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Invariant variational problems on homogeneous spaces
Journal of Geometry and Physics 90 (2015) 104–110
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Invariant variational problems on homogeneous spaces Cornelia Vizman Department of Mathematics, West University of Timişoara, Bd. V. Pârvan 4, 300223-Timişoara, Romania
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Article history: Received 27 August 2014 Accepted 27 October 2014 Available online 4 November 2014
abstract Covariant Euler–Poincaré equations on homogeneous spaces are studied, including the special case of strands on homogeneous spaces. Space–time strands on homogeneous spaces are treated also dynamically, using affine Euler–Poincaré reduction. © 2014 Elsevier B.V. All rights reserved.
MSC: 70S05 70H30 58E30 58E40 Keywords: Homogeneous space Lagrangian Euler–Poincaré equation Covariant reduction Dynamic reduction
1. Introduction Classical Euler–Poincaré equations on the Lie algebra g of the Lie group G arise via reduction of the variational principle for a G-invariant Lagrangian L : TG → 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 [1]. The variational reduction program is extended in [2] to the setting of a principal G-bundle P over M, see also [3]. The Ginvariant Lagrangian density is defined on J 1 P, the first jet bundle of sections of P. Its quotient space J 1 P /G is identified with the bundle of principal connections on P. The reduced equations that are obtained can be seen as generalized EP equations for field theory. Zero curvature conditions on the reduced solutions are imposed for the existence of the original solutions. The case when the manifold M is sliced leads to the reduced equations for space–time strands, also called sliced covariant EP equations [4,5]. Special cases are the G-strands [6]. When M is sliced, the covariant EP reduction can be reformulated as a classical dynamic reduction. Two approaches have been proposed in [4] and in [3]. We shall follow the approach of [3] based on affine EP reduction. See also [5] for more explanation of the link between G-strands, covariant EP, and affine EP reductions. Reduction theory for the principal fiber bundle P by a subgroup H ⊂ G of symmetries is performed in [7]. The configuration bundle of the reduced problem is now J 1 P /H. A fixed connection on the principal bundle P → P /H is used for splitting the reduced equation in two equations, the first of them containing the Euler–Lagrange operator applied to sections of the fiber bundle P /H over M (maps from M to G/H in the trivial bundle case). In this paper we focus on reduced variational problems on homogeneous spaces G/H for a more restrictive class of Lagrangians. Because of these stronger invariance requirements the second equation in the splitting mentioned above
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.geomphys.2014.10.012 0393-0440/© 2014 Elsevier B.V. All rights reserved.
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vanishes identically. That is why our approach does not require to fix a connection. We emphasize the special invariance properties of the reduced equations that come up after lifting the Lagrangian to the Lie group G as in [8]. We pass from the classical case of curves to strands and then to more general maps, especially space–time strands, all of them taking values in homogeneous spaces. Reduced equations for G-invariant parameter dependent Lagrangians, called Euler–Poincaré equations for symmetry breaking, are introduced in [9]. Reduction can be performed for parameter dependent Lagrangians on homogeneous spaces too [10]. We use the same approach to adapt to homogeneous spaces a result from [3], where space–time strands on Lie groups are treated dynamically. To an invariant field theoretic Lagrangian density on the first jet bundle of a principal bundle one associates a gauge group invariant dynamic Lagrangian on the tangent bundle of the gauge group depending parametrically on a connection. Covariant and dynamic reduction lead to equivalent reduced equations. We treat only the trivial principal bundle case. The plan of the paper is the following. In Section 2 we recall the Euler–Poincaré equations on homogeneous spaces [8] and the parameter dependent version which leads to Euler–Poincaré equations for symmetry breaking [10]. In Section 3 we treat covariant Euler–Poincaré equations for maps into homogeneous spaces. In the last section we compare covariant reduction for space–time strands in homogeneous spaces with dynamic reduction, as done in [3] for Lie groups. 2. Euler–Poincaré equations on homogeneous spaces In this section we recall the Euler–Poincaré equations on homogeneous spaces [8] and the parameter dependent version which leads to Euler–Poincaré equations for symmetry breaking [10]. 2.1. Classical Euler–Poincaré equations The Euler–Lagrange (EL) equations for a right G-invariant Lagrangian L : TG → R, written for the reduced Lagrangian ℓ : g → R on the Lie algebra g of G, are the Euler–Poincaré (EP) equations [1] δℓ d δℓ + ad∗ξ = 0. (2.1) dt δξ δξ Here ξ is a curve in g, the right logarithmic derivative of a curve g in G, i.e. ξ = g˙ g −1 , and δℓ/δξ denotes the functional δℓ derivative: δξ , ζ = dtd t =0 ℓ(ξ + t ζ ) for all ζ ∈ g. Logarithmic derivative and homogeneous spaces. The right logarithmic derivative for curves in a homogeneous space G/H can be seen as a multivalued map by considering the logarithmic derivatives of all the lifted curves g in G of a given curve g¯ in G/H. All of them belong to the same orbit of the left action of C ∞ (I , H ) on C ∞ (I , g): h · ξ = Adh ξ + δ r h,
(2.2)
because δ r (hg ) = Adh δ r g + δ r h. Thus we can define [8]
δ¯ r : C ∞ (I , G/H ) → C ∞ (I , g)/C ∞ (I , H ),
δ¯ r g¯ = C ∞ (I , H ) · δ r g .
(2.3)
Similarly, a right logarithmic derivative for homogeneous space valued maps on a smooth manifold M can be given by
δ¯ r : C ∞ (M , G/H ) → Ω 1 (M , g)/C ∞ (M , H ), δ¯ r g¯ = C ∞ (M , H ) · δ r g , (2.4) r −1 where δ g = dgg is the right logarithmic derivative of g and the action of C ∞ (M , H ) on Ω 1 (M , g) is the gauge action h · σ = Adh σ + δ r h,
(2.5)
restriction of the action of the gauge group C ∞ (M , G) on the space of connections for the trivial principal G-bundle over M. The tangent bundle TG carries a natural group multiplication. Given a Lie subgroup H of G, its tangent bundle TH is a Lie subgroup of TG and there is a canonical diffeomorphism between TG/TH and T (G/H ). The following are equivalent data: a right G-invariant Lagrangian L¯ on T (G/H ), a left TH-invariant and right G-invariant Lagrangian L on TG, as well as an hinvariant and Ad(H )-invariant reduced Lagrangian ℓ on g. Proposition 2.1 ([8]). Given a reduced Lagrangian ℓ : g → R that satisfies ℓ(Adh ξ + η) = ℓ(ξ ) for all h ∈ H and η ∈ h, the EP equations d δℓ dt δξ
+ ad∗ξ
δℓ =0 δξ
(2.6)
are C ∞ (I , H )-invariant for the action (2.2). In other words (2.6) being an equation for C ∞ (I , H )-orbits in C ∞ (I , g), is an equation for the multivalued right logarithmic derivative for curves in the homogeneous space (2.3). A special EL equation is the geodesic equation for a right G-invariant Riemannian metric on G/H, i.e. Euler equation on homogeneous spaces [11]. Examples are the Hunter–Saxton equation and its multidimensional version, geodesic equations on Diff(S 1 )/S 1 and Diff(M )/Diffvol (M ).
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2.2. Euler–Poincaré equations for symmetry breaking In this section we treat the case of a parameter dependent Lagrangian. When the action on the parameter space is linear, one gets the EP equations with advected parameters [12]. Replacing the linear action with an affine action leads to the affine EP equations [13]. For an arbitrary action of G on a parameter manifold M, the EP equations for symmetry breaking are obtained [9]. These results were adapted to the case of homogeneous spaces in [10]. Let G act on the smooth manifold Q from the left, and let ξQ ∈ X(Q ) denote the infinitesimal generator of ξ ∈ g. The cotangent action is Hamiltonian with equivariant momentum map J : T ∗ Q → g∗ given by (J (αq ), ξ ) = (αq , ξQ (q)) for all αq ∈ Tq∗ Q . Any G-invariant Lagrangian L : TG × Q → R is determined by its reduced Lagrangian ℓ : g × Q → R, namely L(vg , q) = ℓ(vg g −1 , g · q). For fixed q0 ∈ Q , the Lagrangian Lq0 : TG → R is invariant only under the isotropy subgroup Gq0 . Theorem 2.2 ([9]). The EL equations for the Lagrangian Lq0 : TG → R become the EP equations for symmetry breaking for the reduced Lagrangian ℓ: d δℓ dt δξ
+
ad∗ξ
δℓ =J δξ
δℓ , δq
q˙ = ξQ (q).
(2.7)
is a g-dependent section of T ∗ Q . Here δℓ δq
Let L : TG × Q → R be the pull-back of a right G-invariant Lagrangian L¯ : T (G/H ) × Q → R, hence L is right G-invariant, but also left TH-invariant in the first argument. The associated reduced Lagrangian ℓ : g × Q → R is both H- and h-invariant:
ℓ(Adh ξ + η, h · q) = ℓ(ξ , q),
h ∈ H , η ∈ h.
(2.8)
Proposition 2.3 ([10]). Given a reduced Lagrangian ℓ : g × Q → R that has the invariance property (2.8), the EP equation for symmetry breaking (2.7) is invariant under the C ∞ (I , H )-action on C ∞ (I , g × Q ): h · (ξ , q) = (Adh ξ + δ r h, h · q).
(2.9)
It is indeed an equation on G/H because h · (δ r g , g · q0 ) = (δ r (hg ), (hg ) · q0 ) for all curves h in H and g in G. 2.3. Affine Euler–Poincaré reduction Let us consider the special case of an affine left G-action on the vector space V ∗ of parameters:
θg (a) = ρg∗ (a) + c (g ),
(2.10)
with c : G → V a group 1-cocycle for the linear G-action ρ on V . Let dc : g → V be the associated Lie algebra 1-cocycle and let dc ⊤ : V → g∗ be defined by ⟨dc ⊤ (v), ξ ⟩ = ⟨dc (ξ ), v⟩. If the diamond operation : V × V ∗ → g∗ is given by ∗
∗
⟨v a, ξ ⟩ := ⟨ξ a, v⟩,
∗
∗
for all ξ ∈ g,
then the cotangent momentum map for the affine action (2.10) of G on V ∗ has the expression J : T ∗ V ∗ = V ∗ × V → g∗ ,
J (a, v) = v a + dc ⊤ (v).
(2.11)
As a special case of Theorem 2.2 one gets the affine EP equations associated to a right G-invariant Lagrangian L : TG×V ∗ → R with reduced Lagrangian ℓ : g × V ∗ → R d δℓ dt δξ
+ ad∗ξ
δℓ δℓ = a + (dc )⊤ δξ δa
δℓ , δa
a˙ = ξ a + dc (ξ ),
(2.12)
introduced in [13] to describe complex fluids. Corollary 2.4 ([10]). Given a reduced Lagrangian ℓ : g × V ∗ → R that has the invariance property ℓ(Adh ξ +η, θh (a)) = ℓ(ξ , a) for all h ∈ H and η ∈ h, the reduced equation (2.12) is invariant under the C ∞ (I , H )-action on C ∞ (I , g × V ∗ ): h · (ξ , a) = (Adh ξ + δ r h, θh (a)),
(2.13)
hence it is an equation on the homogeneous space G/H. 3. Covariant Euler–Poincaré equations In this section we recall the passage from the G-invariant variational problem for curves in G to the analogous problem for G-strands (i.e. maps R2 → G) as well as more general G-valued maps. Then we pass to maps with values in a homogeneous space of G.
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G-strands. G-invariant Lagrangians defined on the Whitney sum of tangent bundles L : TG ⊕ TG → R lead to reduced equations for G-strands. These are written for the reduced Lagrangian ℓ : g × g → R as [6]
δℓ δℓ d δℓ + + ad∗η = 0, dt δξ δξ ds δη δη where ξ = ∂t gg −1 and η = ∂s gg −1 . d δℓ
+ ad∗ξ
(3.1)
Unlike the classical case, when direct integration of the Euler–Poincaré equations provides solutions of the variational problem, now the reduced variational problem is no longer free: a set of compatibility equations are needed for the reconstruction of the original problem, namely a zero curvature equation,
∂s ξ − ∂t η = −[ξ , η],
(3.2)
that has to be added in order to ensure the reconstruction of the original problem for G-valued maps. Strands on matrix groups lead often to integrable systems, e.g. the principal chiral model [14]. A preferred form for the reduced Lagrangian ℓ : g × g → R, both for strands on matrix groups and for strands on diffeomorphism groups, is ℓ(ξ , η) = 21 |ξ |2 − 21 |η|2 . Covariant EP reduction. We recall EP reduction for G-valued maps on a compact manifold M [3], the trivial bundle version of the results performed in [2] for arbitrary principal G-bundles P → M. The space J 1 (M , G) of 1-jets of maps in C ∞ (M , G) can be identified in this case with the vector bundle L(TM , TG), while J 1 (M , G)/G = L(TM , g) → M is the bundle of principal connections of the trivial bundle. A Lagrangian density is a bundle map L : J 1 (M , G) → Λm M, where m = dim M. By fixing a volume form µ on M, we identify the Lagrangian density with the map L : J 1 (M , G) → R. A right G-invariant Lagrangian density L is completely determined by the reduced Lagrangian ℓ : L(TM , g) → R. Then, for any map g ∈ C ∞ (M , G), the function L(j1 g ) on M coincides with the function ℓ(σ ) on M, where σ = δ r g ∈ Ω 1 (M , g). Remark 3.1. Special cases are curves and strands in G. For M = R the jet space is J 1 (R, G) = R × TG, so we have a time dependent Lagrangian on TG. For M = R2 the jet space is J 1 (R2 , G) = R2 × TG ⊕ TG. If the Lagrangian does not depend on the two real parameters t and s, we are in the G-strands setting mentioned above. δℓ Given σ ∈ Ω 1 (M , g) (section of T ∗ M ⊗ g), the functional derivative δσ is identified with a section of TM ⊗ g∗ , i.e. a g∗ -valued vector field. Using the pairing between tangent and cotangent vectors, together with the coadjoint action δℓ on M. Also a divergence operator can be defined for g∗ -valued vector ad∗ : g × g∗ → g∗ , one gets a g∗ -valued function ad∗σ δσ fields, the result being again in C ∞ (M , g∗ ). We consider now a variation gε ∈ C ∞ (M , G) of g and σε = δ r gε ∈ Ω 1 (M , g). Then, by the Maurer–Cartan equation for the logarithmic derivative of the variation of g, we find that the variation of σ satisfies the relation
δσ =
σε = dζ + [ζ , σ ], dε d
0
where ζ =
gε g −1 ∈ C ∞ (M , g), dε d
0
so we compute
δℓ ℓ(σ )µ = , δσ µ δσ M M M δℓ δℓ δℓ ∗ δℓ , dζ µ + , [ζ , σ ] µ = − div + adσ , ζ µ, = δσ δσ δσ δσ M M M
0 = δ
L(j1 g )µ = δ
thus proving the following specialization to trivial bundles of Theorem 3.1 from [2]. Proposition 3.2 ([3]). Let L : J 1 (M , G) → R be a right G-invariant Lagrangian density with reduced Lagrangian ℓ : L(TM , g) → R, and let µ be a fixed volume form on M. The following are equivalent: 1. The map g ∈ C ∞ (M , G) satisfies the covariant EL equations for L. 2. The variational principle δ M L(j1 g )µ = 0 holds for arbitrary variations. 3. The variational principle δ M ℓ(σ )µ = 0 holds for variations δσ = dζ + [ζ , σ ]. 4. The covariant EP equations hold: div
δℓ δℓ + ad∗σ = 0. δσ δσ
(3.3)
When the compact manifold M has a nonempty boundary, then one considers variations of g with fixed boundary points, hence ζ vanishes on the boundary. When the manifold M is not compact, then one works with compactly supported objects on M. Remark 3.3. The zero curvature condition on the solution σ ∈ Ω 1 (M , g) of (3.3), i.e. the Maurer–Cartan equation dσ + 21 [σ , σ ] = 0,
(3.4)
is necessary for the existence of g ∈ C ∞ (M , G) such that σ = δ r g. It is also sufficient, provided the manifold M is simply connected. For M = R2 this condition becomes the zero curvature relation (3.2).
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The case of homogeneous spaces. We consider a G-invariant Lagrangian density L¯ on the first jet bundle J 1 (M , G/H ) of maps M → G/H. By composition with the canonical push-forward map J 1 (M , G) → J 1 (M , G/H ) we get a G-invariant Lagrangian density L : J 1 (M , G) → R that enjoys a J 1 (M , H )-invariance too (recall that J 1 (M , H ) is a subgroup of J 1 (M , G)). The reduced Lagrangian ℓ : L(TM , g) → R inherits an Ad(H )- and a L(TM , h)-invariance from the J 1 (M , H )-invariance of L, i.e.
ℓ(Adh ϕ + ψ) = ℓ(ϕ),
ϕ ∈ L(Tx M , g), ψ ∈ L(Tx M , h), h ∈ C ∞ (M , H ).
(3.5)
The following result generalizes Proposition 2.1. Proposition 3.4. If the reduced Lagrangian ℓ : L(TM , g) → R satisfies the invariance properties (3.5), then the covariant EP equation (3.3) is invariant under the gauge action h · σ = Adh σ + δ r h of C ∞ (M , H ) on Ω 1 (M , g). Proof. For all h ∈ C ∞ (M , H ) and σ ∈ Ω 1 (M , g) we get
δℓ δℓ (h · σ ) = Ad∗h−1 (σ ) δσ δσ by the following computation for arbitrary ζ ∈ g:
δℓ (h · σ ), ζ δσ
d δℓ r ∗ = ℓ(Adh σ + δ h + t ζ ) = ℓ(σ + tAdh−1 ζ ) = Adh−1 (σ ), ζ . dt dt δσ d
0
0
Using also the identities ad∗Adh ξ Ad∗h−1 ν = Ad∗h−1 ad∗ξ ν,
div(Ad∗h−1 ν) = −ad∗δ r h Ad∗h−1 ν
that hold for all sections ν of TM ⊗ g∗ , we obtain that
δℓ δℓ div + ad∗σ δσ δσ
δℓ δℓ (h · σ ) = div Adh−1 + ad∗Adh σ +δr h Ad∗h−1 δσ δσ δℓ ∗ ∗ δℓ = Adh−1 div + adσ . δσ δσ
∗
This shows that the covariant EP equation is invariant under the gauge action of C ∞ (M , H ).
Proposition 3.4 ensures that the covariant EP equation (3.3), being an equation for C ∞ (M , H )-orbits in Ω 1 (M , g), is an equation for right logarithmic derivatives (2.4) for maps into homogeneous spaces. One can specialize Proposition 3.4 to G-strands. Corollary 3.5. If the reduced Lagrangian ℓ : g × g → R satisfies the invariance property ℓ(Adh ξ + ζ1 , Adh η + ζ2 ) = ℓ(ξ , η) for all h ∈ H and ζ1 , ζ2 ∈ h, then the reduced G-strand Eq. (3.1) is invariant under the action h · (ξ , η) = (Adh ξ + ∂t hh−1 , Adh η + ∂s hh−1 ) of C ∞ (R2 , H ) on C ∞ (R2 , g2 ). Example 3.6. EP equations for strands on the homogeneous spaces of right cosets Diff(S 1 )/S 1 and Diff(M )/Diffvol (M ) generalize the Hunter–Saxton equation and its multidimensional version [15]. With the reduced Lagrangian
ℓ : X( S 1 ) × X( S 1 ) → R ,
ℓ(ξ , η) =
1 2
S1
(ξ ′ )2 dx −
1 2
S1
(η′ )2 dx
we are led to the equations
∂t u′′ + 2u′ u′′ + uu′′′ = ∂s v ′′ + 2v ′ v ′′ + vv ′′′ with zero curvature relation ∂t v − ∂s u = [u, v]. Its multidimensional version is
∂t d(div u) + dLu (div u) + (div u)d(div u) = ∂t d(div v) + dLv (div v) + (div v)d(div v) an EP equation for strands on the homogeneous space Diff(M )/Diffvol (M ), for the reduced Lagrangian ℓ(u, v) 1 (div u)2 µ − 12 M (div v)2 µ, u, v ∈ X(M ). 2 M
=
4. Covariant versus dynamic reduction Time dependent maps R×M → G can also be written dynamically as curves in C ∞ (M , G). Performing covariant reduction for the G-invariant Lagrangian density L : J 1 (R × M , G) → R one gets the EP equations for space–time strands. Performing dynamic reduction for a gauge invariant dynamic Lagrangian L : R × TC ∞ (M , G)× Ω 1 (M , g) → R depending parametrically on a connection, one gets affine EP equations for spin systems. It is shown in [3] that, if L is appropriately built from L, then the resulting reduced equations are equivalent.
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Spin systems. In [13], it is shown that EP reduction for a gauge invariant Lagrangian L : TC ∞ (M , G) × Ω 1 (M , g) → R depending parametrically on a connection can be used in describing spin systems. The action of the gauge group C ∞ (M , G) on the parameter space Ω 1 (M , g) of principal connections on the trivial bundle M × G is
θg (γ ) = Adg γ − dgg −1 .
(4.1)
The 1-cocycle c : C (M , G) → Ω (M , g) is in this case minus the right logarithmic derivative c (g ) = −dgg , so dc (ξ ) = −dξ for all ξ ∈ C ∞ (M , g). The infinitesimal action involves the covariant derivative dγ ξ = dξ + [γ , ξ ], namely ξΩ 1 (M ,g) (γ ) = −dγ ξ . Since a volume form on M is fixed, 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 (2.11) becomes J (γ , α) = −ad∗γ α − div α = ∞
−1
1
−divγ α , since dc ⊤ (α) = div α and the diamond map is α γ = −ad∗γ α . Hence the affine EP equations for the reduced Lagrangian ℓ : C ∞ (M , g) × Ω 1 (M , g) → R are the equations for spin systems δℓ δℓ ∂ δℓ + ad∗ξ = −divγ , ∂ t δξ δξ δγ
γ˙ + dγ ξ = 0.
(4.2)
Space–time strands. We recall the reduced equations for space–time strands g : R × M → G as in [4,5]. These are also called sliced covariant EP equations because the sliced manifold R × M is considered instead of M in the covariant EP equations. Let L : J 1 (R × M , G) → R be a G-invariant Lagrangian density with reduced Lagrangian ℓ : L(T (R × M ), g) → R. Then both δℓ the g-valued 1-form σ = δ r g and the g∗ -valued vector field δσ can be decomposed according to the slicing:
δℓ d δℓ δℓ = + , δσ δξ dt δγ
σ = ∂t gg −1 dt + dgg −1 = ξ dt + γ ,
where ξ (t ) ∈ C ∞ (M , g), γ (t ) ∈ Ω 1 (M , g). Taking the differential respectively the divergence in the sliced manifold R × M, we obtain dσ =
dξ +
d dt
γ
dt + dγ ,
div
δℓ d δℓ δℓ = + div . δσ dt δξ δγ
The covariant EP equations (3.3) become the EP equations for space–time strands on G d δℓ dt δξ
+ ad∗ξ
δℓ δℓ δℓ = −div − ad∗γ , δξ δγ δγ
(4.3)
with zero-curvature identities
γ˙ − dξ = [ξ , γ ] and dγ = [γ , γ ].
(4.4)
When M is just a real interval, these are the reduced equation for G-strands. Covariant versus dynamic reduction. Given a field theoretic Lagrangian density L : J 1 (R × M , G) = R × TG ×G J 1 (M , G) → R defined on the first jet bundle of the trivial principal bundle over R × M, one assigns a new family of real valued dynamic Lagrangians on the tangent bundle of the gauge group depending parametrically on a connection: LL : R × TC ∞ (M , G) × Ω 1 (M , g) → R L (t , ξg , γ ) = L
L(t , ξg (x), dx g − g (x)γ (x))µ,
ξg ∈ Tg C ∞ (M , G).
M
If the Lagrangian density L is G-invariant, one assigns a reduced Lagrangian density ℓ : L(T (R × M ), g) = R × g × L(TM , g) → R that completely determines L. One can check that the dynamic Lagrangian LL is C ∞ (M , G)-invariant and its reduced Lagrangian is given by
ℓ : R × C (M , g) × Ω (M , g) → R, L
∞
1
ℓ (t , ξ , γ ) = L
ℓ(t , ξ , −γ )µ.
(4.5)
M
Proposition 4.1 ([3]). Covariant reduction for the G-invariant Lagrangian density L and dynamic reduction with affine advected parameters for the C ∞ (M , G)-invariant Lagrangian LL lead to the same reduced equations. Indeed, the reduced equations (4.3) together with (4.4) on one side and (4.2) on the other side coincide, up to the zero curvature condition for the connection γ . This inconvenience can be removed by restricting the parameter space Ω 1 (M , g) to the gauge-invariant subset of flat connection. Indeed, if γ is a flat connection, then θg (γ ) is also flat for the gauge action θ from (4.1). The case of homogeneous spaces. Let H be a Lie subgroup of G and let h be its Lie algebra. In this last paragraph we show that the above passage from the Lagrangian density L to the connection depending density LL can be performed also for G/H-valued maps.
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Lemma 4.2. If the reduced Lagrangian density ℓ : R × g × L(TM , g) → R is invariant under the adjoint action of H and under the addition action of L(TM , h), then the reduced dynamic Lagrangian ℓL : R × TC ∞ (M , G) × Ω 1 (M , g) → R is C ∞ (M , H )and C ∞ (M , h)-invariant. Proof. We use the relation Eq. (4.5) between ℓ and ℓL to check the C ∞ (M , H )-invariance:
ℓL (t , Adh ξ , h−1 · γ ) =
ℓ(t , Adh(x) ξ (x), (dhh−1 )(x) − Adh(x) γ (x))µ(x) ℓ(t , ξ , −γ )µ = ℓL (t , ξ , γ ). ℓ(t , Adh(x) ξ (x), −Adh(x) γ (x))µ(x) = = M
M
M
The C ∞ (M , h)-invariance of ℓL is shown similarly.
The analogue of Proposition 4.1 holds for homogeneous spaces too. Proposition 4.3. Covariant reduction for the G-invariant Lagrangian density L¯ : J 1 (R × M , G/H ) → R and dynamic reduction with affine advected parameters for the C ∞ (M , G)-invariant Lagrangian L¯ L : R × TC ∞ (M , G/H ) × Ω 1 (M , g) → R lead to the same reduced equations. Proof. If the G-invariant Lagrangian density L : J 1 (R × M , G) → R is the pullback of a density on J 1 (R × M , G/H ), then the reduced Lagrangian density ℓ : R × g × L(TM , g) → R is invariant under the adjoint action of H and under the addition action of both h and L(TM , h). By the previous lemma, the reduced dynamic Lagrangian ℓL : R × TC ∞ (M , G) × Ω 1 (M , g) → R is C ∞ (M , H )- and C ∞ (M , h)-invariant, so the time dependent C ∞ (M , G)-invariant Lagrangian LL on C ∞ (M , G), depending also on a connection in Ω 1 (M , g), is the pullback of a time dependent Lagrangian on C ∞ (M , G/H ) with the same parameter space. Example 4.4. This example is inspired by an application from [13]. We consider a bilinear map b : g × g → R that is H-invariant under the adjoint action and its kernel contains the Lie subalgebra h. We choose a Riemannian metric g and a volume form µ on the smooth manifold M. Now we define a reduced Lagrangian density
ℓ : L(T (R × M ), g) = R × g × L(TM , g) → R,
ℓ(t , ξ , γx ) =
1 2
| b(ξ , γx ) |2 ,
where the norm of b(ξ , γx ) ∈ Tx∗ M is computed with the Riemannian metric g. It is invariant under the adjoint action of H and under the addition action of both h and L(TM , h), so it corresponds to a Lagrangian density L¯ : J 1 (R × M , G/H ) → R. The associated reduced Lagrangian is
ℓL : R × C ∞ (M , g) × Ω 1 (M , g) → R,
ℓL (t , ξ , γ ) =
1 2
| b(ξ , γ ) |2 µ.
M
Acknowledgments The author is grateful to François Gay-Balmaz for extremely helpful suggestions. This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0921. References [1] J.E. Marsden, T. Ratiu, Introduction to Mechanics and Symmetry, second ed., Springer, 1999. [2] M. Castrillon Lopez, T.S. Ratiu, S. Shkoller, Reduction in principal fiber bundles: covariant Euler–Poincaré equations, Proc. Amer. Math. Soc. 128 (2000) 2155–2164. [3] F. Gay-Balmaz, T.S. Ratiu, A new Lagrangian dynamic reduction in field theory, Ann. Inst. Fourier 16 (2010) 1125–1160. [4] M. Castrillon Lopez, J.E. Marsden, Covariant and dynamical reduction for principal bundle field theories, Ann. Global Anal. Geom. 34 (2008) 263–285. [5] F. Gay-Balmaz, Clebsch variational principles in field theories and singular solutions of covariant EPDiff equations, Rep. Math. Phys. 71 (2013) 231–277. [6] D.D. Holm, R.I. Ivanov, G-strands and peakon collisions on Diff(R), SIGMA 9 (2013) 027. 14 pages. [7] M. Castrillon Lopez, T.S. Ratiu, Reduction in principal bundles: covariant Lagrange–Poincaré equations, Comm. Math. Phys. 236 (2003) 223–250. [8] F. Tiglay, C. Vizman, Generalized Euler–Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications, Lett. Math. Phys. 97 (2011) 45–60. [9] F. Gay-Balmaz, C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems, Physica D 239 (2010) 1929–1947. [10] C. Vizman, Lagrangian reduction on homogeneous spaces with advected parameters, 2014. arXiv:1408.3019 [math-ph]. [11] B. Khesin, G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math. 176 (2003) 116–144. [12] H. Cendra, D.D. Holm, J. Marsden, T.S. Ratiu, Lagrangian reduction, the Euler–Poincaré equations, and semidirect products, AMS Transl. 186 (1998) 1–25. [13] F. Gay-Balmaz, T.S. Ratiu, The geometric structure of complex fluids, Adv. Appl. Math. 42 (2009) 176–275. [14] D.D. Holm, R.I. Ivanov, J.R. Percival, G-strands, J. Nonlinear Sci. 22 (4) (2012) 517–551. [15] B. Khesin, J. Lenells, G. Misiołek, S. Preston, Geometry of diffeomorphism groups, complete integrability, and geometric statistics, Geom. Funct. Anal. 23 (2013) 334–366.
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Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp
Invariant variational problems on homogeneous spaces Cornelia Vizman Department of Mathematics, West University of Timişoara, Bd. V. Pârvan 4, 300223-Timişoara, Romania
article
info
Article history: Received 27 August 2014 Accepted 27 October 2014 Available online 4 November 2014
abstract Covariant Euler–Poincaré equations on homogeneous spaces are studied, including the special case of strands on homogeneous spaces. Space–time strands on homogeneous spaces are treated also dynamically, using affine Euler–Poincaré reduction. © 2014 Elsevier B.V. All rights reserved.
MSC: 70S05 70H30 58E30 58E40 Keywords: Homogeneous space Lagrangian Euler–Poincaré equation Covariant reduction Dynamic reduction
1. Introduction Classical Euler–Poincaré equations on the Lie algebra g of the Lie group G arise via reduction of the variational principle for a G-invariant Lagrangian L : TG → 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 [1]. The variational reduction program is extended in [2] to the setting of a principal G-bundle P over M, see also [3]. The Ginvariant Lagrangian density is defined on J 1 P, the first jet bundle of sections of P. Its quotient space J 1 P /G is identified with the bundle of principal connections on P. The reduced equations that are obtained can be seen as generalized EP equations for field theory. Zero curvature conditions on the reduced solutions are imposed for the existence of the original solutions. The case when the manifold M is sliced leads to the reduced equations for space–time strands, also called sliced covariant EP equations [4,5]. Special cases are the G-strands [6]. When M is sliced, the covariant EP reduction can be reformulated as a classical dynamic reduction. Two approaches have been proposed in [4] and in [3]. We shall follow the approach of [3] based on affine EP reduction. See also [5] for more explanation of the link between G-strands, covariant EP, and affine EP reductions. Reduction theory for the principal fiber bundle P by a subgroup H ⊂ G of symmetries is performed in [7]. The configuration bundle of the reduced problem is now J 1 P /H. A fixed connection on the principal bundle P → P /H is used for splitting the reduced equation in two equations, the first of them containing the Euler–Lagrange operator applied to sections of the fiber bundle P /H over M (maps from M to G/H in the trivial bundle case). In this paper we focus on reduced variational problems on homogeneous spaces G/H for a more restrictive class of Lagrangians. Because of these stronger invariance requirements the second equation in the splitting mentioned above
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.geomphys.2014.10.012 0393-0440/© 2014 Elsevier B.V. All rights reserved.
C. Vizman / Journal of Geometry and Physics 90 (2015) 104–110
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vanishes identically. That is why our approach does not require to fix a connection. We emphasize the special invariance properties of the reduced equations that come up after lifting the Lagrangian to the Lie group G as in [8]. We pass from the classical case of curves to strands and then to more general maps, especially space–time strands, all of them taking values in homogeneous spaces. Reduced equations for G-invariant parameter dependent Lagrangians, called Euler–Poincaré equations for symmetry breaking, are introduced in [9]. Reduction can be performed for parameter dependent Lagrangians on homogeneous spaces too [10]. We use the same approach to adapt to homogeneous spaces a result from [3], where space–time strands on Lie groups are treated dynamically. To an invariant field theoretic Lagrangian density on the first jet bundle of a principal bundle one associates a gauge group invariant dynamic Lagrangian on the tangent bundle of the gauge group depending parametrically on a connection. Covariant and dynamic reduction lead to equivalent reduced equations. We treat only the trivial principal bundle case. The plan of the paper is the following. In Section 2 we recall the Euler–Poincaré equations on homogeneous spaces [8] and the parameter dependent version which leads to Euler–Poincaré equations for symmetry breaking [10]. In Section 3 we treat covariant Euler–Poincaré equations for maps into homogeneous spaces. In the last section we compare covariant reduction for space–time strands in homogeneous spaces with dynamic reduction, as done in [3] for Lie groups. 2. Euler–Poincaré equations on homogeneous spaces In this section we recall the Euler–Poincaré equations on homogeneous spaces [8] and the parameter dependent version which leads to Euler–Poincaré equations for symmetry breaking [10]. 2.1. Classical Euler–Poincaré equations The Euler–Lagrange (EL) equations for a right G-invariant Lagrangian L : TG → R, written for the reduced Lagrangian ℓ : g → R on the Lie algebra g of G, are the Euler–Poincaré (EP) equations [1] δℓ d δℓ + ad∗ξ = 0. (2.1) dt δξ δξ Here ξ is a curve in g, the right logarithmic derivative of a curve g in G, i.e. ξ = g˙ g −1 , and δℓ/δξ denotes the functional δℓ derivative: δξ , ζ = dtd t =0 ℓ(ξ + t ζ ) for all ζ ∈ g. Logarithmic derivative and homogeneous spaces. The right logarithmic derivative for curves in a homogeneous space G/H can be seen as a multivalued map by considering the logarithmic derivatives of all the lifted curves g in G of a given curve g¯ in G/H. All of them belong to the same orbit of the left action of C ∞ (I , H ) on C ∞ (I , g): h · ξ = Adh ξ + δ r h,
(2.2)
because δ r (hg ) = Adh δ r g + δ r h. Thus we can define [8]
δ¯ r : C ∞ (I , G/H ) → C ∞ (I , g)/C ∞ (I , H ),
δ¯ r g¯ = C ∞ (I , H ) · δ r g .
(2.3)
Similarly, a right logarithmic derivative for homogeneous space valued maps on a smooth manifold M can be given by
δ¯ r : C ∞ (M , G/H ) → Ω 1 (M , g)/C ∞ (M , H ), δ¯ r g¯ = C ∞ (M , H ) · δ r g , (2.4) r −1 where δ g = dgg is the right logarithmic derivative of g and the action of C ∞ (M , H ) on Ω 1 (M , g) is the gauge action h · σ = Adh σ + δ r h,
(2.5)
restriction of the action of the gauge group C ∞ (M , G) on the space of connections for the trivial principal G-bundle over M. The tangent bundle TG carries a natural group multiplication. Given a Lie subgroup H of G, its tangent bundle TH is a Lie subgroup of TG and there is a canonical diffeomorphism between TG/TH and T (G/H ). The following are equivalent data: a right G-invariant Lagrangian L¯ on T (G/H ), a left TH-invariant and right G-invariant Lagrangian L on TG, as well as an hinvariant and Ad(H )-invariant reduced Lagrangian ℓ on g. Proposition 2.1 ([8]). Given a reduced Lagrangian ℓ : g → R that satisfies ℓ(Adh ξ + η) = ℓ(ξ ) for all h ∈ H and η ∈ h, the EP equations d δℓ dt δξ
+ ad∗ξ
δℓ =0 δξ
(2.6)
are C ∞ (I , H )-invariant for the action (2.2). In other words (2.6) being an equation for C ∞ (I , H )-orbits in C ∞ (I , g), is an equation for the multivalued right logarithmic derivative for curves in the homogeneous space (2.3). A special EL equation is the geodesic equation for a right G-invariant Riemannian metric on G/H, i.e. Euler equation on homogeneous spaces [11]. Examples are the Hunter–Saxton equation and its multidimensional version, geodesic equations on Diff(S 1 )/S 1 and Diff(M )/Diffvol (M ).
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2.2. Euler–Poincaré equations for symmetry breaking In this section we treat the case of a parameter dependent Lagrangian. When the action on the parameter space is linear, one gets the EP equations with advected parameters [12]. Replacing the linear action with an affine action leads to the affine EP equations [13]. For an arbitrary action of G on a parameter manifold M, the EP equations for symmetry breaking are obtained [9]. These results were adapted to the case of homogeneous spaces in [10]. Let G act on the smooth manifold Q from the left, and let ξQ ∈ X(Q ) denote the infinitesimal generator of ξ ∈ g. The cotangent action is Hamiltonian with equivariant momentum map J : T ∗ Q → g∗ given by (J (αq ), ξ ) = (αq , ξQ (q)) for all αq ∈ Tq∗ Q . Any G-invariant Lagrangian L : TG × Q → R is determined by its reduced Lagrangian ℓ : g × Q → R, namely L(vg , q) = ℓ(vg g −1 , g · q). For fixed q0 ∈ Q , the Lagrangian Lq0 : TG → R is invariant only under the isotropy subgroup Gq0 . Theorem 2.2 ([9]). The EL equations for the Lagrangian Lq0 : TG → R become the EP equations for symmetry breaking for the reduced Lagrangian ℓ: d δℓ dt δξ
+
ad∗ξ
δℓ =J δξ
δℓ , δq
q˙ = ξQ (q).
(2.7)
is a g-dependent section of T ∗ Q . Here δℓ δq
Let L : TG × Q → R be the pull-back of a right G-invariant Lagrangian L¯ : T (G/H ) × Q → R, hence L is right G-invariant, but also left TH-invariant in the first argument. The associated reduced Lagrangian ℓ : g × Q → R is both H- and h-invariant:
ℓ(Adh ξ + η, h · q) = ℓ(ξ , q),
h ∈ H , η ∈ h.
(2.8)
Proposition 2.3 ([10]). Given a reduced Lagrangian ℓ : g × Q → R that has the invariance property (2.8), the EP equation for symmetry breaking (2.7) is invariant under the C ∞ (I , H )-action on C ∞ (I , g × Q ): h · (ξ , q) = (Adh ξ + δ r h, h · q).
(2.9)
It is indeed an equation on G/H because h · (δ r g , g · q0 ) = (δ r (hg ), (hg ) · q0 ) for all curves h in H and g in G. 2.3. Affine Euler–Poincaré reduction Let us consider the special case of an affine left G-action on the vector space V ∗ of parameters:
θg (a) = ρg∗ (a) + c (g ),
(2.10)
with c : G → V a group 1-cocycle for the linear G-action ρ on V . Let dc : g → V be the associated Lie algebra 1-cocycle and let dc ⊤ : V → g∗ be defined by ⟨dc ⊤ (v), ξ ⟩ = ⟨dc (ξ ), v⟩. If the diamond operation : V × V ∗ → g∗ is given by ∗
∗
⟨v a, ξ ⟩ := ⟨ξ a, v⟩,
∗
∗
for all ξ ∈ g,
then the cotangent momentum map for the affine action (2.10) of G on V ∗ has the expression J : T ∗ V ∗ = V ∗ × V → g∗ ,
J (a, v) = v a + dc ⊤ (v).
(2.11)
As a special case of Theorem 2.2 one gets the affine EP equations associated to a right G-invariant Lagrangian L : TG×V ∗ → R with reduced Lagrangian ℓ : g × V ∗ → R d δℓ dt δξ
+ ad∗ξ
δℓ δℓ = a + (dc )⊤ δξ δa
δℓ , δa
a˙ = ξ a + dc (ξ ),
(2.12)
introduced in [13] to describe complex fluids. Corollary 2.4 ([10]). Given a reduced Lagrangian ℓ : g × V ∗ → R that has the invariance property ℓ(Adh ξ +η, θh (a)) = ℓ(ξ , a) for all h ∈ H and η ∈ h, the reduced equation (2.12) is invariant under the C ∞ (I , H )-action on C ∞ (I , g × V ∗ ): h · (ξ , a) = (Adh ξ + δ r h, θh (a)),
(2.13)
hence it is an equation on the homogeneous space G/H. 3. Covariant Euler–Poincaré equations In this section we recall the passage from the G-invariant variational problem for curves in G to the analogous problem for G-strands (i.e. maps R2 → G) as well as more general G-valued maps. Then we pass to maps with values in a homogeneous space of G.
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G-strands. G-invariant Lagrangians defined on the Whitney sum of tangent bundles L : TG ⊕ TG → R lead to reduced equations for G-strands. These are written for the reduced Lagrangian ℓ : g × g → R as [6]
δℓ δℓ d δℓ + + ad∗η = 0, dt δξ δξ ds δη δη where ξ = ∂t gg −1 and η = ∂s gg −1 . d δℓ
+ ad∗ξ
(3.1)
Unlike the classical case, when direct integration of the Euler–Poincaré equations provides solutions of the variational problem, now the reduced variational problem is no longer free: a set of compatibility equations are needed for the reconstruction of the original problem, namely a zero curvature equation,
∂s ξ − ∂t η = −[ξ , η],
(3.2)
that has to be added in order to ensure the reconstruction of the original problem for G-valued maps. Strands on matrix groups lead often to integrable systems, e.g. the principal chiral model [14]. A preferred form for the reduced Lagrangian ℓ : g × g → R, both for strands on matrix groups and for strands on diffeomorphism groups, is ℓ(ξ , η) = 21 |ξ |2 − 21 |η|2 . Covariant EP reduction. We recall EP reduction for G-valued maps on a compact manifold M [3], the trivial bundle version of the results performed in [2] for arbitrary principal G-bundles P → M. The space J 1 (M , G) of 1-jets of maps in C ∞ (M , G) can be identified in this case with the vector bundle L(TM , TG), while J 1 (M , G)/G = L(TM , g) → M is the bundle of principal connections of the trivial bundle. A Lagrangian density is a bundle map L : J 1 (M , G) → Λm M, where m = dim M. By fixing a volume form µ on M, we identify the Lagrangian density with the map L : J 1 (M , G) → R. A right G-invariant Lagrangian density L is completely determined by the reduced Lagrangian ℓ : L(TM , g) → R. Then, for any map g ∈ C ∞ (M , G), the function L(j1 g ) on M coincides with the function ℓ(σ ) on M, where σ = δ r g ∈ Ω 1 (M , g). Remark 3.1. Special cases are curves and strands in G. For M = R the jet space is J 1 (R, G) = R × TG, so we have a time dependent Lagrangian on TG. For M = R2 the jet space is J 1 (R2 , G) = R2 × TG ⊕ TG. If the Lagrangian does not depend on the two real parameters t and s, we are in the G-strands setting mentioned above. δℓ Given σ ∈ Ω 1 (M , g) (section of T ∗ M ⊗ g), the functional derivative δσ is identified with a section of TM ⊗ g∗ , i.e. a g∗ -valued vector field. Using the pairing between tangent and cotangent vectors, together with the coadjoint action δℓ on M. Also a divergence operator can be defined for g∗ -valued vector ad∗ : g × g∗ → g∗ , one gets a g∗ -valued function ad∗σ δσ fields, the result being again in C ∞ (M , g∗ ). We consider now a variation gε ∈ C ∞ (M , G) of g and σε = δ r gε ∈ Ω 1 (M , g). Then, by the Maurer–Cartan equation for the logarithmic derivative of the variation of g, we find that the variation of σ satisfies the relation
δσ =
σε = dζ + [ζ , σ ], dε d
0
where ζ =
gε g −1 ∈ C ∞ (M , g), dε d
0
so we compute
δℓ ℓ(σ )µ = , δσ µ δσ M M M δℓ δℓ δℓ ∗ δℓ , dζ µ + , [ζ , σ ] µ = − div + adσ , ζ µ, = δσ δσ δσ δσ M M M
0 = δ
L(j1 g )µ = δ
thus proving the following specialization to trivial bundles of Theorem 3.1 from [2]. Proposition 3.2 ([3]). Let L : J 1 (M , G) → R be a right G-invariant Lagrangian density with reduced Lagrangian ℓ : L(TM , g) → R, and let µ be a fixed volume form on M. The following are equivalent: 1. The map g ∈ C ∞ (M , G) satisfies the covariant EL equations for L. 2. The variational principle δ M L(j1 g )µ = 0 holds for arbitrary variations. 3. The variational principle δ M ℓ(σ )µ = 0 holds for variations δσ = dζ + [ζ , σ ]. 4. The covariant EP equations hold: div
δℓ δℓ + ad∗σ = 0. δσ δσ
(3.3)
When the compact manifold M has a nonempty boundary, then one considers variations of g with fixed boundary points, hence ζ vanishes on the boundary. When the manifold M is not compact, then one works with compactly supported objects on M. Remark 3.3. The zero curvature condition on the solution σ ∈ Ω 1 (M , g) of (3.3), i.e. the Maurer–Cartan equation dσ + 21 [σ , σ ] = 0,
(3.4)
is necessary for the existence of g ∈ C ∞ (M , G) such that σ = δ r g. It is also sufficient, provided the manifold M is simply connected. For M = R2 this condition becomes the zero curvature relation (3.2).
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The case of homogeneous spaces. We consider a G-invariant Lagrangian density L¯ on the first jet bundle J 1 (M , G/H ) of maps M → G/H. By composition with the canonical push-forward map J 1 (M , G) → J 1 (M , G/H ) we get a G-invariant Lagrangian density L : J 1 (M , G) → R that enjoys a J 1 (M , H )-invariance too (recall that J 1 (M , H ) is a subgroup of J 1 (M , G)). The reduced Lagrangian ℓ : L(TM , g) → R inherits an Ad(H )- and a L(TM , h)-invariance from the J 1 (M , H )-invariance of L, i.e.
ℓ(Adh ϕ + ψ) = ℓ(ϕ),
ϕ ∈ L(Tx M , g), ψ ∈ L(Tx M , h), h ∈ C ∞ (M , H ).
(3.5)
The following result generalizes Proposition 2.1. Proposition 3.4. If the reduced Lagrangian ℓ : L(TM , g) → R satisfies the invariance properties (3.5), then the covariant EP equation (3.3) is invariant under the gauge action h · σ = Adh σ + δ r h of C ∞ (M , H ) on Ω 1 (M , g). Proof. For all h ∈ C ∞ (M , H ) and σ ∈ Ω 1 (M , g) we get
δℓ δℓ (h · σ ) = Ad∗h−1 (σ ) δσ δσ by the following computation for arbitrary ζ ∈ g:
δℓ (h · σ ), ζ δσ
d δℓ r ∗ = ℓ(Adh σ + δ h + t ζ ) = ℓ(σ + tAdh−1 ζ ) = Adh−1 (σ ), ζ . dt dt δσ d
0
0
Using also the identities ad∗Adh ξ Ad∗h−1 ν = Ad∗h−1 ad∗ξ ν,
div(Ad∗h−1 ν) = −ad∗δ r h Ad∗h−1 ν
that hold for all sections ν of TM ⊗ g∗ , we obtain that
δℓ δℓ div + ad∗σ δσ δσ
δℓ δℓ (h · σ ) = div Adh−1 + ad∗Adh σ +δr h Ad∗h−1 δσ δσ δℓ ∗ ∗ δℓ = Adh−1 div + adσ . δσ δσ
∗
This shows that the covariant EP equation is invariant under the gauge action of C ∞ (M , H ).
Proposition 3.4 ensures that the covariant EP equation (3.3), being an equation for C ∞ (M , H )-orbits in Ω 1 (M , g), is an equation for right logarithmic derivatives (2.4) for maps into homogeneous spaces. One can specialize Proposition 3.4 to G-strands. Corollary 3.5. If the reduced Lagrangian ℓ : g × g → R satisfies the invariance property ℓ(Adh ξ + ζ1 , Adh η + ζ2 ) = ℓ(ξ , η) for all h ∈ H and ζ1 , ζ2 ∈ h, then the reduced G-strand Eq. (3.1) is invariant under the action h · (ξ , η) = (Adh ξ + ∂t hh−1 , Adh η + ∂s hh−1 ) of C ∞ (R2 , H ) on C ∞ (R2 , g2 ). Example 3.6. EP equations for strands on the homogeneous spaces of right cosets Diff(S 1 )/S 1 and Diff(M )/Diffvol (M ) generalize the Hunter–Saxton equation and its multidimensional version [15]. With the reduced Lagrangian
ℓ : X( S 1 ) × X( S 1 ) → R ,
ℓ(ξ , η) =
1 2
S1
(ξ ′ )2 dx −
1 2
S1
(η′ )2 dx
we are led to the equations
∂t u′′ + 2u′ u′′ + uu′′′ = ∂s v ′′ + 2v ′ v ′′ + vv ′′′ with zero curvature relation ∂t v − ∂s u = [u, v]. Its multidimensional version is
∂t d(div u) + dLu (div u) + (div u)d(div u) = ∂t d(div v) + dLv (div v) + (div v)d(div v) an EP equation for strands on the homogeneous space Diff(M )/Diffvol (M ), for the reduced Lagrangian ℓ(u, v) 1 (div u)2 µ − 12 M (div v)2 µ, u, v ∈ X(M ). 2 M
=
4. Covariant versus dynamic reduction Time dependent maps R×M → G can also be written dynamically as curves in C ∞ (M , G). Performing covariant reduction for the G-invariant Lagrangian density L : J 1 (R × M , G) → R one gets the EP equations for space–time strands. Performing dynamic reduction for a gauge invariant dynamic Lagrangian L : R × TC ∞ (M , G)× Ω 1 (M , g) → R depending parametrically on a connection, one gets affine EP equations for spin systems. It is shown in [3] that, if L is appropriately built from L, then the resulting reduced equations are equivalent.
C. Vizman / Journal of Geometry and Physics 90 (2015) 104–110
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Spin systems. In [13], it is shown that EP reduction for a gauge invariant Lagrangian L : TC ∞ (M , G) × Ω 1 (M , g) → R depending parametrically on a connection can be used in describing spin systems. The action of the gauge group C ∞ (M , G) on the parameter space Ω 1 (M , g) of principal connections on the trivial bundle M × G is
θg (γ ) = Adg γ − dgg −1 .
(4.1)
The 1-cocycle c : C (M , G) → Ω (M , g) is in this case minus the right logarithmic derivative c (g ) = −dgg , so dc (ξ ) = −dξ for all ξ ∈ C ∞ (M , g). The infinitesimal action involves the covariant derivative dγ ξ = dξ + [γ , ξ ], namely ξΩ 1 (M ,g) (γ ) = −dγ ξ . Since a volume form on M is fixed, 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 (2.11) becomes J (γ , α) = −ad∗γ α − div α = ∞
−1
1
−divγ α , since dc ⊤ (α) = div α and the diamond map is α γ = −ad∗γ α . Hence the affine EP equations for the reduced Lagrangian ℓ : C ∞ (M , g) × Ω 1 (M , g) → R are the equations for spin systems δℓ δℓ ∂ δℓ + ad∗ξ = −divγ , ∂ t δξ δξ δγ
γ˙ + dγ ξ = 0.
(4.2)
Space–time strands. We recall the reduced equations for space–time strands g : R × M → G as in [4,5]. These are also called sliced covariant EP equations because the sliced manifold R × M is considered instead of M in the covariant EP equations. Let L : J 1 (R × M , G) → R be a G-invariant Lagrangian density with reduced Lagrangian ℓ : L(T (R × M ), g) → R. Then both δℓ the g-valued 1-form σ = δ r g and the g∗ -valued vector field δσ can be decomposed according to the slicing:
δℓ d δℓ δℓ = + , δσ δξ dt δγ
σ = ∂t gg −1 dt + dgg −1 = ξ dt + γ ,
where ξ (t ) ∈ C ∞ (M , g), γ (t ) ∈ Ω 1 (M , g). Taking the differential respectively the divergence in the sliced manifold R × M, we obtain dσ =
dξ +
d dt
γ
dt + dγ ,
div
δℓ d δℓ δℓ = + div . δσ dt δξ δγ
The covariant EP equations (3.3) become the EP equations for space–time strands on G d δℓ dt δξ
+ ad∗ξ
δℓ δℓ δℓ = −div − ad∗γ , δξ δγ δγ
(4.3)
with zero-curvature identities
γ˙ − dξ = [ξ , γ ] and dγ = [γ , γ ].
(4.4)
When M is just a real interval, these are the reduced equation for G-strands. Covariant versus dynamic reduction. Given a field theoretic Lagrangian density L : J 1 (R × M , G) = R × TG ×G J 1 (M , G) → R defined on the first jet bundle of the trivial principal bundle over R × M, one assigns a new family of real valued dynamic Lagrangians on the tangent bundle of the gauge group depending parametrically on a connection: LL : R × TC ∞ (M , G) × Ω 1 (M , g) → R L (t , ξg , γ ) = L
L(t , ξg (x), dx g − g (x)γ (x))µ,
ξg ∈ Tg C ∞ (M , G).
M
If the Lagrangian density L is G-invariant, one assigns a reduced Lagrangian density ℓ : L(T (R × M ), g) = R × g × L(TM , g) → R that completely determines L. One can check that the dynamic Lagrangian LL is C ∞ (M , G)-invariant and its reduced Lagrangian is given by
ℓ : R × C (M , g) × Ω (M , g) → R, L
∞
1
ℓ (t , ξ , γ ) = L
ℓ(t , ξ , −γ )µ.
(4.5)
M
Proposition 4.1 ([3]). Covariant reduction for the G-invariant Lagrangian density L and dynamic reduction with affine advected parameters for the C ∞ (M , G)-invariant Lagrangian LL lead to the same reduced equations. Indeed, the reduced equations (4.3) together with (4.4) on one side and (4.2) on the other side coincide, up to the zero curvature condition for the connection γ . This inconvenience can be removed by restricting the parameter space Ω 1 (M , g) to the gauge-invariant subset of flat connection. Indeed, if γ is a flat connection, then θg (γ ) is also flat for the gauge action θ from (4.1). The case of homogeneous spaces. Let H be a Lie subgroup of G and let h be its Lie algebra. In this last paragraph we show that the above passage from the Lagrangian density L to the connection depending density LL can be performed also for G/H-valued maps.
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Lemma 4.2. If the reduced Lagrangian density ℓ : R × g × L(TM , g) → R is invariant under the adjoint action of H and under the addition action of L(TM , h), then the reduced dynamic Lagrangian ℓL : R × TC ∞ (M , G) × Ω 1 (M , g) → R is C ∞ (M , H )and C ∞ (M , h)-invariant. Proof. We use the relation Eq. (4.5) between ℓ and ℓL to check the C ∞ (M , H )-invariance:
ℓL (t , Adh ξ , h−1 · γ ) =
ℓ(t , Adh(x) ξ (x), (dhh−1 )(x) − Adh(x) γ (x))µ(x) ℓ(t , ξ , −γ )µ = ℓL (t , ξ , γ ). ℓ(t , Adh(x) ξ (x), −Adh(x) γ (x))µ(x) = = M
M
M
The C ∞ (M , h)-invariance of ℓL is shown similarly.
The analogue of Proposition 4.1 holds for homogeneous spaces too. Proposition 4.3. Covariant reduction for the G-invariant Lagrangian density L¯ : J 1 (R × M , G/H ) → R and dynamic reduction with affine advected parameters for the C ∞ (M , G)-invariant Lagrangian L¯ L : R × TC ∞ (M , G/H ) × Ω 1 (M , g) → R lead to the same reduced equations. Proof. If the G-invariant Lagrangian density L : J 1 (R × M , G) → R is the pullback of a density on J 1 (R × M , G/H ), then the reduced Lagrangian density ℓ : R × g × L(TM , g) → R is invariant under the adjoint action of H and under the addition action of both h and L(TM , h). By the previous lemma, the reduced dynamic Lagrangian ℓL : R × TC ∞ (M , G) × Ω 1 (M , g) → R is C ∞ (M , H )- and C ∞ (M , h)-invariant, so the time dependent C ∞ (M , G)-invariant Lagrangian LL on C ∞ (M , G), depending also on a connection in Ω 1 (M , g), is the pullback of a time dependent Lagrangian on C ∞ (M , G/H ) with the same parameter space. Example 4.4. This example is inspired by an application from [13]. We consider a bilinear map b : g × g → R that is H-invariant under the adjoint action and its kernel contains the Lie subalgebra h. We choose a Riemannian metric g and a volume form µ on the smooth manifold M. Now we define a reduced Lagrangian density
ℓ : L(T (R × M ), g) = R × g × L(TM , g) → R,
ℓ(t , ξ , γx ) =
1 2
| b(ξ , γx ) |2 ,
where the norm of b(ξ , γx ) ∈ Tx∗ M is computed with the Riemannian metric g. It is invariant under the adjoint action of H and under the addition action of both h and L(TM , h), so it corresponds to a Lagrangian density L¯ : J 1 (R × M , G/H ) → R. The associated reduced Lagrangian is
ℓL : R × C ∞ (M , g) × Ω 1 (M , g) → R,
ℓL (t , ξ , γ ) =
1 2
| b(ξ , γ ) |2 µ.
M
Acknowledgments The author is grateful to François Gay-Balmaz for extremely helpful suggestions. This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0921. References [1] J.E. Marsden, T. Ratiu, Introduction to Mechanics and Symmetry, second ed., Springer, 1999. [2] M. Castrillon Lopez, T.S. Ratiu, S. Shkoller, Reduction in principal fiber bundles: covariant Euler–Poincaré equations, Proc. Amer. Math. Soc. 128 (2000) 2155–2164. [3] F. Gay-Balmaz, T.S. Ratiu, A new Lagrangian dynamic reduction in field theory, Ann. Inst. Fourier 16 (2010) 1125–1160. [4] M. Castrillon Lopez, J.E. Marsden, Covariant and dynamical reduction for principal bundle field theories, Ann. Global Anal. Geom. 34 (2008) 263–285. [5] F. Gay-Balmaz, Clebsch variational principles in field theories and singular solutions of covariant EPDiff equations, Rep. Math. Phys. 71 (2013) 231–277. [6] D.D. Holm, R.I. Ivanov, G-strands and peakon collisions on Diff(R), SIGMA 9 (2013) 027. 14 pages. [7] M. Castrillon Lopez, T.S. Ratiu, Reduction in principal bundles: covariant Lagrange–Poincaré equations, Comm. Math. Phys. 236 (2003) 223–250. [8] F. Tiglay, C. Vizman, Generalized Euler–Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications, Lett. Math. Phys. 97 (2011) 45–60. [9] F. Gay-Balmaz, C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems, Physica D 239 (2010) 1929–1947. [10] C. Vizman, Lagrangian reduction on homogeneous spaces with advected parameters, 2014. arXiv:1408.3019 [math-ph]. [11] B. Khesin, G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math. 176 (2003) 116–144. [12] H. Cendra, D.D. Holm, J. Marsden, T.S. Ratiu, Lagrangian reduction, the Euler–Poincaré equations, and semidirect products, AMS Transl. 186 (1998) 1–25. [13] F. Gay-Balmaz, T.S. Ratiu, The geometric structure of complex fluids, Adv. Appl. Math. 42 (2009) 176–275. [14] D.D. Holm, R.I. Ivanov, J.R. Percival, G-strands, J. Nonlinear Sci. 22 (4) (2012) 517–551. [15] B. Khesin, J. Lenells, G. Misiołek, S. Preston, Geometry of diffeomorphism groups, complete integrability, and geometric statistics, Geom. Funct. Anal. 23 (2013) 334–366.