, M

90

M.V. Deryabin / Physica D 221 (2006) 84–91

where by w ([i]) we have denoted partial derivatives with the multiindex i: w ([i]) =

∂ [i] w ∂ x1i1 . . . ∂ xnin

,

i = (i 1 , . . . , i n ), [i] = i 1 + · · · + i n .

The functions F˜i j contain both the derivatives of F, and the derivatives of the Green’s functions. Every such term can be integrated by parts (the boundary of M is empty), reducing each summand to Z Z w j F˜ j dn x = w ∧ αj. M

M

We note that integration by parts and studying variational derivatives is a standard tool in finding first integrals and determining stability in 2-dimensional ideal hydrodynamics, see, e.g., [17]. Thus the whole expression can be written as: Z w∧α M

for some (n − 1)-form α. The form α is closed, as the original functional (and thus its variational derivative) is well-defined on cosets. Thus, this variational derivative is a vector from SVect(M), and the theorem is proved.  If we can take for granted the fact that the “typical” case, when our vortex fields do not admit extra symmetries, is everywhere dense in the space of all 1-forms on M in some appropriate topology, then from Theorem 5.3 follows readily that there are no invariants of coadjoint orbits, which are the integrals of local densities, that are linearly independent of the integrals (5.1). Remark. The similar assumption, formulated for continuous first integrals may not be true. Consider for example a perturbed smooth 2-dimensional Hamiltonian system. One can always construct a continuous first integral of such systems, by setting it to constants between the Kolmogorov tori, see, e.g., [20]. It may be possible to imitate this type of construction (for lowdimensional hydrodynamics) using, for example, the Casimir functions of the type Z u ∧ du, C

is transversal to the boundary. In the even-dimensional case the Casimir function Z f (λ)µ M

is well-defined, but its differential, being a divergence-free vector field on M, does not necessarily belong to the Lie algebra SVect(M), as it may be transversal to the boundary ∂ M, as above. However one can still define a “flow” of the vorticity field, which is a one-to-one volume-preserving mapping of the manifold M, defined almost for all points of M. This will be referred to as a generalized flow. This observation is crucial for constructing symmetry fields for the geodesic flow reduced to the group SDiff(M): not necessarily being a diffeomorphism (even a homeomorphism) this “flow” still can serve as a symmetry of the reduced system. If this is the case, then Theorem 4.1 still should be valid, i.e., the equality L η du 0 = 0 should hold. The absence of such generalized symmetries in the “typical” case proves Theorem 5.3 for integrals of local densities in the case when M has a boundary. For simplicity we consider the case when M ∈ R2n+1 is a bounded subset of a Euclidean space. Both the volume form µ and a 2-form du can be smoothly extended from M to some open neighbourhood, which defines the extension of the divergence-free vector field η to this neighbourhood. Consider a subset N ⊂ M, such that all the trajectories of the vector field η, that start in N at t = 0 remain in M at t > 0, and there is τ > 0, such that all these trajectories do not belong to M at t = −τ (but clearly belonging to the above open neighbourhood of M). Proposition 6.1. The measure (defined by the volume form µ) of the set N is zero. Thus almost all phase trajectories that enter M, escape from M after some finite time interval. Propositions similar to Proposition 6.1 are known: it is a slightly modified version of Poincar´e recurrence theorem, and had been proved for various particular cases (see, e.g., an interesting application of this type of results in the problem of one-way oscillating motions in a Kolmogorov model [21]).

where C is a subset of the flow domain M, which is an invariant set of the vorticity vector field for the instantaneous velocity.

Proof. Consider the mapping gητ that consists of the shifts along the phase trajectories of the field η by τ . Then for all n ∈ N,

6. Generalized flows

gηnτ N ⊂ M.

Consider now the case when the boundary ∂ M is nonempty. In the odd-dimensional case, the functional I ([u]) itself is not well-defined: it depends on a particular choice of a 1-form u ∈ [u] (cf. [17]). Indeed, the integral Z dh ∧ (du)n M

is not necessarily zero if the boundary of M is nonempty. The vorticity vector field η on M is defined anyway (as above, by the condition i η µ = (du)n ), however it may not be tangent to the boundary ∂ M. In the typical situation the vorticity vector field

Suppose that the measure of the set N is positive. Since the set M is bounded, there exist k, m ∈ N, m > k, such that the intersection gηkτ N ∩ gηmτ N 6= ∅. Thus N ∩ gη−τ N 6= ∅. But by the conditions of the proposition, N ⊂ M and gη−τ N ∩ M = ∅.  The mapping of the manifold M can be constructed in the following way. Let a phase curve of the vector field η enter M at a point γ1 and leave at a point γ2 , γ1 , γ2 ∈ ∂ M. In the interior of M a point moves along the phase curve until

M.V. Deryabin / Physica D 221 (2006) 84–91

it reaches the boundary at γ2 . Then the point jumps to γ1 and continues moving from it. Thus for any trajectory that intersects the boundary of M twice, we identify the intersection points. Note that as we glue these points together we may not end up with a manifold. Proposition 6.2. The mapping described above preserves the measure µ on M. The mapping that we have defined can be thought of as a generalized flow, see [17,22,23]. A particle that reaches the boundary “splits” into continuum of particles that move along the boundary independently and “gather” at the opposite point of the boundary. 7. Conclusion In this paper we consider reduction of a geodesic flow of a left- or right-invariant metric onto a Lie group. Given a Lie group G and a metric (or, in the general case, a Hamiltonian for Euler equations on the corresponding dual algebra), a coadjoint orbit (a momentum) being fixed, we write down the reduced vector field and describe its “symmetry fields”. These are leftor right-invariant fields on the group that commute with our reduced vector field. The symmetry fields are generated by the isotropy vectors of the coadjoint orbit. This reduction has a remarkable hydrodynamic interpretation: the reduced vector field is a stationary flow with constant pressure of an ideal fluid on the Riemannian manifold G, and the symmetry fields are the vortex vector fields. The above construction admits a straight-forward infinitedimensional generalization. Consider the Euler equations on a Lie group of volume-preserving diffeomorphisms SDiff(M) of a Riemannian manifold M, and perform the above reduction. As an invariant of a coadjoint orbit generates a symmetry field for the reduced flow, and in a “typical” case one can find all such symmetry fields, we get readily a simple proof for nonexistence of new invariants of coadjoint orbits for ideal fluid flows in the class of functionals on the dual algebra, such that the variational derivative of such a functional is a smooth vector field from the Lie algebra SVect(M), if we can take for granted the fact that the “typical” case is everywhere dense in the space of all 1forms on M in some appropriate topology. Acknowledgements The author wishes to thank Yu.N. Fedorov and A.V. Bolsinov for constructive criticism and stimulating discussions, and the reviewers for useful suggestions.

91

References [1] E.T. Whittaker, A Treatise on Analytical Dynamics, 4th ed., Cambridge Univ. Press, Cambridge, 1960. [2] J.E. Marsden, A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974) 120–121. [3] V.I. Arnold, Mathematical Methods in Classical Mechanics, SpringerVerlag, New York, 1989, p. 508. [4] V.V. Kozlov, Hydrodynamics of Hamiltonian systems, Vestnik Moskov. Univ., Ser. I. Mat. Mekh. (6) (1983) 10–22 (Russian). [5] V.V. Kozlov, The vortex theory of the top, Vestnik Moskov. Univ., Ser. I. Mat. Mekh. (4) (1990) 56–62 (Russian). [6] V.V. Kozlov, Dynamical Systems X. General Vortex Theory, SpringerVerlag, Berlin, 2003, p. 184. [7] V.V. Kozlov, Dynamics of variable systems and Lie groups, J. Appl. Math. Mech. 68 (2004) 803–808. [8] M.V. Deryabin, Yu.N. Fedorov, On reductions on groups of geodesic flows with (left-) right-invariant metrics and their fields of symmetry, Dokl. Math. 68 (1) (2003) 75–78. Interperiodica Translation. [9] Yu.N. Fedorov, Integrable flows and B¨acklund transformations on extended Stiefel varieties with application to the Euler top on the Lie group S O(3), J. Nonlinear Math. Phys. 12 (suppl. 2) (2005) 77–94. [10] D. Bao, T. Ratiu, On the geometrical origin and the solutions of a degenerate Monge–Amp`ere equation, Proc. Sympos. Pure Math. 54 (1993) 55–68. [11] B. Khesin, G. Misiołek, Asymptotic directions, Monge–Amp`ere equations and the geometry of diffeomorphism groups, J. Math. Fluid Mech. 7 (2005) 365–375. [12] G. Misiołek, Stability of flows of ideal fluids and the geometry of the group of diffeomorphisms, Indiana Univ. Math. J. 42 (1993) 215–235. [13] B. Palmer, The BaoRatiu equations on surfaces, Proc. R. Soc. London Ser. A 449 (1937) (1995) 623–627. [14] J. Marsden, A. Weinstein, Coadjoint orbits, vortices, and clebsch variables for incompressible fluids, Physica D 7 (1983) 305–323. [15] B.A. Khesin, Yu.Y. Chekanov, Invariants of the Euler equations for ideal or barotropic hydrodynamics and superconductivity in d dimensions, Physica D 40 (1989) 119–131. [16] V.L. Ginzburg, B.A. Khesin, Topology of steady fluid flows, in: H.K. Moffat et al. (Eds.), Topological Aspects of the Dynamics of Fluids and Plasmas, Kluwer Acad. Publ., Dordrecht, 1992, pp. 265–272. [17] V.I. Arnold, B.A. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag, New York, 1998, p. 374. [18] F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, New York, Berlin, 1983, p. 272. [19] B.A. Dubrovin, S.P. Novikov, A.T. Fomenko, Modern Geometry, vol. 2, Springer-Verlag, New York, 1992, p. 468. [20] V.I. Arnold, V.V. Kozlov, A.I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, 1988, 291 pages. [21] L.D. Pustyl’nikov, On the measure of one-way oscillating motions for the Kolmogorov model and its generalization in the n-body problem, Russian Math. Surveys 5 (1998) 1102–1103. [22] Y. Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Math. Soc. 2 (2) (1989) 225–255. [23] A. Shnirelman, Generalized flows, their approximation and applications, Geom. Funct. Anal. 4 (5) (1994) 586–620.