Discrete Euler-Poincaré and Euler-Poincaré-Poisson Equations for Semidirect Products and Principal Bundles

D. Opriş and I.D. Albu

Abstract A variational approach for a discrete Lagrangian on a semidirect product

and on a princpal bundle is developed.

Mathematics Subject Classification: 17B30 Key Words: Semidirect product, principal bundle, discrete Euler-Poincaré equations, reduction-reconstruction.

1 Introduction

Dynamical systems with symmetry play an important role in mathematical modelling for several processes in physics, mechanics, economy etc. In the last time the variational description of systems with symmetries seems to be more interesting ([4], [3],

[2] etc). In [2], [3], the theory is developed for Lagrangian systems on Lie groups. A symmetry of the Lagrangian with respect to a group action leads to a reduced Lagrangian system which is described by the so called Euler-Poincaré equations.

Independently by all appearances, [5] and [1] propose a discrete analogues of Euler-Poincaré reduction theory for systems on Lie groups. Our study is maked knowing only the first paper, [5].

We propose a variational approach for a discrete Lagrangian system whose configuration space is a semidirect product or a principal bundle. By following the construction in [5] we consider a discrete Lagrangian L : S × S ⟶ R, where S = G ◯ V is a semidirect product and we write the discrete Euler-Lagrange equations. If L is left G-invariant, then we define a reduced discrete Lagrangian l : S × V ⟶ R and we deduced the discrete Euler-Poincré equations. In Section 3 a reduction-reconstruction theorem about the algorithms satisfying these discrete equations is formulated. A discrete Euler-Poincaré-Poisson algorithm is discussed in Section 4. In Section 5 a discrete variant of the Kelvin-Noether Theorem is established. The last Section is an attempt in to approach the discrete variational description for a G-invariant Lagrangian on a principal bundle and to obtain dicrete Euler-Poincaré equations and discrete Euler-Poincaré algorithms.

Balkan Journal of Geometry and Its Applications, Vol.4, No.2, 1999, pp. 93-102 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

2 Generalities on semidirect products

Let V be a finite dimensional vector space and assume that a finite dimensional Lie group G acts on the left by linear maps on V and also on its dual space V^{∗}. The action of g ∈ G on v ∈ V is denoted by simple concatenation gv. The semidirect product S = G ◯ V is the Cartezian product G × V endowed with the group multiplication given by (g_{1}, v_{1})(g_{2}, v_{2}) = (g_{1}g_{2}, v_{1} + g_{1}v_{2}). (1)

If e ∈ G is the identity and o ∈ V is the null vector of V, then (e, o) ∈ G × V is the identity element of S. The inverse of the element (g, v) is

(g, v)^{−1} = (g^{−1}, −g^{−1}v). (2)

The Lie algebra of S is the semidirect product Lie algebra S = G ◯ V, whose bracket is given by

[(ξ_{1}, v_{1}), (ξ_{2}, v_{2})] = ([ξ_{1}, ξ_{2}], ξ_{1}v_{2} − ξ_{2}v_{1}), (3)

where the induced action of G on V is also denoted by concatenation. As well the action of S on S is given by

(g, v)(ξ, u) = (gξ, gu − (gξ)v), (4)

where (g, v) ∈ S, (ξ, u) ∈ S and the action of S on S^{∗} = G^{∗} ◯ V^{∗} is given by

(g, v)(μ, a) = (gμ + ρ_{v}^{∗}(ga), ga), (5)

where (μ, a) ∈ S^{∗}, ρ_{v}^{∗} : V^{∗} ⟶ G^{∗} is the dual of the linear map ρ_{v} : G ⟶ V, ρ_{v}(ξ) = ξv. In (4) and (5) we denote gξ = Ad_{g}ξ, gμ = Ad_{g^{−1}}^{∗}μ (the adjoint and coadjoint actions).

For a ∈ V^{∗} and v ∈ V we shall denote ρ_{v}^{∗} = v◇a ∈ G^{∗} and the formula (5) becomes

(g, v)(μ, a) = (gμ + v◇(ga), ga). (6)

Using the concatenation notations for Lie algebra actions on G^{∗} and V^{∗}, ξμ, ξa, where ξ ∈ G, a ∈ V^{∗}, we obtain an alternative definition of v◇a ∈ G^{∗}: for v ∈ V, a ∈ V^{∗}, ξ ∈ G^{∗}, < ξa, v >= − < v◇, aξ >.

Sometimes it is convenient to use the right representations of G on V which changes unessentially the formulas (we note the adjoint and coadjoint actions are left actions).

3 The discrete Euler-Poincaré algorithm

Let L : S × S ⟶ R be a discrete Lagrangian on S = G ◯ V and let A : S^{N+1} ⟶ R be the associated action,

A = ∑_{k=0}^{N−1} L(g_{k}, v_{k}, g_{k+1}, v_{k+1}), (7)

where (g_{k}, v_{k}) ∈ S, k = 0, 1, …, N with (g_{0}, v_{0}), (g_{N}, v_{N}) fixed.

The discrete Euler-Lagrange equations (DEL) are given by

D_{3}L(g_{k−1}, v_{k−1}, g_{k}, v_{k}) + D_{1}L(g_{k}, v_{k}, g_{k+1}, v_{k+1}) = 0,

D_{4}L(g_{k−1}, v_{k−1}, g_{k}, v_{k}) + D_{2}L(g_{k}, v_{k}, g_{k+1}, v_{k+1}) = 0, (8)

where D_{i}L, i = 1, 2, 3, 4, is the matrix containing the partial derivatives with respect to the i−th variable.

Consider the quotient map π : S × S ⟶^{S×S} /_{G} =^{∼} S × V given by

π((g_{k}, v_{k}), (g_{k+1}, v_{k+1}) = (g_{k+1}^{−1}g_{k}, g_{k+1}^{−1}v_{k}, g_{k+1}^{−1}v_{k+1}). (9)

Using the notations f_{k+1k} = g_{k+1}^{−1}g_{k}, u_{k} = g_{k}^{−1}v_{k} it results g_{k+1}^{−1}v_{k} = f_{k+1k}u_{k} and π is given by π((g_{k}, v_{k}), (g_{k+1}, v_{k+1}) = (f_{k+1k}, f_{k+1k}, u_{k}, u_{k+1}). (10)

Assume that the discrete Lagrangian L : S × S ⟶ R is left G−invariant. The projection (10) defines the reduced discrete Lagrangian l : S × V ⟶ R by l ∘ π = L. Consequently

L(g_{k}, v_{k}, g_{k+1}, v_{k+1}) = L(g_{k+1}^{−1}g_{k}, g_{k+1}^{−1}v_{k}, e, g_{k+1}^{−1}v_{k+1}) =

= L(f_{k+1k}, f_{k+1k}u_{k}, e, u_{k+1}) = l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1}). (11)

The action associated to the reduced discrete Lagrangian l is a : (S×V)^{N+1} ⟶ R,

a = ∑_{k=0}^{N−1} l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1}). (12)

The DEL equations for reduced discrete Lagrangian l are called the discrete Euler-Poincaré equations (DEP). Theorem 3.1. Let L be a left G−invariant discrete Lagrangian on S × S and let l be the reduced discrete Lagrangian associated to L. For a finite sequence {(g_{k}, v_{k}), (g_{k+1}, v_{k+1})}, k = 0, 1, …, N − 1, in S × S, with (g_{0}, v_{0}), (g_{N}, v_{N}) fixed we define a corresponding sequence f_{k+1k} = g_{k+1k}^{−1}g_{k} in G and a sequence u_{k} = g_{k}^{−1}v_{k} in V. The following statements are equivalent:

1. The sequence {(g_{k}, v_{k}), (g_{k+1}, v_{k+1})}, k = 0, …, N − 1, is an extremum for the action A for arbitrary variations δg_{k}, δv_{k}, k = 1, 2, …, N − 1.

2. The sequence {(g_{k}, v_{k}), (g_{k+1}, v_{k+1})}, k = 0, …, N − 1, satisfies the DEL(8).

3. The sequence {f_{k+1k}, f_{k+1k}u_{k}, u_{k+1}}, k = 0, …, N − 1, is an extremum for the reduced action a with respect to variations δf_{k+1k}, δu_{k} induced by the variations δg_{k}, δv_{k}.

4. The sequence {f_{k+1k}, f_{k+1k}u_{k}, u_{k+1}}, k = 0, …, N − 1, satisfies the DEP:

D_{1}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1})Ad_{f_{k+1k}}T_{e}R_{f_{k+1k}} − D_{1}l(f_{kk−1}, f_{kk−1}u_{k−1}, u_{k})

T_{e}R_{f_{kk−1}} = Ad_{f_{k+1k}}^{∗}(f_{k+1k}u_{k}◇D_{2}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1})) − f_{kk−1}u_{k−1}

◇D_{2}l(f_{kk−1}, f_{kk−1}u_{k−1}, u_{k}), f_{k+1k}D_{2}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1})+

D_{3}l(f_{kk−1}, f_{kk−1}u_{k−1}, u_{k}) = 0. (13)

Proof. Let (g_{k}, v_{k}) : ε ∈ (−a, a) ↦ (g_{k}(ε), v_{k}(ε)) ∈ S, k = 0, …, N, a family of curves on S such that g_{k}(0) = g_{k}, v_{k}(0) = v_{k}. For this family the action of L is

A(ε) = ∑_{k=0}^{N−1} L(g_{k}(ε), v_{k}(ε), g_{k+1}(ε), v_{k+1}(ε)).

By considering the variations of g_{k}, v_{k},

δg_{k} = \frac{dg_{k}(ε)}{dε}|_{ε=0} , δv_{k} = \frac{dv_{k}(ε)}{dε}|_{ε=0}

and using the discrete analogue of integration by parts it results

δA = \frac{dA(ε)}{dε}|_{ε=0} = ∑_{k=1}^{N−1} [D_{1}L(g_{k}, v_{k}, g_{k+1}, v_{k+1}) + D_{3}L(g_{k−1}, v_{k−1}, g_{k}, v_{k})] δg_{k}

+ ∑_{k=1}^{N−1} [D_{2}L(g_{k}, v_{k}, g_{k+1}, v_{k+1}) + D_{4}L(g_{k−1}, v_{k−1}, g_{k}, v_{k})] δv_{k}.

Since the variations δg_{k}, δv_{k} are arbitrary it follows δA = 0 if and only if the DEL(8) holds.

Now consider the family of curves f_{k+1k}(ε) = g_{k+1}^{−1}(ε)g_{k}(ε), u_{k}(ε) = g_{k}^{−1}(ε)v_{k}(ε), with f_{k+1k}(0) = f_{k+1k}, u_{k}(0) = u_{k}. Using the concatenation notation it results

δf_{k+1k} = \frac{df_{k+1k}(ε)}{dε}|_{ε=0} = δg_{k+1}^{−1}g_{k} + g_{k+1}^{−1}δg_{k} = −g_{k+1}^{−1}δg_{k+1}g_{k+1}^{−1}g_{k}+

g_{k+1}^{−1}δg_{k} = −(g_{k+1}^{−1}δg_{k+1})f_{k+1k} + f_{k+1k}(g_{k+1}^{−1}δg_{k})f_{k+1k}^{−1}f_{k+1k} =
−T_{e}R_{f_{k+1k}}(g_{k+1}^{−1}δg_{k+1}) + Ad_{f_{k+1k}}T_{e}R_{f_{k+1k}}(g_{k}^{−1}δg_{k});

δu_{k} = \frac{du_{k}(ε)}{dε}|_{ε=0} = −g_{k}^{−1}δg_{k}g_{k}^{−1}v_{k} + g_{k}^{−1}δv_{k} = −(g_{k}^{−1}δg_{k})u_{k} + g_{k}^{−1}δg_{k}; (14)

δ(f_{k+1k}u_{k}) = \frac{df_{k+1k}(ε)u_{k}(ε)}{dε}|_{ε=0} = −(g_{k+1}^{−1}δg_{k+1})(f_{k+1k}u_{k})+

[f_{k+1k}(g_{k}^{−1}δg_{k})f_{k+1k}^{−1}] (f_{k+1k}u_{k}) + f_{k+1k}δu_{k} = −(g_{k+1}^{−1}δg_{k+1})(f_{k+1k}u_{k})+

Ad_{f_{k+1k}}(g_{k}^{−1}δg_{k})(f_{k+1k}u_{k}) + f_{k+1k}δu_{k}.

From L = l ∘ π it results a ∘ π = A and \frac{da(ε)}{dε}|_{ε=0} = \frac{dA(ε)}{dε}|_{ε=0} , where

a(ε) = ∑_{k=0}^{N−1} l(f_{k+1k}(ε), f_{k+1k}(ε)u_{k}(ε), u_{k+1}(ε)).

It follows

δa = \frac{da(ε)}{dε}|_{ε=0} = ∑_{k=0}^{N−1}[D_{1}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1})δf_{k+1k}+

+D_{2}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1})δ(f_{k+1k}u_{k}) + D_{3}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1})δu_{k+1}].

Using the discrete analogue of integration by parts we obtain

δa = ∑_{k=0}^{N−1}[D_{1}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1})Ad_{f_{k+1k}}T_{e}R_{f_{k+1k}}(g_{k}^{−1}δg_{k})−

D_{1}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1})T_{e}R_{f_{k+1k}}(g_{k+1}^{−1}δg_{k+1}) + f_{k+1k}u_{k}◇

D_{2}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1})(g_{k+1}^{−1}δg_{k+1}^{−1}) − f_{k+1k}u_{k}◇D_{2}l(f_{k+1k}, f_{+1k}u_{k}, u_{k+1})

(Ad_{f_{k+1k}}g_{k}^{−1}δg_{k}) + f_{k+1k}D_{2}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1})δu_{k}+ (15)

D_{3}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1})δu_{k+1}] = ∑_{k=1}^{N−1}[D_{1}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1})Ad_{f_{k+1k}}

T_{e}R_{f_{k+1k}} − D_{1}l(f_{kk−1}, f_{kk−1}u_{k−1}, u_{k})T_{e}R_{f_{kk−1}} + f_{kk−1}u_{k−1}◇

D_{2}l(f_{kk−1}, f_{kk−1}u_{k−1}, u_{k}) − Ad_{f_{k+1k}}(f_{k+1k}u_{k}◇D_{2}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1}))]

(g_{k}^{−1}δg_{k}) + ∑_{k=1}^{N−1}[f_{k+1k}D_{2}l(f_{k+1k}, f_{k+1k}u_{k}, u_{k+1}) + D_{3}l(f_{kk−1}, f_{kk−1}u_{k−1}, u_{k})]δu_{k}.

Because the variations δf_{k+1k}, δu_{k} are abitrary it follows δa = 0 if and only if the DEP(13) holds.

A map

Φ : ((g_{k−1}, v_{k−1}), (g_{k}, v_{k})) ∈ S^{2} ↦ ((g_{k}, v_{k}), (g_{k+1}, v_{k+1})) ∈ S^{2} (16)

which satisfies (8) is called the discrete Euler-Lagrange algorithm.

A map φ : (f_{kk−1}, f_{kk−1}u_{k−1}, u_{k}) ∈ S × V ↦ (f_{k+1k}, f_{k+1k}u_{k}, u_{k+1}) ∈ S × V which satisfies (13) is called the discrete Euler-Poincaré algorithm. Theorem 3.2. (reduction, reconstruction). The canonical projection π given by (10) applies the DEL algorithm in the DEP algorithm. From the DEP algorithm it can be constructed the DEL algorithm. Proof. The first part of the theorem derives from Theorem 3.1. The DEL algorithm can be reconstructed from the DEP algorithm by

((g_{k−1}, v_{k−1}), (g_{k}, v_{k})) ⟶ ((g_{k}, v_{k}), (g_{k+1}, v_{k+1})) =

((g_{k−1}f_{kk−1}^{−1}, g_{k−1}f_{kk−1}^{−1}u_{k}), (g_{k}f_{k+1k}^{−1}, g_{k}f_{k+1k}^{−1}u_{k+1})), (17)

where (f_{k+1k}, u_{k}) is a solution of the DEP equations given by 13. The algorithm 17 is the DEL algorithm because

g_{k−1}f_{kk−1}^{−1} = g_{k−1}g_{k−1}^{−1}g_{k} = g_{k}g_{k−1}f_{kk−1}^{−1}u_{k} = g_{k−1}g_{k−1}^{−1}g_{k}u_{k} = v_{k}.

4 The discrete Euler-Poincaré-Poisson algorithm

The discrete Euler-Poicaré-Poisson are obtained for Lagrange functions which contain adequate parameters by using the Lagrange-d’Alembert principle.

Let V be a finite dimensional vector space, G a finite dimensional Lie group acting on the left on V. The action is denoted by gv ∈ V, for g ∈ G, v ∈ V. The left action on the cartesian product G×V is given by h(g, v) = (hg, hv), ∀(g, v) ∈ G×V, ∀h ∈ G. The action of G on V^{∗} is given by

(ha)(v) = a(hv), ∀a ∈ V^{∗}, ∀h ∈ G, ∀v ∈ V.

Let L : G × G × V^{∗} ⟶ R be a left G−invariant function, that is

L(g_{1}, g_{2}, a) = L(hg_{1}, hg_{2}, ha), ∀h ∈ G, (g_{1}, g_{2}) ∈ G × G, a ∈ V^{∗}.

For a_{0} ∈ V^{∗} fixed, we define L_{a_{0}} : G × G ⟶ R by L_{a_{0}}(g_{1}, g_{2}) = L(g_{1}, g_{2}, a_{0}). L_{a_{0}} is left G−invariant with respect to the left action of G_{a_{0}} on G, where G_{a_{0}} = {h ∈ G|ha_{0} = a_{0}} is the isotropy group of a_{0}. The invariance of L permits us to define the reduced Lagrangian l : G × V^{∗} ⟶ R by

l(g_{2}^{−1}g_{1}, g_{2}^{−1}a_{0}) = L(g_{1}, g_{2}, a_{0}) = L(g_{2}^{−1}g_{1}, e, g_{2}^{−1}a_{0}). (18)

If we consider a sequence (g_{k}, g_{k+1}) ∈ G × G, k = 0, …, N − 1, then we can form the sequences f_{k+1k} = g_{k+1}^{−1}g_{k} ∈ G and a_{k} = g_{k}^{−1}a_{0} ∈ V^{∗}, k = 0, …, N − 1. Theorem 4.1. With the preceding notation, the following are equivalent:

1. The sequence (g_{k}, g_{k+1}), k = 0, …, N − 1, is an extremum for the action A_{a_{0}} : G^{N+1} ⟶ R associated to L_{a_{0}} : G^{2} ⟶ R with a_{0} fixed.

2. The sequence (g_{k}, g_{k+1}), k = 0, …, N − 1 satifies the DEL for L_{a_{0}} and for arbitrary variation δg_{k}, k = 1, …, N − 1, δg_{0} = δg_{N} = 0.

3. The sequence (f_{k+1k}, a_{k}), k = 0, …, N − 1 satisfies the Lagrange-d’Alembert on G × V^{∗}, associated to l : G × V^{∗} ⟶ R for the variations δf_{k+1k}, δa_{k}.

4. The sequance (f_{k+1k}, a_{k}), k = 0, …, N −1, satisfies the discrete Euler-Poincaré-Poisson equations (DEPP):

D_{1}l(f_{k+1k}, a_{k})Ad_{f_{k+1k}}T_{e}R_{f_{k+1k}} − D_{1}l(f_{kk−1}, a_{k−1})T_{e}R_{f_{kk−1}} = (19)

= D_{2}l(f_{k+1k}, a_{k})◇a_{k}.

Proof. Let g_{k} : ε ∈ (−a, a_{}↦ g_{k}(ε) ∈ G be with g_{k}(0) = g_{k}. The action of L_{a_{0}} for g_{k}(ε) is

A_{a_{0}}(ε) = ∑_{k=0}^{N−1} L_{a_{0}}(g_{k}(ε), g_{k+1}(ε)).

Denoting δg_{k} = \frac{dg_{k}(ε)}{dε}|_{ε=0} and using the discrete analogue of the integration by parts it results that δA_{a_{0}} = 0, for arbitrary variations δg_{k}, δg_{0} = δg_{N} = 0 if, and only if the DEL hold: D_{2}L_{a_{0}}(g_{k−1}, g_{k}) + D_{1}L_{a_{0}}(g_{k−1}, g_{k+1}) = 0. (20)

If f_{k+1k}(ε) = g_{k+1}^{−1}(ε)g_{k}(ε), then we denote δf_{k+1k} = \frac{df_{k+1k}(ε)}{dε}|_{ε=0} and using the concatenation operations it results δf_{k+1k} = −T_{e}R_{f_{k+1k}}(g_{k+1}^{−1}δg_{k+1}) + Ad_{f_{k+1k}}T_{e}R_{f_{k+1k}}(g_{k}^{−1}δg_{k}). (21)

From a_{k} = g_{k}^{−1}a_{0} for a_{k}(ε) = g_{k}^{−1}(ε)a_{0}, we have

δa_{k} = δg_{k}^{−1}a_{0} = −(g_{k}^{−1}δg_{k}g_{k}^{−1})a_{0} = −(g_{k}^{−1}δg_{k})a_{k}. (22)

We shall prove 3) is equivalent to 4). The Lagrange-d’Alembert principle, for l : G × V^{∗} ⟶ R and the variations δf_{k+1k}, δa_{k}, is

∑_{k=0}^{N−1} D_{1}l(f_{k+1k}, a_{k})δf_{k+1k} + D_{2}l(f_{k+1k})δa_{k} = 0.

Substituting (21) and (22) we obtain:

∑_{k=1}^{N−1}[D_{1}l(f_{k+1k}, a_{k})Ad_{f_{k+1k}}T_{e}R_{f_{k+1k}} − D_{1}l(f_{kk−1}, a_{k−1})T_{e}R_{f_{kk−1}}−

D_{2}l(f_{k+1k}, a_{k})◇a_{k}](g_{k}^{−1}δg_{k}) = 0. (23)

For each variation of the form g_{k}^{−1}δg_{k}, from (23) follows (19). The equivalence 1) ⟺ 3) results from the G−invariance of L and from the fact that the variation of g_{k}a_{k} = a_{0} cancels, which proves that L_{a_{0}} depends only of g_{k}, g_{k+1}.

5 The Kelvin-Noether theorem

Let L_{a_{0}} : G × G ⟶ R depending on a parameter a_{0} ∈ V^{∗} as above. Consider a manifold C on which G acts to the left and an equivariant map K : C × V^{∗} ⟶ G. A discrete Kelvin-Noether quantity associated to the discrete reduced Lagrangian l is the map I : C × G × V^{∗} ⟶ R given by:

I(c_{k}, f_{kk−1}, a_{k}) = D_{1}l(f_{kk−1}, a_{k−1})T_{e}R_{f_{kk−1}}K(c_{k}, a_{k}). (24)

Theorem 5.1. (discrete Kelvin-Noether). Let c_{0} ∈ C fixed, f_{k+1k}, a_{k} and g_{k} the solution of the equation g_{k+1} = g_{k}f_{k+1k}^{−1} with g_{0} = e ∈ G. c_{k} = g_{k}^{−1}c_{0} and I_{k} = I(c_{k}, f_{kk−1}, a_{k}) the following relatio holds:

I_{k+1} − I_{k} = (D_{2}l(f_{k+1k}, a_{k})◇a_{k})(K(c_{k}, a_{k})). (25)

Proof. Applying to K(c_{k}, a_{k}) the operator (25), it results

D_{1}l(f_{k+1k}, a_{k})Ad_{f_{k+1k}}T_{e}R_{f_{k+1k}}K(c_{k}, a_{k}) − D_{1}l(f_{kk−1}, a_{k−1})T_{e}

R_{f_{kk−1}}K(c_{k}, a_{k}) = [D_{2}l(f_{k+1k}, a_{k})◇a_{k}] (K(c_{k}, a_{k})). (26)

K is equivariant, therefore (26) implies

D_{1}l(f_{k+1k}, a_{k})T_{e}R_{f_{k+1k}}K(f_{k+1k}c_{k}, f_{k+1k}a_{k}) − D_{1}l(f_{kk−1}, a_{k−1})T_{e}

R_{f_{kk−1}}K(c_{k}, a_{k}) = [D_{2}l(f_{k+1k}, a_{k})◇a_{k}] (K(c_{k}, a_{k})). (27)

Since f_{k+1k}c_{k} = g_{k+1}^{−1}c_{0} = c_{k+1}; f_{k+1k}a_{k} = a_{k+1}, from (27) we deduce (25).

100 D.Opriş and I.D.Albu

6 The discrete Euler-Poincaré equations

for principal bundles

Let (Q, M, π, G) be a principal G−bundle, where M =^{Q} /_{G} is the quotient manifold of Q by the action of the Lie group G on Q nd π : Q ⟶ M is the canonical projection. A M−morphism (F, φ) : (Q × Q, M × M, π × π, G × G) ⟶ (Q, M, π, G) is gives by the maps F : Q × Q ⟶ Q, φ : G × G ⟶ G, such that

F (q_{k}, q_{k+1}) = B(q_{k}, q_{k+1})q_{k},

φ(g, h) = h^{−1}g, (28) where B : Q × Q ⟶ G satisfies the condition B(gq_{k}, hq_{k+1}) = h^{−1}gB(q_{k}, q_{k+1})g−1. (29) Indeed, from the M−morphism property F ((g, h)(q_{k}, q_{k+1})) = φ(g, h)F (q_{k}, q_{k+1}) (30) it follows B(gq_{k}, hq_{k+1})gq_{k} = h^{−1}gB(q_{k}, q_{k+1})q_{k}, therefore (29).

By considering a local trivalization on Q the element q_{k} ∈ Q has the form q_{k} = (r_{k}, g_{k}), where r_{k} = π(q_{k}), q_{k} = g_{k}(r_{k}, e). From (29) it results

B(q_{k}, q_{k+1}) = B((r_{k}, g_{k}), (r_{k+1}, g_{k+1})) = B(g_{k}(r_{k}, e), g_{k+1}(r_{k+1}, e)) = g_{k+1}^{−1}g_{k}B((r_{k}, e), (r_{k+1}, g_{k+1}))g_{k}^{−1} = f_{k+1k}b(r_{k}, r_{k+1})g_{k}^{−1} = ω_{k+1k}g_{k}^{−1}, (31) where f_{k+1k} = g_{k+1}^{−1}g_{k}, g(r_{k}, r_{k+1}) = B((r_{k}, e), (r_{k+1}, e)). (32) Let π^{∼}: Q × Q ⟶ Q × M be the map given by

π^{∼} (q_{k}, q_{k+1}) = (B(q_{k}, q_{k+1})q_{k}, π(q_{k+1})). (33)

It follows

π^{∼} (q_{k}, q_{k+1}) = (ω_{k+1k}g_{k}^{−1}(r_{k}, g_{k}), (r_{k+1}, e)) = ((r_{k}, ω_{k+1k}), r_{k+1}). (34)

Now let L : Q × Q ⟶ R be a G−invariant discrete Lagrangian and let l be the reduced Lagrangian defined by l∘ π^{∼}= L. Thus l(r_{k}, ω_{k+1k}, r_{k+1}) = L((r_{k}, g_{k+1}^{−1}g_{k}), (r_{k+1}, e)) = L((r_{k}, ω_{k+1k}b(r_{k}, r_{k+1})^{−1})(r_{k+1}, e)). Theorem 6.1. The following statements are equivalent:

1. The sequence {q_{k} = (r_{k}, g_{k})}, k = 0, …, N, is an extremum for the action A : Q^{N+1} ⟶ R associated to L, where

A(q_{0}, …, q_{N}) = ∑_{k=0}^{N−1} L((r_{k}, g_{k}), (r_{k+1}, g_{k+1})).

2. The sequence {q_{k} = (r_{k}, g_{k})}, k = 0, …, N,, satisfies the DEL for L and for any variation δq_{k} = (δr_{k}, δg_{k}), with δr_{0} = δr_{N} = 0, δg_{0} = δg_{N} = 0.

3. The sequence (r_{k}, ω_{k+1k}, r_{k+1}), k = 0, …, N − 1, is an extremum for the action a : (^{Q×Q}/_{G})^{N+1} ⟶ R associated to l;

a(r_{0}, ω10, r_{1}, …, r_{N−1}, ω_{NN−1}, r_{N}) = ∑_{k=0}^{N−1} l(r_{k}, ω_{k+1k}, r_{k+1}).

4. The sequence (r_{k}, ω_{k+1k}, r_{k+1}), k = 0, …, N − 1, satisfies the discrete Euler-Poncaré equations (DEP):

D_{1}l(r_{k}, ω_{k+1k}, r_{k+1}) + D_{3}l(r_{k−1}, ω_{kk−1}, r_{k}) + D_{2}l(r_{k}, ω_{k+1k}, r_{k+1})
ω_{1}(k + 1, k) + D_{2}l(r_{k−1}, ω_{kk−1}, r_{k})ω_{2}(k, k − 1) = 0
D_{2}l(r_{k}, ω_{k+1k}, r_{k+1})T_{e}R_{ω_{k+1k}}Ad_{ω_{k+1k}}b^{−1}(r_{k}, r_{k+1})−

D_{2}l(r_{k−1}, ω_{kk−1}, r_{k})T_{e}R_{ω_{kk−1}} = 0, (35)

where D_{i}l(r_{k}, ω_{k+1k}, r_{k+1}), i = 1, 2, 3, represents the matrix of the partial derivatives with respect to the argument i and

ω_{1}(k + 1, k) = T_{e}L_{ω_{k+1k}}T_{(r_{k},)}L_{b^{−1}(r_{k},r_{k+1})}D_{1}b(r_{k}, r_{k+1}),

ω_{2}(k + 1, k) = T_{e}L_{ω_{k+1k}}T_{(r_{k},)}L_{b^{−1}(r_{k},r_{k+1})}D_{2}b(r_{k}, r_{k+1}), (36)

represent the 1−forms associated to the M−morphism (F, φ). Proof. From the properties of this type for differentiable manifolds it results that the statements 1) and 2) are equivalent. Consider the family ω_{k+1k}(ε) = f_{k+1k}(ε) · b(r_{k}(ε), r_{k+1}(ε)) with f_{k+1k}(ε) = g_{k+1}^{−1}(ε)g_{k}(ε), ε ∈ (−a, a) and ω_{k+1k}(0) = ω_{k+1k}. By

denoting δω_{k+1k} = \frac{dω_{k+1k}(ε)}{dε}|_{ε=0}, we have

δω_{k+1k} = −T_{e}R_{ω_{k+1k}}(g_{k+1}^{−1}δg_{k+1}) + T_{e}R_{ω_{k+1k}}Ad_{ω_{k+1k}b^{−1}(r_{k},r_{k+1})}(g_{k}^{−1}δg_{k})+
+T_{e}L_{ω_{k+1k}}T_{(r_{k},e)}L_{b^{−1}(r_{k},r_{k+1})}D_{1}b(r_{k}, r_{k+1})(δr_{k}) + T_{e}L_{ω_{k+1k}}T_{r_{k+1}}L_{b^{−1}(r_{k},r_{k+1})}
D_{2}b(r_{k}, r_{k+1})(δr_{k+1}).

We deduce

δa = \frac{da(ε)}{dε}|_{ε=0} = ∑_{k=1}^{N−1}[D_{1}l(r_{k}, ω_{k+1k}, r_{k+1}) + D_{3}l(r_{k−1}, ω_{kk−1}, r_{k})+

+D_{2}l(r_{k}, ω_{k+1k}, r_{k+1})ω_{1}(k + 1, k) + D_{2}l(r_{k−1}, ω_{kk−1}, r_{k})ω_{2}(k − 1, k)](δr_{k})+

+ ∑_{k=1}^{N−1}[D_{2}l(r_{k}, ω_{k+1k}, r_{k+1})T_{e}R_{ω_{k+1k}}, r_{k+1})T_{e}R_{ω_{k+1k}}

Ad_{ω_{k+1k}b^{−1}(r_{k},r_{k+1})} − D_{2}l(r_{k−1}, ω_{k−1k}, r_{k})T_{e}R_{ω_{kk−1}}](g_{k}^{−1}δg_{k}).

Thus δa = 0 iff the relations (35) holds. Therefore 3) and 4) are equivalent. The equivalence of 1) and 3) follows from the G−invariance of L.

A map Φ : (q_{k−1}, q_{k}) ∈ Q^{2} ↦ (q_{k}, q_{k+1}) ∈ q^{2} satisfying the relation

D_{1}L(q_{k}, q_{k+1}) + D_{2}L(q_{k−1}, q_{k}) = 0 (37)

is called a discrete Euler-Lagrange algorithm (DELA).

A map φ : (r_{k−1}, ω_{kk−1}, r_{k}) ∈^{Q×Q} /_{G} ↦ (r_{k}, ω_{k+1k}, r_{k+1}) ∈^{Q×Q} /_{G} satisfaying the relations 35 is called a discrete Euler-Poincaré algorithm (DEPA). Theorem 6.2. (reduction, reconstruction). The canonical projection π^{∼} given by (33) applies the algorithm DELA on the algorithm DEPA. The algorithm DELA can be reconstructed from the algorithm DEPA. Proof. The first statement of the theorem results from Theorem 6.1.. The algorithm DELA is constructed from the algorithm DEPA as follows: let (r_{k}, ω_{k+1k}) be a solution of the algorithm DEPA and f_{k+1k} = ω_{k+1k} b^{−1}(r_{k}, r_{k+1}); since f_{k+1k} = g_{k+1}^{−1}g_{k} for g_{k} fixed, it results g_{k+1} = g_{k}f_{k+1k}^{−1}. In this way the algorithm DELA is given by

((r_{k−1}, g_{k−1}), (r_{k}, g_{k})) ⟶ ((r_{k}, g_{k}), (r_{k+1}, g_{k+1})) = ((r_{k}, g_{k−1}f_{kk+1}^{−1}), (r_{k+1}, g_{k}f_{k+1k}^{−1}).

Remark. If the map B has the property B(q_{k}, q_{k+1}) = e ∈ G∀(q_{k}, q_{k+1}) ∈ Q^{2}, then the EDP are

D_{1}l(r_{k}, f_{k+1k}, r_{k+1}) + D_{3}l(r_{k−1}, f_{kk−1}, r_{k}) = 0,
D_{2}l(r_{k}, f_{k+1k}, r_{+1})T_{e}R_{f_{k+1k}} · Ad_{f_{k+1k}} − D_{2}l(r_{k−1}, f_{kk−1}, r_{k})T_{e}R_{f_{kk−1}} = 0,

where f_{k+1k} = g_{k+1}^{−1}g_{k}. Acknowledgements. A version of this paper was presented at the Workshop on Electromagnetic Flows and Dynamics, Oct 15-19, 1999, University Politehnica of Bucharest.

Supported by MEN Grant No 21815/28.09.1999, CNCSU-31.

References

[1] A. Bobenko, Y. B. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaré equations and semidirect products, Lett. Math. Phys. 49(1999), 79-93.

[2] H. Cendra, D. D. Holm, J. E. Marsden, T. Ratiu, Lagrangian reduction, the Euler-Poincaré equations and semidirect products, AMS Arnold Volume II (1999).

[3] D. D. Holm, J. E. Marsden, T. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. (1998).

[4] J. E. Marsden, G. Patrick, S. Shkoller, Multisymplectic geometry, variational integrators and nonlinear PDE´ s, Comm. Math. Phys. (1998).

[5] J. E. Marsden, S. Pekarsky, S. Shkoller, Discrete Euler-Poincaré and Lie-Poisson equations, (1998) (15 pag.)

Universitatea de Vest din Timişoara, Catedra de Algebră Bulevardul V. Pârvan 4, 1900 Timişoara E-mail addresss: opris@tim1.uvt.ro E-mail addresss: albud@tim1.uvt.ro