Variational integrators for reduced field equations

In the reduction of field theories in principal G-bundles, when a subgroup H ⊂ G acts by symmetries of the Lagrangian, each of the H-reduced unknown fields decomposes as a flat principal connection and a parallel H-structure. A suitable variational principle with differential constraints on such fields leads to necessary criticality conditions known as Euler-Poincaré equations. We model constrained discrete variational theories on a simplicial complex and generate from the smooth theory, in a covariant way, a discrete variational formulation of H-reduced field theories. Critical fields in this formulation are characterized by a corresponding discrete version of Euler-Poincaré equations. We present a numerical integration algorithm for discrete Euler-Poincaré equations, that extends integration algorithms for Euler-Poincaré equations in mechanics to the case of field theories and, also, extends integration algorithms for Euler-Lagrange equations in discrete field theories to the case of variational principles with constraints. For regular reduced discrete Lagrangians, this algorithm allows to univocally propagate initial condition data, on an initial condition band, into a solution of the corresponding equations for the discrete variational principle.


Introduction
In the community working on numerical methods for ordinary and partial differential equations arising in physics, it has become clear in the last years that discretising the variational principles, that originally generated the equations, is a very fruitful method, when we are looking for some discrete version of the equations and integrators for this discrete version.In such a way that, the usual geometrical considerations in the smooth theory also make sense for the discretized versions of the equations and for its solutions.Following this guideline one usually is rewarded with integrators with simplecticity and discrete conservation laws.
It is in the area of field theory that many aspects on the discretization of variational principles remain unsolved.A common approach is to treat the discrete field theory with large-dimensional functional spaces, like in the theory of finite elements [1].In many other situations the corresponding field theory has a natural time variable and its discretization allows for a treatment similar to that of discrete mechanical systems [14,15].These examples (x 1 , . . ., x n , y 1 , . . ., y m ) on Y .Mappings y : U → Y defined on open subsets U ⊆ X and such that π • y = Id U (local sections of the bundle) are called local fields and can be expressed in such a system of local coordinates as a family of smooth functions y i (x 1 , . . ., x n ).We denote by Γ(Y ) the set of local sections and by Γ(K, Y ) the subset of local sections whose domain contains K ⊆ X.
Consider the first jet bundle JY of sections, associated to the bundle Y , a bundle with local coordinates (x ν , y i , y i ν ).Local sections y ∈ Γ(Y ) of π : Y → X determined by equations y i = y i (x 1 , . . ., x n ) induce a corresponding jet extension j 1 y ∈ Γ(JY ), a local section of jπ : JY → X determined by equations y i = y i (x 1 , . . ., x n ), y i ν = (∂y i /∂x ν )(x 1 , . . ., x n ).Consider the vertical bundle π V Y : V π Y → Y determined by all tangent vectors δ y ∈ T Y that are vertical for the projection dπ : T Y → T X (when the projector under consideration is clear, we shall simply write V Y instead of V π Y ).This is a vector bundle on Y and any system of fibred coordinates (x 1 , . . ., x n , y 1 , . . ., y m ) induces linear coordinates α i , so that each element δ y ∈ V Y can be written as δ y = α i (∂/∂y i ) y (we use Einstein's convention for summation on repeated indices).In a similar way as local sections have a natural extension to the jet bundle, also local sections of π • π V Y : V Y → X have a natural extension to V (JY ).For any given vertical vector field δ y ∈ Γ(V Y ) defined on a local section y = π V Y • δ y ∈ Γ(Y ), with local expression α i = α i (x 1 , . . ., x n ) and y i = y i (x 1 , . . ., x n ), there is a naturally induced local section D j 1 y = j 1 δ y ∈ Γ(V (JY )), projecting to δ y when we use dπ J : V (JY ) → V Y , projecting to j 1 y ∈ Γ(JY ) when we use π V (JY ) : V (JY ) → JY , and locally given as D j 1 y = α i (∂/∂y i ) j 1 y + β i ν (∂/∂y i ν ) j 1 y with: this section D j 1 y = j 1 δ y is called the vertical infinitesimal contact transformation associated to δ y ∈ Γ(V Y ).
We should recall that π • π V Y : V Y → X is not a vector bundle.However for any fixed section y ∈ Γ(Y ), we have a vector bundle y * V Y → X defined as the pull-back (or restriction) of the bundle V Y → Y along the local section y : U → Y .Observe that a smooth variation of y 0 ∈ Γ(Y ), that is, a smooth mapping y ϵ (x 1 , . . . ,x n ) : [0, ϵ max ] × U → Y where y ϵ ∈ Γ(Y ) have a common domain U ⊂ X, allows to determine the associated infinitesimal variation δ y0 = (d/dϵ) 0 (y ϵ (x 1 , . . ., x n )) ∈ Γ(y * 0 V Y ), and its associated vertical infinitesimal contact transformation fulfills: Variational problems on a bundle π : Y → X arise when we fix a smooth function L on JY (the Lagrangian) and a smooth volume form vol X ∈ Ω n (X) on X.The product Lvol X is a horizontal n-form on JY , and determines an action functional L, a real-valued mapping that associates to any compact domain K ⊂ X and local field y ∈ Γ(K, Y ), defined on an open neighbourhood of the domain, a value called the action associated to the field on the domain K: Variational problems try to characterize critical points for the action functional, with respect to certain admissible variations.Indeed, if (y ϵ ) 0≤ϵ≤ϵmax ⊂ Γ(K, Y ) is a smooth variation of a section y 0 ∈ Γ(K, Y ) for some compact domain K ⊂ X, derivating inside of the integral symbol and using the chain rule together with (1) we have the expression for the first derivative of L K (y ϵ ): where dL ∈ Γ(T * (JY )) determines d v y0 L ∈ Γ((j 1 y 0 ) * V * (JY )) using the restriction to the section j 1 y 0 and the restriction to V (JY ) ⊂ T (JY ), and where ⟨•, •⟩ represents the duality product with elements in Γ((j 1 y 0 ) * V (JY )).
The integration ∫ K (j 1 y) * Lvol X in the action functional is a real value if we assume that we are working with a compact domain K ⊆ X (for example we may assume K = X in the case that X is compact).If we do not fix a compact domain K ⊂ X, we do not have a well defined value L(y) for every y ∈ Γ(Y ).However, the linear functional d y L : δ y ∈ Γ(y * V Y ) → ∫ X ⟨d v y0 L, j 1 δ y ⟩vol X ∈ R makes sense with the simple condition that δ y ∈ Γ(y * V Y ) is chosen with a support that is compact in the domain of y ∈ Γ(Y ), no matter that this domain might be a non-compact subset U ⊂ X.

Definition 2.1
For any local section y : U → Y of the bundle π : Y → X, we call differential at y of the action functional L, associated to a Lagrangian density Lvol X , the linear functional: where the subspace Γ c (y * V Y ) is given by all sections of the bundle y * V Y → U whose support is a compact subset K ⊂ U .
A general formulation of a variational problem with differential constraints on a fibred manifold π : Y → X (see [18]) tries to determine which local sections y ∈ Γ(y) are critical, in the following sense: Definition 2.2 Let π : Y → X be a bundle on an n-dimensional manifold, oriented by a volume element vol X ∈ Ω n (X).A constrained variational problem is determined by: • A smooth submanifold S ⊆ JY (the constraint submanifold).
(A local field y ∈ Γ(Y ) is called admissible for the variational problem if j 1 y takes values on S. The set of admissible local fields shall be denoted by Γ S (Y ).)For the particular case of S = JY and AV y = Γ c (y * V Y ) (a situation that we denote as fixed boundary variations) criticality is equivalent to the annihilation of the Euler-Lagrange tensor EL(y) ∈ Γ(y * V * Y ), that for the case vol X = dx 1 ∧ . . .∧ dx n takes the classical form: )) the appearance of (d/dx ν ) in an object depending on j 1 y shows that equations EL(y) = 0 represents a system of second order partial differential equations on the unknown y ∈ Γ(Y ).
For a different choice of admissible infinitesimal variations, however, critical sections can be characterized by a different system of partial differential equations.This is, for example, the case of Euler-Poincaré equations in the case of gauge field theories invariant by the action of a certain group H.These equations arise as necessary criticality conditions for a variational problem with constraints that we introduce next.

H-structures on a principal G-bundle
An action (also called action on the left) of a group G on a set M is a group morphism λ : ).We reserve the expression "left action" for the case M = G, to represent the specific where l h (g) = hg, and the expression "right action" for the specific mapping Both of them determine actions (on the left) of the group G on the set G. In the case that G, M are smooth manifolds, the group morphism λ determines a mapping We say the action is smooth if (ac, π M ) is smooth, say the action is free if (ac, π M ) is injective and say the action is proper if (ac, π M ) is proper (inverse image of compact sets are compact sets).
We will be concerned with Lie groups: groups with a smoth manifold structure, such that both the left and right actions l, r −1 of G on itself are smooth actions.For any Lie group (G, •) there exists a corresponding reverse Lie group G r = (G, •) using G itself as manifold and the reverse product g • h = h • g.We shall avoid talking of actions of G on the right on M , that can be defined as actions of the reverse group G r on M .Smooth free proper actions of Lie groups on smooth manifolds will appear in this work and the existence of the quotient manifold for such actions is a classical result, with main properties expressed in the appendix.In particular, we shall introduce principal G-bundles with the action on the left convention: Let G be a Lie group.A principal G-bundle is a smooth manifold P , together with a free proper smooth action of a Lie group G on P .
The fact that the smooth free action is proper warrants that there exists a unique quotient manifold structure P/G, such that the quotient mapping π G : P → P/G is a surjective submersion.This submersion is usually called the fiber bundle structure, and the quotient manifold X = P/G is called the base manifold of the principal bundle.
We should observe that for any given closed subgroup H ⊆ G, the same manifold P has also a principal Hbundle structure considering the restriction of the action to H ⊆ G (the mapping (λ, Id) is still proper because H is a closed subset).The quotient morphism π H : P → P/H represents the fiber bundle structure of P seen as a principal H-bundle.Fixing a principal G-bundle π G : P → P/G = X and a closed subgroup H ⊆ G, we get a principal H-bundle π H : P → HStr on the quotient manifold HStr = P/H, the manifold of H-structures.
We observe that elements on the same H-orbit project to a unique point by π G : P → X, therefore π G factors by a morphism π HStr : HStr = P/H → X.This is called the bundle of H-structures of the principal G-bundle P → X.The principal H-bundle structure determined by P as manifold, H a group acting by λ, and HStr as base manifold shall be called the principal H-bundle P HStr induced by the principal G-bundle P → X, on the manifold of H-structures HStr.
Elements in a particular fiber π −1 HStr (x) = HStr x are H-orbits q = Hp ∈ P/H contained in an specific G-orbit P x ⊂ P .A section q : x ∈ X → q x ∈ HStr of the bundle π HStr : HStr → X can be seen as a smoothly varying choice of H-orbits q x ⊂ P x for all possible x ∈ X, therefore as a choice of a smooth sub-bundle Q → X of the fiber bundle P → X, where each fiber is a single H-orbit.Therefore, sections q ∈ Γ(HStr) of the bundle of Hstructures are simply sub-bundles of P → X whose fibers are orbits by the restricted action of the subgroup H.These are called reductions of the principal bundle P to the subgroup H, or H-structures contained in P .
Consider V π G P ⊂ T P , the space of tangent vectors in P that project as zero by dπ G : T P → T X.The action λ g on P induces an action dλ g on V π G P .The quotient manifold (V π G P )/G is then a vector bundle on P/G = X, called adjoint bundle π Ad : Ad P → X. Giving an element a x ∈ Ad P x of this quotient over a point x ∈ X is the same as giving a vertical vector at some point p ∈ P x together with all the vectors obtained by application of dλ g .This is the same as giving a vector field on the fiber P x , that is invariant for all the automorphisms λ g : P x → P x of this fiber.Any such a G-invariant vector field a x on a single fiber P x generates a 1-parameter flow, a family of automorphisms (exp ϵa x ) : P x → P x on the G-orbit P x .All of them commute with all actions λ g .Moreover, by definition of the flow, all tangent vectors: are a representative of the vector field a x on P x .That is, all these tangent vectors project in V P/G as the element a x ∈ V P/G = Ad P .
In a similar way if we consider V π H P the space of tangent vectors that project as zero by dπ H , and the action λ h on P , we have an induced action by dλ h on V π H P .The quotient manifold (V π H P )/H is then a vector bundle on P/H = HStr, which by definition represents the adjoint bundle Ad P HStr → HStr of the principal H-bundle π H : P → P/H (a bundle that we denote as P HStr → HStr).Any element a q ∈ P HStr q on the fiber associated to q ∈ HStr also induces a 1-parameter family of automorphisms exp ϵa q : Q x → Q x on the H-orbit Q x ⊂ P x determined by q ∈ HStr x .These automorphisms commute with each λ h for h ∈ H.

Proposition 2.1
Consider a principal G-bundle P and for any closed subgroup H ⊆ G the induced principal H-bundle P HStr on the manifold of H-structures.There exists an exact sequence of vector bundles on HStr 0 → Ad P HStr → HStr× X Ad P → V HStr → 0 Proof Observe that π H : P → P/H determines a natural projector pr : HStr× X (V π G P )/G → V (P/H), that associates to each H-orbit q x ∈ HStr x and each G-covariant vertical vector field A on the fiber P x the tangent vector at q x ∈ P/H obtained by π H -projection dπ H (A px ) ∈ T qx HStr, for any representative p x of the H-orbit q x .The condition that A is G-covariant, and in particular H-covariant, shows that this mapping does not depend on the particular choice of p x in the H-orbit q x .The fact that dπ H is linear and surjective shows that this morphism is also linear and surjective.
As π G = π H • π HStr , there exists a natural inclusion V π H P ⊆ V π G P .There also exists a natural projection V π H P → P → HStr = P/H.The combination of both leads to a morphism: We observe that two elements in V π H P mapping to the same element in HStr× X Ad P are necessarily one the image of the other by dλ g for some g ∈ G (because they determine the same element in (V π G P )/G), and that this g ∈ G belongs in fact to the subgroup H (because they determine the same element in HStr = P/H).We conclude that fibers of f are empty or a whole orbit in V π H P by the action dλ h for h ∈ H. Therefore f determines a natural immersion: We still have to prove that the exactness in the central term in the sequence: As elements in V π H P project as zero into HStr = P/H, in this sequence the image of i is contained in the kernel of pr.Taking into account that we have vector bundles on HStr with ranks dim H, dim G, dim HStr = dim G − dim H, and linear morphisms, we conclude that the image of i must coincide with the kernel of pr and this is an exact sequence.

Corollary 2.1
Consider the adjoint bundles Ad P → X and Ad P HStr → HStr associated to the principal G-bundle P → X and H-bundle P HStr → HStr.There exists a natural identification V HStr ≃ (HStr× X Ad P )/ Ad P HStr and, for each H-structure q : X → HStr a natural identification q * V HStr ≃ Ad P/q * Ad P HStr .The corresponding dual bundle can be identified as a sub-bundle q * V * HStr ⊂ Ad * P of linear forms on Ad P that vanish when applied to q * Ad P HStr ⊂ Ad P .

Euler-Poincaré equations for H-reduced fields on principal G-bundles
Let π : P → X be a principal G-bundle and denote λ g the left-translation by any element g ∈ G.If we denote by π J : JP → P the associated first jet bundle, the group G acts on JP by the induced mappings jλ g : JP → JP .
An Ehresmann connection on the bundle π : P → X is any linear immersion χ : π * T X → T P , section of the natural linear projector dπ : T P → π * T X. Giving a connection is then equivalent to giving a section χ : P → JP of the associated first jet bundle (consult the theory of Ehresmann's connections in [23]).
The connection χ is called principal connection if it is invariant by the linear morphisms induced by λ g on the bundle T P ⊗ π * T * X. Principal connections are equivalent to sections χ : P/G → JP/G of the so-called bundle of principal connections π CP : CP = JP/G → P/G = X.
For any fixed Ehresmann connection on a bundle π : P → X there exists a natural (horizontal) lift of vector fields on X to vector fields on P .To be precise, given a vector field A : X → T X, it induces Id ×A : P × X X → P × X T X = π * T X, which composed with χ determines A χ : P = P × X X → T P , a vector field on P called the horizontal lift of A ∈ Γ(X, T X).For any pair of vector fields A, B ∈ Γ(X, T X) the commutator [A χ , B χ ] ∈ Γ(P, T P ) of the corresponding horizontal lifts A χ , B χ can be compared to the horizontal lift [A, B] χ of the commutator [A, B] ∈ Γ(X, T X).Both of these fields on P project to the manifold X as [A, B] ∈ Γ(X, T X) and the difference is then a vertical vector field: In fact, one can prove that Curv χ is a bilinear alternating form, leading then to Curv χ ∈ Γ(π * (Λ 2 T * X) ⊗ V P ), called the curvature tensor associated to the Ehresmann connection χ.
In the case that the connection on P is a principal connection (invariant by translations λ g ), its associated curvature tensor will also be invariant by translations and can be described as a 2-form on the base manifold, with values on the adjoint bundle π Ad : Ad P = V P/G → X.A connection is called flat when its curvature vanishes.
As a remark, observe that π H • π J : JP → P/H and π G : JP → (JP )/G determine a smooth surjective mapping JP → (P/H)× X (JP/G).Moreover, two elements in the same fiber by this mapping must differ by some transformation jλ g (because they determine the same element in (JP )/G) and necessarily for some g ∈ H (because they determine the same element in P/H).Fibers of the morphism are precisely orbits on JP by the action jλ h for h ∈ H.There exists a naturally induced isomorphism defined on the quotient manifold JP/H: For any principal G-bundle π : P → X and closed subgroup H ⊆ G, we call local H-reduced field any local section of the bundle Y = JP/H ≃ HStr× X CP.We call H-reduced Lagrangian function any function ℓ on some open sub-bundle of JP/H = Y → X, or equivalently, an H-invariant function on some open sub-bundle of JP → X.
We recall now how a principal connection determines parallelism notions on associated bundles.Consider any manifold E → X and smooth action ac : (e, g) ∈ E × G → eg ∈ E of the Lie group G on the right on E. Using the left action g • (e, p) = (eg −1 , gp) on the manifold E × P , there exists a quotient manifold Q E = (E × P )/G, called the associated bundle for the principal bundle P and the G-space E. For a given principal connection χ on P , and any associated bundle, there exists an induced Ehresmann connection χ Q on the bundle We recall that a connection As both linear morphisms are sections of dπ Q : T X → q * T Q, the difference is a linear mapping T X → q * V Q and determines a notion of covariant derivative of q ∈ Γ(Q) with respect to the connection χ Q : Stat., Optim.Inf.Comput.Vol. 6, March 2018 Using E = G/H (with left cosets G/H = {Hg} g∈G ) and the natural action on the right of the group G on this space, from χ ∈ Γ(CP) we obtain a connection on the bundle of H-structures π HStr : HStr = E × P = P/H → X.A principal connection χ determines a connection χ HStr on this bundle and we have a notion of covariant derivative of any local H-structure q ∈ Γ(HStr), with respect to the principal connection χ.
One should observe that any local field p ∈ Γ(P ) naturally induces, by projection π H : JP → JP/H, a local Hreduced field.However, there may exist local H-reduced fields (q, χ) ∈ Γ(HStr× X CP) that are not projection of any local field on P .There are differential and topological obstructions in this regard [10].Whenever the Hreduced field (q, χ) is obtained projecting a jet extension jp ∈ Γ(JP ), the principal connection component χ is flat (has vanishing curvature), and the H-structure component q ∈ Γ(HStr) is χ-parallel, that is, d χ q = 0.In fact, this condition can be stated saying that the principal connection χ on the principal G-bundle P is reducible to the principal sub-H-bundle Q → X determined by q ∈ Γ(HStr).That is, horizontal lifts at points In particular, using E = Lie G with the right adjoint action, the associated bundle is the adjoint bundle and any principal connection χ on P induces an Ehresmann connection χ Ad on the adjoint bundle π Ad : Ad P → X.We get a notion of covariant derivative of any local section of the adjoint bundle a ∈ Γ(Ad P ) with respect to the principal connection χ: where the identification V a(x) Ad P ≃ Ad P x is using the natural identification of tangent vectors at any point a(x) of a linear space Ad P x with elements of the linear space itself.

Definition 2.5
A local H-reduced field (q, χ) ∈ Γ(HStr× X CP) is called admissible if χ has vanishing curvature and q is χparallel (equivalently, the connection χ is flat and reducible to the specific H-structure q).This determines the following constraint submanifold on J(HStr× X CP): This submanifold S determines a space of admissible jet configurations, the first element needed in definition 2.2 to characterize a constrained variational principle.
Finally for any admissible local H-reduced field (q, χ) ∈ Γ(Y ) we may consider two linear operators: here we are using that CP is an affine bundle modelled on Ad P ⊗ T * X, and we are considering the principal H-bundle π H : P → P/H = HStr, that we denote as P HStr , its associated adjoint bundle Ad P HStr → HStr and the projection a → [a] in the natural exact sequence: Using these operators P q , P χ (the first one a linear projection and the second one a first order differential operator) we have corresponding spaces of admissible variations:

Definition 2.6
For any admissible local H-reduced field (q, χ) ∈ Γ(HStr× X CP) we call space of admissible infinitesimal variations at (q, χ) the linear space: As indicated in [10] any element (δ q , δ χ ) ∈ AV (q,χ) has an associated infinitesimal contact transformation j 1 (δ q , δ χ ) ∈ Γ(J(HStr× X CP)) that is tangential to the constraint submanifold S ⊂ J(HStr× X CP).The constraint submanifold S and the family of spaces of admissible infinitesimal variations AV (q,χ) serve as compatible data to define a constrained variational problem on HStr× X CP, as described in definition 2.2.
The constrained variational problem on H-reduced fields is finally fixed when we consider some H-reduced Lagrangian density, a function on S that we will consider as determined by the values of the field and not of its jet extension j 1 x (q, χ) ∈ S ⊂ J(HStr× X CP).We shall restrict our variational problems to the following type of Lagrangian functions: We call zero-order H-reduced Lagrangian function any function ℓ : HStr× X CP → R. Giving a zero-order Hreduced Lagrangian function is the same as giving a first order Lagrangian function Zero-order H-reduced Lagrangian functions determine a Lagrangian function S → R, as demanded in definition 2.2 of constrained variational problems.In this situation we have the following fundamental result: ]) Let π : P → X be a principal G-bundle on a manifold X with volume element vol X .Let H ⊆ G be a closed subgroup, and denote by π H : JP → HStr× X CP the natural quotient morphism.Let ℓ : HStr× X CP → R be a zero-order H-reduced Lagrangian function and L = ℓ • π H : JP → R the associated first order Lagrangian function.For any local field p ∈ Γ(P ), and for the induced local H-reduced field π H • j 1 p = (q, χ) ∈ Γ(HStr× X CP), the following are equivalent: 1.The local field p is critical for the variational problem with fixed boundary variations and Lagrangian function L. 2. The local field p ∈ Γ(P ) satisfies the system of Euler-Lagrange equations 0 = EL(p) for L : JP → R, where EL(p) is described by (4).3. The local H-reduced field is critical for the constrained variational problem described on HStr× X CP by the constraint submanifold S, admissible infinitesimal variations AV (q,χ) , and Lagrangian function ℓ • π J : J(HStr× X CP) → R. 4. The H-reduced field (q, χ) ∈ Γ(HStr× X CP) satisfies the system of Euler-Poincaré equations 0 = EP(q, χ) for ℓ : HStr× X CP → R, where EP(q, χ) is described by: In this result ∂ℓ/∂χ ∈ Γ(χ * V * CP) ≃ Γ(Ad * P ⊗ T X) (recall that CP is an affine bundle on Ad P ⊗ T * X), ∂ℓ/∂q ∈ Γ((Ad P/q * Ad P HStr ) * ) (recall that q * V HStr ≃ Ad P/q * Ad P HStr ), P * q is the natural immersion Γ((Ad P/q * Ad P HStr ) * ) ⊂ Γ(Ad * P ) as the subspace of linear forms that vanish on q * Ad P HStr ⊂ Ad P , and finally the divergence operator div χ : Γ(Ad * P ⊗ T X) → Γ(Ad * P ) associated to χ represents the differential operator adjoint to d χ , in the sense: here i θ vol X stands for the Ad P -valued n − 1-alternating form obtained by contraction of θ with the volume element X and d χ stands for the covariant differential on alternating forms with values on Ad P and on Ad * P obtained as extension of d χ : Γ(Ad P ) → Γ(Ad P ⊗ T * X).This theorem relates the variational problem with fixed boundary variations determined by the Lagrangian function L = ℓ • π H on JP , with the constrained variational problem on HStr× X CP with admissible infinitesimal variations AV (q,χ) , determined by the Lagrangian function ℓ • π J : S → R on the constraint submanifold in J(HStr× X CP).We stress, however, that this does not imply that both variational problems are equivalent.There may exist admissible reduced fields that are not obtained as projection of a jet extension for any section of the principal bundle.

Discretization through Forward difference operators
We aim to obtain an appropriate discretization of the variational problems with constraints on local Hreduced fields.That is, if we have a principal G-bundle π : P → X and an H-reduced Lagrangian function ℓ : HStr× X CP → R, we want to substitute the infinite-dimensional space of smooth H-reduced fields by some finite-dimensional (at least locally) space of discrete H-reduced fields.We also need to relate H-reduced fields with a discrete counterpart, and to derive a variational principle on these discrete fields from the given smooth H-reduced Lagrangian function ℓ.
Discretization mechanisms substitute partial derivatives of a smooth field with some kind of difference between the configurations of the field at two given points.In this process it is desirable that the main symmetries and geometrical properties of the smooth theory are retained in the discretized formulation.This was achieved in [6], where the corresponding discrete notions and discretization mechanism were introduced using forward difference operators in a covariant way.In the reduction and discretization of elements of the variational theory on a principal G-bundle P , a relevant object was Ehresmann's groupoid of fiber-to-fiber endomorphisms.
On a principal bundle π : P → X the natural mapping (p, g) ∈ P × G → (p, gp) ∈ P × X P establishes a diffeomorphism, whose inverse mapping takes the form (p, p) ∈ P × X P → (p, pp −1 ) ∈ P × G determining the group difference map (p, p) ∈ P × X P → pp −1 ∈ G, characterized by (pp −1 )p = p.This way of computing differences, however, can be performed only on pairs belonging to the same G-orbit and the difference is not preserved when we apply left translations λ g : P → P on the pair.
For any pair of elements p 0 ∈ P x0 , p 1 ∈ P x1 , if we want to fix some G-covariant morphism ψ : P → P such that ψ(p 0 ) = p 1 , there might be several choices, but on the x 0 -fiber the choice is totally determined, indeed for any p ∈ P x0 we may write p = (pp −1 0 )p 0 and consequently

Definition 2.8
For any pair of elements p 0 ∈ P x0 , p 1 ∈ P x1 we call fiber-to-fiber endomorphism induced by (p 0 , p 1 ), denoting it as p −1 0 p 1 , the unique G-covariant mapping ψ : P x0 → P whose domain is the G-orbit of p 0 and such that ψ(p 0 ) = p 1 .This mapping takes the form p ∈ P x0 → (pp −1 0 )p 1 ∈ P x1 , for the element pp −1 0 ∈ G defined as the group difference of p 0 , p ∈ P x0 , and has the following properties: Definition 2.9 We call fiber-to-fiber endomorphism on the principal G-bundle π : P → X any G-covariant morphism ψ : whose domain is a single fiber P x ⊆ P .The mapping s : End P → X taking a fiber-to-fiber endomorphism into its domain Dom(ψ) = P x ∈ P/G ≃ X is called the source mapping on End P .By G-covariance, the image of the G-orbit P x must be another G-orbit.The mapping t : End P → X taking a fiber-to-fiber endomorphism into its image Img(ψ) = P x1 ∈ P/G ≃ X is called the target mapping on End P .
For any fiber-to-fiber endomorphism ψ and for any element in its source fiber p 0 ∈ P s(ψ) , the image of p 0 by ψ shall be denoted as p 0 ψ.This element lies in P t(ψ) ⊂ P .

Remark 2.2
As proved in [6], the mapping (p 0 , p 1 ) ∈ P × P → p −1 0 p 1 ∈ End P that transforms any pair into its associated fiber-to-fiber endomorphism is surjective and its fibers are the orbits in P × P by the diagonal action λ g × λ g .Therefore there exists an identification (P × P )/G ≃ End P and the set End P has a natural smooth structure.The projectors s : End P → X and t : End P → X are smooth and (s, t) : End P → X × X is called Ehresmann's gauge groupoid.
There is a natural product (composition) of any fiber-to-fiber endomorphism with target x 1 with any fiber-tofiber endomorphism with source x 1 , determining a smooth groupoid product (ψ 0 , ψ 1 ) ∈ End P × (t,s) End P → ψ 1 • ψ 0 ∈ End P .The manifold End P has a groupoid structure using (s, t) : End P → X × X as anchor mapping and • as product.
For notational convention the image of p ∈ P x by ψ ∈ (End P ) x will be denoted pψ ∈ P t(ψ) and the composition ψ 1 • ψ 0 as ψ 0 • ψ 1 .In this notation • represents the reverse product structure on the set End P .We have, moreover: The isotropy group bundle associated to a groupoid bundle (s, t) : End P → X × X is defined as the subset of elements that have a coincident source and target.This determines a bundle Gau P → X called the gauge group bundle, whose elements are G-covariant automorphisms defined on a single fiber P x .Recall that the flow associated to G-invariant vector fields on a fiber is always given by G-covariant automorphisms of this fiber.Therefore the flow ϵ ∈ R → ϕ ϵ ∈ Aut(P x ) of a x ∈ Ad P x takes values in the gauge group bundle, and determines an exponential mapping (ϵ, a x ) ∈ R × Ad P x → exp ϵa x ∈ Gau P x that allows to interpret Ad P x as the Lie algebra of the Lie group Gau P x .
Discretization will replace partial derivatives at a single point, with a sample of configurations at several points.This leads to a pair of relevant definitions: Definition 2.10 For any set X and each k ∈ N we denote X ×k = X × . . .× X the direct product of k + 1 copies of the set X. Hence X ×0 = X, X ×1 = X × X and so on.Elements in X ×k can be seen as finite ordered sequences in X consisting of k + 1 terms.An ordered sequence (x 0 , x 1 , . . ., x k ) ∈ X ×k without repeated terms shall be called an ordered abstract simplex (being an abstract simplex any finite subset of X with exactly k + 1 elements).
In a similar way, for any bundle s : E → X we denote by E × s k = E× s . . .× s E the fibered product of k + 1 copies of the bundle s : E → X. Elements in E × s k can be seen as a point x ∈ X together with a finite ordered sequence (e 0 , e 1 , . . ., e k ) consisting of k + 1 terms on the fiber E x associated to the given point, that is, for the projection s : One of the most relevant results in [6] is that fixing a locally defined mapping ∆ G : End P → T P/G with specific particular properties (projectable faithful reduced forward difference operator), delivers all the mechanisms that we shall use to describe a discrete H-reduced variational problem.Specifically from ∆ G one may derive: • An open domain X ⊂ X ×n (domain of regular facet configurations) and a smooth function vol : X → R (the discrete volume function) associated to any volume form vol X ∈ Ω n (X).• An open domain JP ⊂ P ×n (discrete jet space) projecting to X by π ×n and an injective local diffeomorphism J P X : JP → JP × (jπ,π0) X ×n (forward Jacobi operator) relating this domain with a corresponding open domain on the pull-back of the jet bundle jπ : JP → X by the projector π 0 : (x 0 , . . . ,x n ) ∈ X ×n → x 0 ∈ X.This determines a smooth function L d : JP → R (the discrete Lagrangian function) associated to any Lagrangian density Lvol X , for L ∈ C ∞ (JP ).
• An open domain CP ⊂ (End P ) × s n−1 (discrete connection space) projecting to X by (s, t ×n−1 ) and an injective local diffeomorphism J CP : CP → CP× (πCP,π0) X ×n (reduced forward jacobi operator) relating this domain with a corresponding open domain on the pull-back of the bundle of principal connections π CP : CP → X by the projector π 0 : X ×n → X.This determines a smooth function ℓ d : HStr× X CP → R (the discrete H-reduced Lagrangian function) associated to any H-reduced Lagrangian density ℓvol X for ℓ ∈ C ∞ (HStr× X CP).
Moreover, both the forward jacobi and reduced forward jacobi operators are covariant with respect to the naturally induced actions of any gauge transformation ϕ : P → P on the bundles P , JP , HStr, CP and End P (see [6]).

Discretization of space
The foundations of differential calculus on abstract manifolds lie on choices of local charts, that is, locally defined topological immersions x : R n → X, which may be restricted to open hypercubic domains R ⊆ R n .Stating that a point x ∈ X has coordinates r = (r 1 , . . ., r n ) ∈ R ⊂ R n in this local chart is the same as stating that x = x(r 1 , . . ., r n ).Any function f : X → R can be expressed in a local chart as a real function in n real variables f (r 1 , . . ., r n ) = f • x : R n → R, defined on an open subset and differential calculus is then performed on these real functions with several real variables.A geometrical treatment of differential calculus warrants that even though computations depend on local coordinates, we may define notions with an intrinsic meaning, independent of a possible change of one local coordinate chart to another using as coordinate transformations any diffeomorphism between any two arbitrary open subsets on R n .
To obtain a difference calculus on an abstract n-dimensional manifold a classical choice is to substitute the smooth local chart and fix a cartesian grid on the manifold, a particular (possibly locally defined) immersion x : Z n → X that associates to each vertex v ∈ Z n a node x(v) ∈ X.In this way, for any function f ∈ C ∞ (X) we have a discrete counterpart f d = f • x and the corresponding difference notions at any node (v) where for each i = 1 . . .n, t i : Z n → Z n stands for the unit translation in the i-th direction of Z n .
For the purposes of variational calculus, where we will deal with Lagrangian densities and connections, we must extend our interest to include also edges and facets and not just vertices.That is, we will not work with a simple set of vertices but with a richer structure, that of an abstract cellular complex [4,5].More specifically, in order to take advantage of (reduced) forward jacobi operators we shall work on a simple structure: abstract simplicial complexes V modelled on Z n .

Definition 3.1 Consider the integer interval
array with diameter N .We call array with diameter N on an n-dimensional manifold X any immersion x : V → X.
Elements in the n-dimensional array of diameter N are points (k 1 , . . ., k n ), where k i ∈ {1, . . ., N }, ∀i = 1 . . .n.An array with diameter N on X is therefore a choice of N n points on X, indexed as (x v ) v∈V for array values v = (k 1 , . . ., k n ) ∈ V .To make a distinction, we shall talk of vertices when using elements of the n-dimensional array V and nodes when we use points x v ∈ X, in the image of some array on X.

Remark 3.1
For simplicity we shall assume that a given total order is fixed on the set V , for example the lexicographical order.In this way, subsets of V can be identified with monotone sequences (v 0 , v 1 , . . ., v m ) on V , and subsets of a set can be given as subsequences (v j0 , v j1 , . . ., v j k ), where 0 ≤ j 0 < j 1 < . . .< j k ≤ m.
To introduce a variational formalism for sections of a discrete bundle Y d → V we need additional structure on V , which will allow to talk of discrete analogues of differential forms and its integration on compact domains.This is achieved using a cellular complex that uses V as set of vertices (see [4]).In the present work we shall construct a cellular complex using abstract polytopes.For a geometrical description of polytopes, see the appendix.

Definition 3.2
A nonempty finite subset of points β = {v 0 , v 1 , . . ., v k } ⊂ R n is called an abstract polytope if none of its points is a convex combination of the remaining ones.An abstract polytope is called an abstract simplex if none of the points is an affine combination of the remaining ones.For abstract simplices with k + 1 elements, the affine space generated by them has dimension k.
For an abstract simplex β we call dim β = ♯β − 1.For arbitrary polytopes we call dimension of the abstract polytope the dimension of the affine space generated by those points, dim β = dim⟨β⟩.Abstract polytopes with dimension 0 are precisely subsets with a single point and are called vertices, abstract polytopes with dimension 1 are precisely subsets with two points and are called edges.For abstract polytopes with dimension higher than 1, the abstract polytope might not be an abstract simplex.Abstract polytopes in R n with dimension n are called facets.
For two abstract polytopes α, β ⊂ R n we say that α is a face of β and write α ≺ β if the convex hull [α] is a face of the convex hull [β].For abstract simplices, all its nonempty subsets are faces.
It is known that the relation of being a face is a symmetric, transitive, and anti-reflexive relation on the abstract polytopes.Moreover, it defines a T 0 topology: We say a subset of abstract polytopes U is closed if for each of its abstract polytopes β ∈ U , the associated faces also belong to U:

Definition 3.4
We call n-dimensional polytopal complex any finite family V of polytopes on R n that is closed and whose maximal elements (with respect to ≺) are n-dimensional abstract polytopes.
We say the abstract polytopal complex is modelled on a family of points V ⊂ R n if all vertices of the family belong to V .The subset of k-dimensional abstract polytopes of the complex is denoted by As V is closed, we may always assume that it is modelled on V 0 .In the following, we shall write V instead of V 0 to represent the set of vertices of a polytopal complex V.
It is clear from the definition that, in order to give an n-dimensional polytopal complex it suffices to give an arbitrary finite family V n (that we call facets of the complex) representing its n-dimensional abstract polytopes.The complex V is then given as the closure of V n , the family of all faces of the facets β ∈ V n .

Definition 3.5
For a given abstract polytope of an abstract polytopal complex α ∈ V, we call star associated to α the family of abstract polytopes of the complex that have α as face: The star can be decomposed using the dimension, as the disjoint union of the k-dimensional stars Let V be an abstract polytopal complex on a set V .We call domain on V any subset of facets K ⊂ V n .Any such domain generates a corresponding sub-complex K ⊂ V.
An abstract polytpe α ∈ V is said to be adherent to the domain K if it is a face of some facet in K.The set of adherent polytopes of a domain K shall be denoted by K.
said to be a frontier polytope of the domain K if it is adherent and non-interior to K. The set of frontier polytopes of a domain K shall be denoted by fr Starting from the n-dimensional array V with diameter N we shall construct two abstract polytopal complexes: the Cartesian complex, and the Coxeter-Freudenthal-Kuhn simplicial complex.These two abstract complexes represent a discrete model for the space [1, N ] n ⊆ R n , described identifying abstract cells with simplices or hypercubes obtained when we split R n using a particular family of hyperplanes.Fix r 1 , . . ., r n as canonical coordinates in R n .

Proposition 3.1
Consider the hyperplanes r i = k on R n , one hyperplane for each i = 1 . . .n and each k ∈ Z.Consider the set H ⊂ R n formed as the union of all these hyperplanes.
The connected components of R n \ H are the open hypercubes: where k i = ⌊r i ⌋ (here ⌊•⌋ stands for the floor function).The closure of these hypercubes, Cv = {v + (ϵ 1 , ϵ 2 , . . ., ϵ n ) : 1 ≥ ϵ k ≥ 0, ∀k = 1 . . .n}, are compact convex sets whose extremal points are: We call abstract cartesian hypercube determined by an element v ∈ Z n the set Consider V the n-dimensional array with diameter N .We call cartesian complex V cart on V the abstract polytopal complex generated by all abstract cartesian hypercubes contained in V .
Enlarging our family of hyperplanes we may do a partition of these hypercubes into simplices, a partition that appears in different applications (see [37] for example) and had independent origins by Coxeter, Freudenthal and Kuhn: Proposition 3.2 Consider the hyperplanes r i = k and the hyperplanes r i1 − r i2 = k on R n , one hyperplane for each k ∈ Z and each integer i = 1 . . .n or pair of integers 1 ≤ i 1 < i 2 ≤ n.Consider the set H ⊂ R n formed as the union of all these hyperplanes.
The connected components of R n \ H are the open affine simplices: where k i = ⌊r i ⌋ and σ is the permutation that determines the decreasing order of the fractional components The closure of these affine simplices, Kv,σ = {v + ϵ 1 e σ(1) + . . .+ ϵ n e σ(n) , 1 ≥ ϵ 1 ≥ ϵ 2 ≥ . . .≥ ϵ n > 0}, are compact convex sets whose extreme points are: That is, all extremal points of the affine simplex Kv,σ are obtained following a path starting at v and sequentially adding all vectors of the canonical basis, following the order determined by σ.

Proof
Recall that when we remove any hyperplane from R n this space is disconnected into two convex subsets (open half-spaces).Moreover, intersection of convex subsets is again convex.Points in R n \ H are those elements r ∈ R n that have noninteger (condition r i / ∈ Z) coordinates (r 1 , . . ., r n ), and whose fractional components f i = r i − ⌊r i ⌋ are all distinct (condition r i1 − r i2 / ∈ Z).Consider the following sets Each of these sets is contained in R n \ H, as none of its points have integer coordinates or coincident fractional component on any pair of coordinates.Conversely, any point r / ∈ H belongs to some K v,σ for some v = (k 1 , . . . ,k n ) and σ ∈ Sym n , it suffices to take k i = ⌊r i ⌋ and σ the permutation that re-arranges the fractional values 0 < f i = r i − k i < 1 in decreasing order.Moreover, if we call ϵ i = f σ(i) we have: by this isomorphism, and the extreme points are the image of (0, . . ., 0), (1, 0, . . ., 0), (1, 1, 0 . . ., 0) . . .by this isomorphism.The closure and extreme points are then the ones given in our statement.
As we have a decomposition of R n \ H as the disjoint union of several open convex (and therefore connected) sets, we conclude that these sets are the connected components in R n \ H.

Definition 3.8
We call abstract CFK simplex determined by an element v ∈ Z n and a permutation σ, the set Consider V the n-dimensional array with diameter N .We call CFK complex on V the abstract polytopal complex V CF K generated by all abstract CFK simplices contained in V .
In figure 1 we represent the CFK simplicial partition of a single hypercube, in R 3 and in R 2 :

Discrete H-reduced variational principles on the CFK simplicial complex
On the discrete manifold V , seen as a totally disconnected 0-dimensional smooth manifold, all usual objects of bundle theory are available, with the particularity that we are dealing with non-connected manifolds and that certain tangent spaces are the trivial null space.In particular we may consider bundles and principal G-bundles on V .

Definition 4.1
We call discrete bundle on a set V any surjective mapping π : We say the discrete bundle π : P d → V is a discrete principal G-bundle if its fibers are the G-orbits for some proper, free, smooth action λ : G → Aut(P d ) of the Lie group G on P d .
Observe that a discrete bundle is, in particular, a non-connected smooth manifold.As the base manifold is discrete, any tangent vector in the manifold Y d is always vertical with respect to the projector π : Y d → V .We shall call vertical bundle π V : V Y d → Y d associated to the discrete bundle Y d , the vector bundle whose fiber at y v ∈ Y d is the tangent space at y v of the fiber This is a smooth vector bundle with a smooth base manifold Y d .
Moreover, the restriction of a smooth bundle on an n-dimensional manifold X to any particular array with diameter N on this manifold generates a discrete bundle: Definition 4.2 Consider x : V → X an array with diameter N on an n-dimensional manifold X.For any bundle π : Y → X, we call discrete bundle induced by Y on the array x, the bundle π x : Y x = x * Y → V .As a manifold, Y x is the disjoint union of the fibers Y xv and the projector onto V is the determination of the vertex v ∈ V associated to π(y) ∈ X.
Any section y ∈ Γ(Y ) of the bundle π : Y → X naturally induces a section y x = y • x ∈ Γ(Y x ), of the discrete bundle induced by the array x : V → X.
Considering that discrete bundles are ordinary bundles where the base manifold is a discrete 0-dimensional manifold, for a discrete principal G-bundle we may use most of the main objects in principal bundle theories: For any discrete principal G-bundle P d → V and any closed subgroup H of G, there exists an induced discrete bundle, the bundle of discrete H-structures π HStr : P d /H = HStr d → V .This is a smooth manifold with several connected components, and P d itself is a (non-discrete) principal H-bundle using HStr d as (non-discrete) base manifold.We denote π H : P HStr → HStr d this smooth principal H-bundle.

Definition 4.4
The vertical bundles V π G P d → P d and V π H P HStr → P HStr determine the corresponding quotient bundles, a discrete one Ad P d = (V P d )/G → V and a smooth one Ad P HStr = (V P HStr )/H → HStr d , adjoint bundles of the principal bundles P d → V and P HStr → HStr.
There is a natural inclusion Ad P HStr ⊆ HStr× V Ad P d .

Definition 4.5
A discrete principal bundle π : P d → V determines a corresponding discrete Ehresmann bundle End P d → V × V , and for a fixed closed subgroup H ⊆ G, also a corresponding smooth Ehresmann bundle End P HStr → HStr × HStr.

Remark 4.1
From [6] we recall that: • The differential of the gauge difference mapping π G : P d × P d → End P d in the first component determines an isomorphism (source trivialisation of V End P d ): where the pull-back of Ad P d → V is performed using s : End P d → V .In this identification, (ψ α , a s(α) ) ∈ End P d × (s,π) Ad P d is identified with the tangent vector at ϵ = 0 of the trajectory exp(−ϵa) • The differential of the quotient mapping π H : P d → HStr d determines a projector: with a pull-back performed using π HStr : HStr d → V .The kernel of this projector is the sub-bundle Ad P HStr ⊆ π * HStr Ad P d .In this identification, (q v , a v ) ∈ HStr d × V Ad P d is identified with the tangent vector at ϵ = 0 of the trajectory q v exp ϵa ∈ HStr v .
We introduce now the specific facets and edges of our simplicial complex V, which includes additional information not available in the discrete bundle P d → V .In the following, V ⊂ Z n will be the n-dimensional array with diameter N , where we consider the Coxeter-Freudenthal-Kuhn simplicial complex.For simplicity we shall assume that the lexicographical order is fixed on V ⊂ Z n and all subsets of V will be given as (lexicographically) monotone sequences of vertices.Edges will then be given as ordered pairs (v 0 , v 1 ), where the vertex v 0 precedes in the lexicographical order the vertex v 1 .The ordering choice leads to an identification of V 1 as a particular subset of V × V , and in general an identification of V k as a subset of V ×k .Moreover, the projectors π i : For simplicial facets and the case i 0 < i 1 , we know that π i0i1 takes values on V 1 ⊂ V × V .In a similar way we may define projectors π i0...i k : For any sequence of indices For any k ≥ 1, we call Ehresmann bundle on k-simplices, For any sequence of indices 0 < i 1 < . . .< i k ≤ j there exists a projector π End 0i1...i k : End j P d → End k P d transforming the sequence (ψ 01 , ψ 02 , . . ., ψ 0j ) on the simplex (v 0 , v 1 , . . ., v j ) ∈ V j ⊂ V ×j into the sequence (ψ 0i1 , ψ 0i2 , . . . ,ψ 0i k ) on the simplex For any sequence of indices 0 < i 0 < i 1 . . .< i k ≤ j, there also exists a projector π End i0i1...i k : End j P d → End k P d transforming the sequence (ψ 01 , ψ 02 , . . ., ψ 0j ) on the simplex Particularly relevant will be Ehresmann bundle on edges We have just introduced all the relevant elements for the formulation of variational principles on discrete Hreduced fields.

Definition 4.8
For a given discrete principal G-bundle P d → V and closed subgroup H ⊆ G, we call discrete H-reduced field, any pair (q, ψ) determined by a section q ∈ Γ(HStr d ) of the associated bundle of discrete H-structures HStr d = P d /H → V on vertices, and another section ψ ∈ Γ(End 1 P d ) of the associated Ehresmann bundle (s, t) : End 1 P d → V 1 on edges of the CFK complex.
Observe that being all fibers of HStr d and End 1 P d finite-dimensional manifolds, and being both V and V 1 finite sets, the space Γ(HStr d ) × Γ(End 1 P d ) of H-reduced fields is a manifold with finite (but large) dimension Using that the tangent space of a product is the direct sum of tangent spaces of its components, the tangent space at a given point (q, ψ) ∈ Γ(HStr d ) × Γ(End 1 P d ) has a canonical identification: Definition 4.9 We call space of infinitesimal variations of an H-reduced discrete field (q, ψ) ∈ Γ(HStr d ) × Γ(End 1 P d ) the vector space Γ(q * V HStr d ) ⊕ Γ(ψ * V End 1 P d ).Its elements shall be denoted by (δq, δψ), with components Moreover, using for each ψ α ∈ End P d the natural source trivialisation V ψα End P d ≃ Ad P s(ψα) and for each q v ∈ HStr d the natural identification V qv HStr d ≃ Ad P πHStrqv / Ad P HStr qv we have an identification: The first component is a section of a vector bundle Ad P d → V on vertices.The second component is a section of a vector bundle s * Ad P d → V 1 on edges.
For the dual space: where the inclusion is as elements whose first component (related to vertices) are linear forms on Ad P v that vanish on the corresponding subspaces Ad P HStr qv ⊆ Ad P v .

Definition 4.10
We call functional on the space of discrete H-reduced fields, any smooth function L defined on an open subset of the manifold Γ(HStr d ) × Γ(End 1 P d ).A discrete H-reduced field (q, ψ) is said to be strongly critical for the functional L, if the differential of this function at this point vanishes.
Using the representation of the tangent space as Γ(q * V HStr d ) ⊕ Γ(ψ * V End 1 P d ), the differential of L at any point is characterized by its components d v (q,ψ) L ∈ T * qv HStr v (one component for each v ∈ V ) and d α (q,ψ) L ∈ T * ψα End 1 P α (one component for each α ∈ V 1 ).It seems, however, more convenient to express them as linear forms on Ad P d (we get an object not depending on the particular point (q, ψ)).
Using the identification (6), for any point (q, ψ) ∈ Γ(HStr d ) × Γ(End 1 P d ) there exist unique sections ∂ 0 (q,ψ) L ∈ Γ(Ad * P d ) and ∂ 1 (q,ψ) L ∈ Γ(s * Ad * P d ) such that ∂ 0 (q,ψ) L vanishes on Γ(q * Ad P d ), and for which holds: , and where a 0 ∈ Γ(Ad P v ) stands for any representative in the class [a 0 ] ∈ Γ(Ad P d /q * Ad P HStr ).The duality product of sections of a bundle and sections of the dual bundle is performed as the summation of duality applied at all of the fibers.

Proof
Criticality is by definition the annihilation of d (q,ψ) L, which means the annihilation of all its components.The result is obtained by the natural identifications or immersions of these components in fibers of the dual adjoint bundle.
However we shall not consider arbitrary functionals on the space of discrete H-reduced fields, but just action functionals derived from a discrete Lagrangian function.Moreover, we shall not be concerned with strongly critical points of this functional, but only with critical points, with respect to a certain constrained variational principle.We begin with the introduction of action functionals.Recall that given a smooth bundle Y → X, a Lagrangian density Lvol X and a compact domain K ⊂ X, we can define a functional L K : y ∈ Γ(Y ) → L K (y) ∈ R on the space of fields y ∈ Γ(Y ) using the following steps: We want to parallel this process for the case of discrete H-reduced fields following a scheme: where ex : Γ(HStr d ) × Γ(End 1 P d ) → Γ(RJP d ) should be the discrete operator that extends a given section to a new section, representing the corresponding discrete analogue of a H-reduced jet rj ∈ Γ(JP/H).

Definition 4.11
For any discrete principal G-bundle P d → V and closed subgroup H ⊆ G, we call In the smooth theory JP/H = HStr× X CP, hence an H-reduced jet is composed of a point x ∈ X, a single Hstructure q x ∈ HStr x , and a principal connection element χ x ∈ CP x at this point, which represents a determination of horizontal lifts δ ∈ T x X → χ x (δ) ∈ (T P/G) x for n independent directions δ ∈ T x X.For our notion of discrete H-reduced jet configuration, the information needed at any fixed facet β is a single H-structure at the vertex π 0 (β) ∈ V , and n fiber-to-fiber endomorphisms for n edges π 0i (β) ∈ V 1 .
There are natural projectors defined in RJP d .We denote π RJP 0 There exists a natural identification of the vertical bundle

Proof
Using the definition of RJP d as a fibered product, the projectors to each of its components determine an isomorphism: using now the natural identification V q HStr d ≃ Ad P π(q) / Ad P HStr q and the source trivialisation V ψ End 1 P d ≃ Ad P s(ψ) , one obtains the result for the vertical bundle.
In the dual vertical bundle, it suffices to observe that the quotient morphism E → E/F of any vector bundle E by some sub-bundle F induces a corresponding immersion of dual bundles (E/F ) * → E * , identifying (E/F ) * as the sub-bundle of linear forms on E that vanish when applied to F ⊂ E. Observe that every discrete H-reduced field (q, ψ) ∈ Γ(HStr d ) × Γ(End 1 P d ) determines a corresponding discrete H-reduced jet ex(q, ψ) ∈ Γ(RJP d ), namely a section β ∈ V n → ex β (q, ψ) ∈ RJP d defined by: ) and has then a smooth manifold structure.We may denote by rj 0 β ∈ HStr π0(β) and by rj 0i β ∈ End 1 P π0i(β) the corresponding components of each section rj ∈ Γ(RJP d ).Consequently, the tangent space of Γ(RJP d ) at any rj ∈ Γ(RJP d ) is a direct sum: The mapping ex : Γ(HStr d ) × Γ(End 1 P d ) → Γ(RJP d ) is smooth.The differential at any given (q, ψ) is described in components (δq v , δψ α ) → (δrj 0 β , δrj 0i β ) as:

Proof
The mapping ex is merely a re-ordering (with possible repetitions) of the several components q v , ψ α in (q, ψ) to give the different components of ex(q, ψ).It can be described as: As a consequence, it is a smooth mapping, and the induced mapping on the tangent spaces is the one given in the statement.
We may observe from (7) that sections rj ∈ Γ(RJP d ) that are obtained in the form ex(q, ψ) must always satisfy the relations: In the same manner as not all sections of a jet bundle are a jet extension of a field, also not all sections of the discrete bundle of H-reduced jet configurations are a reduced jet extension of some discrete H-reduced field (q, ψ).  1 and an H-reduced Lagrangian function can be obtained considering a single smooth function ℓ : HStr× X (End P ) × s n−1 → R and its corresponding restriction to a finite family of fibers (x(v 0 ), . . ., x(v n )) ∈ X ×n , for all choices (v 0 , . . ., v n ) ∈ V n ⊂ V ×n .In the case of a trivial principal G-bundle this class of H-reduced Lagrangian functions can be written in the form ℓ(x 0 , . . . ,x n , Hg, g 01 , . . . ,g 0n ) for some smooth function on jet configurations).We say a discrete H-reduced field (q, ψ) belongs to the regular domain if its H-reduced jet extension ex(q, ψ) ∈ Γ(RJP d ) takes values in the regular domain RJP d .In this case, we say that the discrete action functional L d determined by ℓ d takes at (q, ψ) the following value: We say an H-reduced field (q, ψ) is strongly critical for the H-reduced Lagrangian function ℓ d , if it is strongly critical for the associated functional L d .
Observe that, in the same way that there might not exist globally defined smooth sections of a bundle, in the discrete case, where the existence of globally defined sections is guaranteed, the condition that the extension of the section to facets should lie on a certain domain of regular configurations might be too strong.For certain choices, the domain of regular configurations could be so small that the discrete action functional cannot be computed on any global section.
To compute strongly critical points for some H-reduced Lagrangian function ℓ d , one must compute the derivative of the locally defined function Using the chain rule, this is performed computing the differential of each function ℓ β , for each facet β ∈ V n , and the differential of ex, given in proposition (4.3), leading to: ), of the discrete action functional L d , associated to some discrete H-reduced Lagrangian density ℓ d = (ℓ β ) β∈V n , has components depending on ℓ β through: Corollary 4.1 When the action functional is defined by a discrete H-reduced Lagrangian function ℓ : RJP d → R, strongly critical discrete H-reduced fields will be characterized by: However we shall not seek for H-reduced fields that ate strongly critical.The discrete variational principle will consider only a certain subset of discrete H-reduced fields (admissible fields) and a certain subspace of (admissible) infinitesimal variations for these fields.

Constrained variational principles on discrete H-reduced fields
In a similar way as was done in the smooth theory of Euler-Poincare reduction, we shall look for global H-reduced fields, that are critical points for the discrete action functional, but only with respect to some restricted set of variations.

Definition 5.1
We say that (q, ψ) ∈ Γ(HStr d ) × Γ(End 1 P d ) is an admissible H-reduced field if for any edge (v 0 , v 1 ) ∈ V 1 ⊂ V × V there holds q v0 ψ v0v1 = q v1 and for any 2-simplex This definition recovers the notion of parallel H-structure and flat connection assumed for admissible Hreduced fields in the smooth theory.In order to determine a constrained variational principle on discrete Hreduced fields, we still need to fix compatible subspaces of admissible infinitesimal variations, for each admissible (q, ψ) ∈ Γ(HStr d ) × Γ(End 1 P d ).

Definition 5.2
For any admissible discrete H-reduced field (q, ψ) ∈ Γ(HStr) × Γ(End 1 P ), we call space of admissible variations AV (q,ψ) , the image of the linear operator: We are in situation to define the constrained variational problem in discrete H-reduced fields.It is given as: • An abstract (ordered) simplicial complex V modelled on a finite set of vertices V 0 = V .
• A discrete principal G-bundle P d → V , its associated discrete Ehresmann groupoid on edges π End : End 1 P d → V 1 and discrete adjoint bundle π Ad : Ad P d → V on vertices.In this situation we may define: • The manifold of discrete H-reduced fields Γ(HStr d ) × Γ(End 1 P d ), and the subset of admissible fields, characterized for having a flat discrete connection that can be reduced to the H-structure (we may also say that the H-structure is parallel).• For each admissible H-reduced field (q, ψ), a corresponding space of admissible infinitesimal variations AV (q,ψ) ⊂ Γ(q * V HStr d ) ⊕ Γ(ψ * V End 1 P d ).
• An action functional L d : Γ(HStr d ) × Γ(End 1 P d ) → R determined by the H-reduced Lagrangian function ℓ d .

Definition 5.3
Se say an admissible discrete H-reduced field (q, ψ) ∈ Γ(HStr d ) × Γ(End 1 P d ) is critical for the constrained variational problem determined by the discrete H-reduced Lagrangian function ℓ d , if the differential d (q,ψ) L d of the action functional vanishes on the subspace of admissible infinitesimal variations AV (q,ψ) .
Obviously, in the case that there exists a strongly critical, admissible H-reduced field (q, ψ) for ℓ d , it will be critical for the constrained variational problem determined by ℓ d .
As any section a ∈ Γ(Ad P d ) is a finite sum of sections of the given form, we conclude that criticality is equivalent to the vanishing of EP(q, ψ) ∈ Γ(Ad * P d ) defined by: To express this element in terms of the discrete H-reduced Lagrangian function ℓ d : RJP d → R, we may use (8) to obtain: Ad * ψ π 0i (β) ∂ 0i rj β ℓ β

Integrators for discrete Euler-Poincaré equations
Observe the structure of discrete Euler-Poincaré equations (9).Suppose we have an admissible discrete H-reduced field (q, ψ) ∈ Γ(HStr d ) × Γ(End 1 P d ), that is critical for a certain discrete Lagrangian function ℓ d .
Let us decompose the space Z n into slices S c defined by equations k 1 + k 2 + . . .+ k n = c.Consider a vertex u ∈ Z n in a given slice S c+n .Assume that all values q w , ψ v0w are known for vertices w in the region k 1 + . . .+ k n < c + n.What can be said about the values q u and ψ v0u ?
If we decompose Euler-Poincaré equations at v into components that depends on rj β for π 0 (β) = v and another one that depends on rj β for π i (β) = v, i = 1 . . .n, (9) is equivalent to: When q w , ψ v0w are known in the region k 1 + . . .+ k n < c + n, the right hand side in the equations is known, and most components on the left hand side are also known, except for the particular component ψ vu with u = v + (1, . . ., 1).If dim G = m (and consequently dim Ad * P v = m, dim End 1 P uv = m) we have a system of m equations with ψ vu as unknown that taking into account the dimensions, in some regular cases, will determine a unique solution.
For any subset of indices S ⊂ [n] = {1, 2, . . ., n}, denote e S the vector with component 1 for any index in S and component 0 for any index not belonging to S. e S = (c 1 , . . . ,c n ), For the CFK simplicial complex we easily observe for a fixed vertex:

Definition 6.1
We call bundle of discrete H-reduced forward configurations, Forw H d → V , the bundle whose fiber on a vertex v ∈ V is the following End 1 P α(v,S) We call bundle of discrete H-reduced backward configurations, Back H d → V , the product bundle of all H-reduced jet configurations at facets containing v but not with source v: We observe that the decomposition of discrete Euler-Poincaré equations lead to a component where rj β = (q v , (ψ α ) α≺β ).
If these points are affinely independent points (none of them is an affine combination of the remaining ones), the abstract convex polytope is called an abstract simplex and the polytope is called a simplex.
We call dimension of an abstract polytope the dimension of the affine space generated by this set of points.For a simplex with m + 1 points, its dimension is m.Observe that all 1-dimensional polytopes are in fact segments [r, s] determined by two extreme points and therefore are simplices.
We say an abstract polytope α is a face of another abstract polytope β, if the associated convex hull ᾱ is a face of β.This determines an order relation α ≺ β in the set of abstract polytopes.
In the case of an m-dimensional simplex generated by {v 0 , . . ., v m }, the convex polytopes generated by any subset are faces of the simplex.In the case of an arbitrary polytope generated by {v 0 , . . ., v m }, a given subset of points generates a smaller polytope, but not necessarily a face.
be called admissible infinitesimal variations at the admissible local field y ∈ Γ S (Y ).) • A smooth function L : S → R. (Determining a horizontal n-form Lvol X called Lagrangian density, the corresponding action functional L K : Γ S (Y ) → R associated to any compact subset K ⊆ X, defined by (2) on the subset of admissible local fields, and the differential d y L : AV y → R of the action functional, well defined by (3) on the subspace AV y of admissible infinitesimal variations AV y at any admissible local field y ∈ Γ S (Y ).)An admissible local field y ∈ Γ S (Y ) is called critical for the variational problem if d y L vanishes on the space of admissible infinitesimal variations AV y .

Proposition 4. 2
Consider s : RJP d → V the source mapping and s * Ad P d → RJP d the pull-back of the adjoint bundle Ad P d → V to RJP d .Consider the sub-bundle (s * Ad P d ) HStr ⊂ s * Ad P d characterized by: (s * Ad P d ) HStr rj = Ad P HStr q
bundle of H-reduced jet configurations associated to P d .The bundle RJP d is a discrete bundle on V n whose fiber on any facet β = (v 0 , v 1 , . .., v n ) ∈ V n ⊂ V ×n is:RJP β = HStr v0 × (End n P ) β = HStr v0 × (End 1 P ) v0v1 × . . .× (End 1 P ) v0vn V RJP d → RJP d as: V RJP d ≃ (s * Ad P d )/(s * Ad P d ) HStr ⊕ (s * Ad P d )There exists an induced natural immersion of the dual vertical bundle V * RJP d → RJP d as sub-bundle of (s * Ad * P d ) ⊕n → RJP d (direct sum of n + 1 copies of the dual adjoint bundle).The fiber of this sub-bundle at any H-reduced jet configuration rj = (q, ψ 01 , . . ., ψ 0n ) ∈ RJP d with source s(rj) = v is: Definition 4.12For any smooth function ℓ : RJP d → R, the n + 1 components in Ad * P s(rj) determined by its differential at a given point rj ∈ RJP d are denoted by ∂ 0 VARIATIONAL INTEGRATORS FOR REDUCED FIELD EQUATIONS Definition 4.13 We call discrete H-reduced Lagrangian function, any locally defined function ℓ d : RJP d → R, whose domain is an open sub-bundle RJP d ⊂ RJP d → V n with non-empty fibers.Elements in the domain shall be called regular H-reduced jet configurations.As RJP d is a discrete union of its fibers RJP β , any discrete H-reduced Lagrangian function ℓ d can be seen as a family of smooth functions {ℓ β } β∈V n , each of them defined on a nonempty open subset of a fiber RJP β ⊂ RJP β .Following Definition 4.12, the components of the differential dℓ β at a certain point rj ∈ RJP β with s(rj) = π 0 (β) = v ∈ V , using the natural immersion V * rj RJP d ⊆ (Ad * P v ) ⊕n , shall be denoted by ∂ 0 rj ℓ β , ∂ 01 rj ℓ β ,. .., ∂ 0n rj ℓ β ∈ Ad * P v .Remark 4.3 In the case that the discrete principal bundle P d → V is the restriction P d = x * P ⊂ P of some smooth bundle P → X to some array x : V → X, we have HStr d = P d /H ⊂ P/H = HStr and End 1 P d ⊂ End P d ⊂ End P , therefore End n or in the simplest case ℓ has an expression ℓ(Hg, g 01 , . . ., g 0n ), smooth function defined on a manifold G/H × G × . . .× G. Consider a discrete H-reduced Lagrangian function ℓ d : RJP d → R, defined on a certain open domain RJP d ⊂ RJP d (of regular H-reduced • A closed subgroup H ⊆ G determining the associated discrete H-structure bundle π HStr : HStr d → V , principal H-bundle π H : P HStr → HStr d and the bundle of H-reduced jet configurations RJP d → V n .• A smooth, locally defined, function (discrete H-reduced Lagrangian) ℓ d : RJP d → R.
EP 1 defined on Forw H d and a second one EP 2 defined on Back H d .Definition 6.2 We call Legendre transformation associated to a discrete H-reduced Lagrangian ℓ d , the mapping Leg : Forw H d → Ad * P d defined on each fiber by: