Lagrangian Reduction on Homogeneous Spaces with Advected Parameters

We study the Euler-Lagrange equations for a parameter dependent $G$-invariant Lagrangian on a homogeneous $G$-space. We consider the pullback of the parameter dependent Lagrangian to the Lie group $G$, emphasizing the special invariance properties of the associated Euler-Poincar\'e equations with advected parameters.


Introduction
The Euler-Poincaré (EP) equations arise via reduction of the variational principle for a right G-invariant Lagrangian L : T G → R. With a restricted class of variations, the extremals of the integral of the reduced Lagrangian ℓ : g → R correspond to extremals of the original variational problem for L [9]. The EP equations are written for the right logarithmic derivative ξ =ġg −1 = δ r g of the curve g in G as Here δℓ/δξ denotes the functional derivative of ℓ, which depends on the choice of a space g * in duality with g, that is δℓ δξ , η = d dt t=0 ℓ(ξ + tη) for all η ∈ g. The case of a parameter dependent G-invariant Lagrangian L : T G × V * → R is studied in [2]. The parameter space V * is a linear representation space of the Lie group G and the associated EP equations include an advection equation for the parameter. These EP equations with advected parameters are applied to continuum theories in [6]. To integrate complex fluids in this setting, the case of an affine G-action on the parameter space V * is treated in [4]. The more general case when the parameter space is a smooth manifold M acted on by G is considered in [5], applied there to nematic particles. The reduced equations, called EP equations for symmetry breaking, written for the reduced Lagrangian ℓ : g × M → R, involve the cotangent momentum map J : T * M → g * : d dt δℓ δξ + ad * ξ δℓ δξ = J δℓ δm .
In this paper we generalize the Lagrangian reduction with avdected parameters from the Lie group setting to the homogeneous space setting. We use the approach from [10] that features the special invariance properties of the reduced equations written for the pullback Lagrangian to the Lie group G. Starting with a right G-invariant Lagrangian L : T (G/H) → R, the reduced Lagrangian ℓ : g → R coming from its pullback to T G must be invariant under the adjoint action of the Lie subgroup H and under the addition action of its Lie algebra h. As a consequence, the EP equations (1.1) are now invariant under the following action of the group C ∞ (I, H) on C ∞ (I, g): (1. 2) The geodesic equations for invariant Riemannian metrics on Lie groups (Euler equations) correspond to reduced Lagrangians ℓ that are quadratic; a famous example is the ideal fluid flow as geodesic equations on the group of volume preserving diffeomorphisms [1] (more geodesic equations on diffeomorphism groups can be found for instance in [11]). The extension of Euler equations from Lie groups to homogeneous spaces is done in [7].
The plan of the paper is the following. In section 2 we review a kind of logarithmic derivative for homogeneous spaces. Then we consider parameter dependent G-invariant Lagrangians on G/H. We treat the case of a linear action on the parameter space in section 3. We devote section 4 to the EP equations for symmetry breaking, obtained for general actions on arbitrary parameter spaces. We obtain the affine EP equations as a special case. The examples are mainly on infinite dimensional homogeneous spaces, such as Diff(S 1 )/S 1 , Diff(M)/ Diff vol (M), Diff(M)/ Diff iso (M), and C ∞ (M, G)/G.

Logarithmic derivative and EP equations
The Euler-Lagrange (EL) equations associated to right invariant Lagrangians on Lie groups lead to the Euler-Poincaré (EP) equations involving reduced Lagrangians and written for right logarithmic derivatives of curves in the Lie group. In this section we recall how this works in homogeneous spaces of right cosets, following [10].
Given a smooth curveḡ : I → G/H, we compare the right logarithmic derivatives of two smooth lifts g 1 , g 2 : I → G ofḡ. Because there exists a smooth curve h : I → H such that g 2 = hg 1 , the logarithmic derivative δ r g 2 = Ad h δ r g 1 + δ r h is obtained from δ r g 1 via the left action (1.2). Hence the right logarithmic derivative for homogeneous spaces is multivalued: where g is any lift ofḡ. The tangent bundle T G of a Lie group G carries a natural group multiplication. Given a subgroup H of G, its tangent bundle T H is a subgroup of T G and there is a canonical diffeomorphism between T G/T H and T (G/H). The following are equivalent data: right G-invariant LagrangianL on T (G/H), left T H-invariant and right G-invariant Lagrangian L on T G, as well as h-invariant and Ad(H)-invariant reduced Lagrangian ℓ on g. This proposition admits a generalization to Lagrangians that are not necessarily right G-invariant. First we note the following property of a left T H-invariant Lagrangian L : T G → R: if the curve g in G is a solution of the corresponding EL equation, then the curve hg is also a solution of the EL equation, for any smooth curve h in H. Indeed, each variation g ε of g with fixed endpoints corresponds to a variation (ε, t) → h(t)g ε (t) of hg with fixed endpoints. Proof. A variationḡ ε ofḡ in G/H with fixed endpoints can be lifted to a variation g ε of g in G, but it doesn't necessarily have fixed endpoints. We can achieve g ε (0) = g(0), but only g ε (1) ∈ Hg(1). Still, there is is another lift of the variationḡ ε that is a variation of g with fixed endpoints: (ε, t) → g(1)g εt (1) −1 g ε (t). This proves the first assertion. The second assertion is clear, since a variation of g in G with fixed endpoints always descends to a variation in G/H with fixed endpoints.
A special case is the geodesic equation for a right G-invariant Riemannian metric on G/H, i.e. Euler equation on homogeneous spaces [7]. The next examples are both of this type. The geodesic equation is the multidimensional Hunter-Saxton equation and the Hunter-Saxton equation (2.3) is invariant under it.
Example 2.5 Let (M, g) be a Riemannian manifold and Diff iso (M) its group of isometries. The homogeneous space of right cosets Diff(M)/ Diff iso (M) admits a right invariant metric induced by the following degenerate inner product on X(M) [8]: The geodesic equation is the EP equation on the homogeneous space Diff(M)/ Diff iso (M) for the reduced Lagrangian ℓ(u) = 1 2 M |L u g| 2 µ on X(M) that has the required Diff vol (M)and X vol (M)-invariance properties. The associated EP equation is invariant under the action (2.4) of C ∞ (I, Diff iso (M)) on C ∞ (I, X(M)).

Advected EP equations: linear action
Now we look at parameter dependent Lagrangians. First we treat the Lie group case, following [2], then we pass to homogeneous spaces.

The case of Lie groups
We consider a linear right action ρ of the Lie group G on the vector space V and its dual left action ρ * on V * . The corresponding Lie algebra actions on V and V * are , then a(t) = ρ * g(t) (a 0 ) is the unique solution of the differential equation with time-dependent coefficientsȧ = ξa, a(0) = a 0 . The diamond operation ⋄ : V × V * → g * is given by v ⋄ a, ξ := ξa, v , for all ξ ∈ g.
A right G-invariant Lagrangian L : T G × V * → R (including the linear action on the second argument) has a reduced Lagrangian ℓ : g × V * → R so that ℓ(v g g −1 , ρ * g (a)) = L(v g , a), v g ∈ T g G. For fixed a 0 ∈ V * the Lagrangian L a 0 : T G → R is right invariant only under the isotropy subgroup G a 0 of a 0 ∈ V * . Theorem 3.1 [2] The EL equations for L a 0 on G given by Hamilton's variational principle δ t 2 t 1 L a 0 (g(t),ġ(t))dt = 0 can be expressed as EP equations on g × V * with advected parameter: for the reduced Lagrangian ℓ.
The main examples are the heavy top and the ideal compressible fluid. For the heavy top G = SO(3) and the parameter Γ ∈ V * = R 3 is the unit vector in the gravity direction in body representation. For the ideal compressible fluid G = Diff(M), with M a Riemannian manifold, and the parameter ρ ∈ V * = C ∞ (M) * is the fluid density in spatial representation. The reduced Lagrangians are ℓ(Ω, Γ) = 1 2 I Ω · Ω − Γ · λ for Ω ∈ so(3) = R 3 in the first example, and ℓ(u, ρ) = 1 2 M ρ|u| 2 µ for u ∈ X(M) in the second one.

The case of homogeneous spaces
This proves the next proposition.
Proposition 3.2 The reduced Lagrangian ℓ : g × V * → R associated to the pullback of a parameter dependent right G-invariant Lagrangian on G/H is H-and h-invariant: The functional derivatives of the reduced Lagrangian ℓ : g × V * → R that has the invariance property (3.2) are equivariant: Proof. We compute for ζ ∈ g: Similarly we get that This action has the property h · (δ r g, ρ * g a) = (δ r (hg), ρ * hg a) for any curve g ∈ C ∞ (I, G). Proof. We need the G-equivariance of the diamond operation: Using also the following identities for α ∈ g * : This ensures the C ∞ (I, H)-invariance of the EP equation with advected parameters.

EP equations for symmetry breaking
One can replace the linear action of G on a parameter vector space V * with an arbitrary action of G on a parameter manifold M. This generalization of the EP equations with advected parameters, called EP equations for symmetry breaking, are presented in [5]. In this section we adapt these results to the case of homogeneous spaces.

The case of Lie groups
Let a Lie group G act on the smooth manifold M from the left, and let ξ M ∈ X(M) denote the infinitesimal generator of ξ ∈ g. Given a curve g in G starting at the identity, the curve m(t) = g(t) · m 0 is the unique solution of the differential equation with time- where ξ(t) = δ r g(t).
The cotangent momentum map J : T * M → g * , defined by (J(α m ), ξ) = (α m , ξ M (m)) for all α m ∈ T * m M, is G-equivariant for the cotangent and coadjoint actions: The functional derivative δℓ δξ takes values in g * , while δℓ δm is a g-dependent section of T * M.
where j and λ are constants, and k the external force field. Proof. The equivariance property of the functional derivative δℓ δξ from lemma 3.3 holds, but also the following equivariance property of δℓ δm :

The case of homogeneous spaces
Indeed, for any curve c in M with c(0) = h · m and c ′ (0) = w, we get: Using also the equivariance of the cotangent momentum map, we compute This shows the required invariance of the equation (4.1).

Affine EP equations
Now we consider the special case of an affine left G-action on a parameter space V * : where c : G → V * is a group 1-cocycle for the action ρ * , i.e.
Remark 4.5 Let dc ⊤ : V → g * be defined by dc ⊤ (v), ξ = dc(ξ), v . Then the cotangent momentum map for the affine action (4.4) of G on V * can be written as because for all ξ ∈ g, The following result for a right G-invariant Lagrangian L : T G × V * → R with reduced Lagrangian ℓ : g × V * → R is a special case of theorem 4.1 and a generalization of theorem 3.1.
Theorem 4.6 [4] The EL equations for L a 0 : T G → R can be expressed as affine EP equations for the reduced Lagrangian ℓ: Spin systems. In [4] is shown that the affine EP equations for the action of the gauge group C ∞ (M, G) on the space V * = Ω 1 (M, g) of principal connections on the trivial bundle M × G θ g (γ) = Ad g γ − dgg −1 can be used in the description of spin systems. The 1-cocycle is in this case the right logarithmic derivative so dc(ξ) = −dξ for all ξ ∈ C ∞ (M, g). The infinitesimal action involves the covariant derivative d γ ξ = dξ + [γ, ξ], namely ξ V * (γ) = −d γ ξ.
We fix a volume form on M, so C ∞ (M, g * ) is a dual space to the gauge Lie algebra C ∞ (M, g), while X(M, g * ) is a dual space to the parameter space Ω 1 (M, g). The cotangent momentum map (4.6) becomes J(γ, α) = − ad * γ α − div α = div γ α, since dc ⊤ (α) = div α and that the diamond map is α ⋄ γ = − ad * γ α. We can write now the affine EP equation on C ∞ (M, g) × Ω 1 (M, g) as For M = R 3 and G = SO(3) one gets a macroscopic description of spin glasses [4]. For M a real interval and G = SE(3), the Euclidean group of rigid motions, one gets an affine EP formulation of Kirchhoff's theory of rods (the Cosserat rod) in the case of potential forces [3].
Homogeneous spaces. Let L : T G × V * → R be now the pull-back of a G-invariant LagrangianL : T (G/H)×V * → R. Because L is left T H-invariant and right G-invariant, its reduced Lagrangian ℓ : g × V * → R is both H-and h-invariant: ℓ(Ad h ξ + η, θ h (a)) = ℓ(ξ, a), h ∈ H, η ∈ h. (4.10) Proposition 4.7 Given a reduced Lagrangian ℓ : g × V * → R that has the invariance property (4.10), the affine EP equation Proof. It is a consequence of proposition 4.3, but it can be shown also directly, as in the proof of proposition 3.4, using the expression of the failure of dc to be G-equivariant: dc(Ad g ξ) − ρ * g dc(ξ) = c(g) Ad g ξ.