THE PROBLEM OF LAGRANGE ON PRINCIPAL BUNDLES UNDER A SUBGROUP OF SYMMETRIES

. Abstract. Given a Lagrangian density L v deﬁned on the 1-jet extension J 1 P of a principal G -bundle π : P → M invariant under the action of a closed subgroup H ⊂ G , its Euler-Poincar´e reduction in J 1 P/H = C ( P ) × M P/H ( C ( P ) → M being the bundle of connections of P and P/H → M being the bundle of H -structures) induces a Lagrange problem deﬁned in J 1 ( C ( P ) × M P/H ) by a reduced Lagrangian density l v together with the constraints Curv σ = 0 , ∇ σ ¯ s = 0, for σ and ¯ s sections of C ( P ) and P/H respectively. We prove that the critical section of this problem are solutions of the Euler-Poincar´e equations of the reduced problem. We also study the Hamilton-Cartan formulation of this Lagrange problem, where we ﬁnd some common points with Pontryagin’s approach to optimal control problems for σ as control variables and ¯ s as dynamical variables. Finally, the theory is illus- trated with the case of aﬃne principal ﬁber bundles and its application to the modelisation of the molecular strands on a Lorentzian plane.

σ ∈ Γ(C(P )) and H-structuress ∈ Γ(P/H) by a reduced Lagrangian density lv (projection of Lv), the constraints Curvσ = 0 and ∇ σs = 0, and the representation of infinitesimal gauge transformations η ∈ Γ(g) as the set of admissible infinitesimal variations (g → M being the adjoint bundle of π). Due to the gauge functoriality of the curvature and the covariant derivative, given an admissible section (σ,s) ∈ Γ(C(P ) × M P/H), i.e. a section satisfying Curvσ = 0 ∇ σs = 0, the 1-jet extension j 1 (δσ, δs) of an infinitesimal variation (δσ, δs) of the reduced problem is tangent along j 1 (σ,s) to the submanifold x (σ,s)|Curvσ = 0, ∇ σs = 0} ⊂ J 1 (C(P ) × M P/H). Therefore, we obtain a subspace of the space of admissible infinitesimal variations along an admissible section of the Lagrange problem defined in J 1 (C(P ) × M P/H) by the reduced density lv and the constraint submanifold S (see [8] and [9] for recent geometric descriptions of the Lagrange problem, and also [1] for the point of view of the vakonomic formulation of contrained Field Theories). As a nice consequence we get that critical sections of the Lagrange problem are also solutions of the Euler-Poincaré equations div σ δl δσ − P + s δl δs = 0, of the reduced problem (with the notation introduced along the article). We first tackled this problem in [3] for H = G, where the reduction morphism is simply J 1 P → (J 1 P )/G = C(P ) and no H-structures occur. The reduced problem and its induced Lagrange problem with constraint Curvσ = 0 on connections σ ∈ Γ(C(P )) define the same set of solutions only under some cohomology condition. On the other hand, in [5] we studied the particular case with arbitrary H ⊂ G but M = R, where the condition Curvσ = 0 is now trivial and the only constraint is ∇ σs = 0. In particular, from the local expression of ∇ σs , the Lagrange problem can be regarded as an optimal control problem where the dynamic variable iss and the control variable is σ (see [2] and [8] for a recent version of optimal control problems). In this present article, we give a solution of the general case for arbitrary manifolds M and any subgroup H ⊂ G.
The structure of the work is as follows. Section 2 gives the definitions and the local expressions of some basic operators appearing in the Euler-Poincaré formalism that will be used in the following. In Section 3 we define the problem of Lagrange associated to an Euler-Poincaré reduction on a principal bundle by a subgroup of the structure group. We also compare explicitly the space of admissible infinitesimal variations of both problems (Theorem 3.3 and Corollary 1). In section 4, the Euler-Lagrange equations of the Lagrange problem are obtained through the Lagrange multipliers rule and we compare them with the Euler-Poincaré equations. Section 5 provides the Hamilton-Cartan formulation of the Lagrange problem showing its similarities with the Pontryagin formulation of optimal control problems with connections as optimal variables and H-structures and the field variables. Finally, in Section 6 the theory is illustrated with the case of affine principal bundles with the linear component of the structure group of the affine group as the subgroup H. This framework is applied to the model of a molecular strand on a Lorentzian plane.
2. Some definitions and local formulas. Let π : P → M be a G-principal bundle and let H be a subgroup of G. On the one hand, we consider the bundle of connections C(P ) → M , the sections σ of which are identified to principal G-connections of P . Recall that any automorphim (or in particular, gauge transformation) Φ : P → P of P induces a bundle morphism Φ C : C(P ) → C(P ) such that for a section σ ∈ Γ(C(P )), the connection of P identified to Φ C • σ is the pushforward by Φ of the connection identified to σ. On the other hand, we consider the quotient P/H → M , which can be also seen as the associated bundle P × G (G/H) where G acts upon the homogenous space G/H by the left, the identification being Again, an automorphism of P (or in particular, a gauge transformation) Φ : P → P induces a bundle morphism Φ P/H : Infinitesimal gauge transformations are π-vertical G-invariant vector fields on P . They can be seen as sections of the adjoint bundleg =P × G g, where G act upon g by the adjoint action. Given a section η ∈ Γ(g), the identification is η u = d/dε| ε=0 u · exp εB ∈ T u P , where η(π(u)) = [u, B] G , u ∈ P . Since the flow of an infinitesimal gauge transformation is a uniparametric subgroup of gauge transformation, the infinitesimal version of the representation Φ → Φ C and Φ → Φ P/H described above induces a representation of the gauge algebra Γ(g) into the set of vertical vector fields in C(P ) and P/H respectively that we denote by In the following, we will need this representation along sections. More precisely, given a section (σ,s) of the bundle C(P ) × M P/H → M , we define the operator P (σ,s) : Γ(g) −→ Γ(σ * V (C(P ))) × Γ(s * V (P/H)) η → (P σ (η), Ps(η)) as P σ (η) = η C | σ = ∇ σ η, where we have taken into account that η C along the section σ is exactly the covariant derivative of η with respect to the connection defined by σ. Note that ∇ σ η is a 1-form on M taking values ing, which is consistent with fact that σ * V (C(P )) T * M ⊗g since the bundle of connections is an affine bundle modeled on that vector bundle. It is easy to see that the operator Ps is surjective and its kernel is which in addition is isomorphic to sections ofs * h , via the map whereh → P/H is the adjoint bundle of the H-principal bundle P → P/H. In the following, we will need the notion of covariant derivative ∇ σs of sections s of the bundle P/H → M with respect to a connection σ ∈ Γ(C(P )) on P . The horizontal distribution of σ on P projects to a distribution on P/H of subspaces complementary to the vertical distribution V y (P/H) , y ∈ P/H. Then, the covariant derivative ∇ σs ∈ Γ(T * M ⊗s * V (P/H)) is a 1-form on M taking values ins * V (P/H), and it is defined as for any w ∈ T M , where w h stands for the horizontal lift of w with respect to the horizontal distribution on P/H just mentioned. We now introduce a coordinate system (x 1 , ..., x n ), n = dim M , over a domain U ⊂ M such that π is trivializable, that is π −1 (U ) U × G. We also choose a basis {B 1 , ..., B m }, m = dim G, of the Lie algebra g. Given an element of the Lie algebra B ∈ g, the vector fieldB ∈ X(π −1 (U )) defined asB u = d/dε| ε=0 (x, exp εB · g), u = (x, g), is an infinitesimal gauge transformation. The bundle of connections is endowed with natural coordinates (x i , A α i ), i = 1, ...m, α = 1, ..., m, over U such that the horizontal lift with respect to a connection σ ∈ Γ(C(P )) at a point With these coordinates, one easily finds that the expression of the curvature Ω σ of a connection σ ∈ Γ(C(P )) is Any infinitesimal gauge transformation η ∈ Γ(g) on U can be written as η = η αB α , for some functions η α ∈ C ∞ (U ). The covariant derivative of η with respect to a connection reads On the other hand, let U ×(G/H) → U be the trivialization of the bundle P/H = are coordinates on G/H, then the projection (B α ) P/H of the vector fieldB α can be expressed as for some functions Ψ j α ∈ C ∞ (G/H). In this situation, for everys ∈ Γ(U, G/H) and every η = η αB α ∈ Γ(g), we have Finally, the expression of the covariant derivative of a sections of P/H → M along U is Indeed, given the vector and we get (5) from (2), since ds(∂/∂x i ) = ∂/∂x i + ∂(y j •s)/∂x i (∂/∂y j ).
3. Lagrange problem associated to an Euler-Poincaré reduction on a principal bundle by a subgroup of the structure group. For a G-principal bundle P → M and a subgroup H ⊂ G, we consider the bundle In the jet bundle J 1 (C(P ) × M (P/H)) we define the submanifold , (constraint) and the so-calledl set of admissible sections x (σ,s) ∈ S, ∀x ∈ M }, that is, sections such thats is parallel with respect to σ (∇ σs = 0) and σ is flat (Curvσ = 0). Definition 3.1. Given (σ,s) ∈ S, we define the tangent space T (σ,s) S as the set of vertical vector fields (δσ, δs) along the sections σ ands (that is, δσ ∈ Γ(σ * V (C(P ))), δs ∈ Γ(s * V (P/H))) such that their 1-jet lifts (δσ) (1) and (δs) (1) We now prove that, although the computation of jet lifts (δσ) (1) and (δs) (1) as vector fields on J 1 C(P ) and J 1 (P/H) respectively requires an extension of δσ and δs to vertical vector fields on the entire bundles C(P ) and P/H, the fact of being tangent to S ⊂ J 1 (C(P ) × M (P/H)) along j 1 σ and j 1s does not depend on the chosen extension, so that the definition above makes sense.
Lemma 3.2. Let σ ands be sections of C(P ) → M and (P/H) → M respectively, and δσ ∈ Γ(σ * V (C(P ))) = Γ(T * M ⊗g), and δs ∈ Γ(s * V (P/H)) vertical vectors along them. Then, given any extension of δσ and δs to vertical vector field on C(P ) and P/H respectively, the restriction of its 1-jet lift is tangent to S if and only if and where, Ps being surjective, η ∈ Γ(g) is such that Ps(η) = (η P/H )s = δs. Curv(σ t ) = 0, Since these conditions are local, we can work on a domain U ⊂ M such that π : P → M is trivial. Under this trivialization, connections σ and variations δσ can be seen g-valued 1-forms on M . Hence and (6) is proved. For the proof of (7) we also assume that U is a chart domain with coordinates (x 1 , ..., x n ) and we have coordinates (y 1 , ..., y r ) on G/H. From the local expression (5), we have where δσ = (δσ) α i dx i ⊗ B α , δs = (δs) k ∂/∂y k . We now take η ∈ Γ(g), η = η αB α , as in the statement. From (4), its local expression is Putting this expression in (10), we get where in the last step we have taken into account that ∇ σs = 0. Hence where we have used the following easy relation and c α βγ are the structure constants of the basis {B 1 , ..., B m }. Hence, the local expression of d/dt| t=0 ∇ σts t is precisely that of (∇ σ η − δσ) P/H alongs and we conclude (7).
Not only is this last Lemma necessary for Definition 3.1 above, but also the fundamental of the following characterization of tangents elements to S, as the following results shows.
Proof. From the previous Lemma, (δσ, δs) belongs to T (σ,s) S if and only if ∇ σ (δσ) = 0 and there exists η ∈ Γ(g) such that (η P/H )s = δs and Ps(∇ σ η − δσ) = 0. The last condition means that ω = δσ − ∇ σ η ∈ Γ(T * M ⊗s * h ), and the first that ∇ σ ω vanishes, since σ is flat. The converse is immediate. Proof. According to Theorem 3.3, δσ = ∇ σ η + ω and δs = η P/H with ∇ σ ω = 0. Then, since H 1 (M,s * h ) = 0, σ is flat and the bundle P → M is trivial there exists ζ ∈ Γ(s * h ) such that ∇ σ ζ = ω. Then we consider η = η + ζ and we get (η P/H )s = Ps(η ) = Ps(η) + Ps(ζ) = Ps(η) = δs, as desired. We now think of l as the reduction of an H-invariant first order Lagrangian L : J 1 P → R as considered in the Introduction, for which we consider the Euler-Poincaré reduction picture. From Theorem 3.3, critical sections with respect to Definition 3.4 are also solutions to the Euler-Poincaré equations, since Euler-Poincaré variations δσ = ∇ σ η, δs = η P/H , are special cases of constrained variations (δσ, δs) ∈ T (σ,s) S. Under the topological assumptions of Corollary 1, this inclusion between sets of solutions is a bijection since in that case the set of variations are the same for both problems. However, if these cohomology condition is not satisfied, this equivalence needs not be true and we have proper inclusion of Euler-Poincaré solution into the set of critical solutions. In the next section we explore with more detail this fact about the equations on the set of critical solutions (σ,s) ∈ S through the method of Lagrange multipliers.
(j 1 σ, j 1s , χ, λ) = l(σ,s) + χ, Curv(σ) + λ, ∇ σs as a free variational problem with no constraints for either the sections (σ,s, χ, λ) or their variations. We first need to introduce the duals of two objects of this work. For the operator Ps : Γ(g) → Γ(s * V (P/H)) its adjoint P + s : Γ(s * V * (P/H)) → Γ(g * ) is defined as the only operator such that for any compact supported sections ∈ Γ(s * V * (P/H)) and η ∈ Γ(g). Moreover, for a connection σ in P , the divergence operator is defined as the only operator such that where div is the standard divergence operator defined by v in M .
Theorem 4.1. The variational equations of the Lagrangian defined above are Proof. The vanishing of the variation of the action is for arbitrary variations δσ, δs, δχ and δλ. Variations on χ and λ give Curv(σ) = 0, and ∇ σs = 0 respectively. On the other hand, for any variation δσ ∈ Γ(σ * V (C(P )) Γ(T * M ⊗g), we have where σ ε = σ + εδσ. For the derivative of the curvature, we easily have d/dε| ε=0 Curv(σ ε ) = ∇ σ δσ. On the other hand, from Lemma 3.2 we have being this formula a special case of the computation of equation (10) for η = 0. Then δ δσ , δσ = δl δσ , δσ + χ, ∇ σ δσ − λ, Ps(δσ) , and, taking into account the adjoint operator P + and the covariant divergence div σ , the variation of the action (14) for δσ reads The last term of the expression above vanishes by Stokes theorem, since the variations are compact supported. Hence, as δσ is free, we get the first equation in the statement. For variations δs ∈ Γ(s * V (P/H)), we have wheres ε is a family of sections withs 0 =s and d/dε| ε=0sε = δs. Again a particular case of formula (10) in Lemma 3.2, now for δσ = 0, gives Hence δ δs , δs = δl δs , Ps(η) + λ, Ps(∇ σ η) , and the action principle (14) for variations δs is The second equation in the statement is obtained since η is free and compact supported.

Remark 1.
It is important to recall the bundles where each of the equations in (13) takes place. The first equation is defined in T M ⊗g * , the second ing * , the third in 2 T M ⊗g * and the forth in T M ⊗ V * (P/H). It is easy to check that every term of these equations, along (σ,s), is a section of the corresponding bundle.
s δl δs where we have taken into account that div σ div σ χ = Curvσ, χ = 0 as σ is flat. We then recover the Euler-Poincaré equations for L. . , x n ) such that v = dx 1 ∧. . .∧dx n , as well as the following induced coordinate systems: and (x i , y j , y j ,i ) in J 1 (P/H) respectively. From formulas (3) and (5), we have that the local expression of the Lagrangian (12) in these coordinate systems is From this formula, the Poincaré-Cartan form (see [7]) of the Lagrangian density v has the following local expression Since ∂ /∂A α i,j = χ ij α , for i < j, ∂ /∂A α i,j = −χ ij α , for i > j and ∂ /∂y i j = λ j i , we have Therefore, the local expression of the multisymplectic form associated to the problem defined by , that is, the differential of the Poincaré-Cartan form, is The Hamilton-Cartan equations defined by this form for sections (σ,s, χ, λ) are given by the condition (σ,s, χ, λ) * i A Ω = 0, for all vector field A vertical with respect to the fibration onto M . From the expression (16), the local expression of these equations are As we know from the general theory on Hamiltonian formulation of variational problems (see, for example [7]), these equations are equivalent to the equations (13).
6. Application to affine principal bundles: the molecular strands. Given a Lie group G acting linearly upon a vector space V , we consider G aff = G V as the Lie group defined by the (semidirect) product is the associated vector bundle to the representation of G, is a principal bundle with structure group G aff . In the particular (but essential) case where P = LM is the frame bundle of M , G = Gl(n, R), n = dimM , and V = R n with the natural action, the bundle P aff → M is the standard bundle of affine frames, a fact that justifies the aff-notation used for the general case. We now explore the constructions of previous sections for P aff → M , taking H = G as the subgroup of the structure group G aff . First, we note that the bundle of connections C(P aff ) of P aff naturally splits as since connections σ aff in P aff split as In particular, we have δL δσ aff = δL δσ , δL δh .
On the other hand, Therefore, the starting point for the Lagrange study in the context of affine bundles is a Lagrangian l : C(P ) × M (T * M ⊗ E) × M E → R, so that the Lagrange multipliers of the corresponding Lagrangian in (12) are Finally, the expression of the adjoint operator where E ⊗ E * is immersed intog * via the mapping (e ⊗ e * )(η) = e * (η · e), e ∈ E, e * ∈ E * , η ∈g.
The notation h⊗ζ ∈ T * M ⊗g is defined as is the wedge product twisted with the action ofg on E. If one solves (21) for λ and put it in (20), then Lemma 6.1. We have that div σ (s ⊗ ζ) = ∇ σs⊗ ζ + div σ ζ ⊗s.
The first component is where h⊗s ∈ Γ(E ⊗ E * ) is obtained by coupling the T M and T * M parts of h ands and E ⊗ E * is immersed intog * as in (18). The second component, taking into account (21), is δL δs − div σ δL δh = 0.
Collecting all these considerations, we have: Proposition 1. The Lagrange problem for affine bundles is given by the following equations δl δσ − δl δh ⊗s − div σ (χ +s ⊗ ζ) = 0 δl δh − div σ ζ − λ = 0 δl δs ⊗s − div σ (λ ⊗s) − h⊗λ = 0 δl δs − div σ δl δh = 0 Reduction in affine principal bundles models molecular strands. For more details on this example, specially the physical meaning of the reduced variables, we refer the reader to [6] (see also [4]). We can consider in this case P = R 2 × SO(3) → R 2 , G = SO(3) and V = R 3 , with the natural action upon it. Then E = R 2 × R 3 → R 2 and P aff = R 2 × SE(3) → R 2 . Since all the bundles are trivial, sections are understood as functions from R 2 to their respective fibers. If we take coordinates (x, t) in R 2 , (x for the parameter of the strand and t for time), we can write for certain Ω, ω ∈ C ∞ (R 2 , so(3)), Γ, γ, λ x , λ t ∈ C ∞ (R 2 , R 3 ). In the following, we will identify so(3) with R 3 and its Lie bracket with the cross product. We consider the Lagrangian defined on a Lorentzian plane (R 2 , g), g = dx ⊗ dx − v 2 dt ⊗ dt, defined as l(t, x, Ω, ω, Γ, γ,s) = Γ, Γ − 1 v 2 γ, γ − U ( s,s ) +l(t, x, Ω, ω), where U is a smooth function defining a (central) potential, andl is a Lagrangian for the connection Ω and ω. This Lagrangian l can be understood as a model for the filament of the strands =s(x, t), interpreted as a kind of "vectorial wave" with v as propagation speed. In this situation, it is a matter of checking that equations (23) take the following form δl δΩ − 2Γ ⊗s − ∂ ∂x + Ω× (χ +s ⊗ ζ) = 0 δl δω