EULER-POINCAR´E REDUCTION FOR SYSTEMS WITH CONFIGURATION SPACE ISOTROPY

. This paper concerns Lagrangian systems with symmetries, near points with conﬁguration space isotropy. Using twisted parametrisations cor- responding to phase space slices based at zero points of tangent ﬁbres, we deduce reduced equations of motion, which are a hybrid of the Euler-Poincar´e and Euler-Lagrange equations. Further, we state a corresponding variational principle.

1. Introduction. It is well known that the presence of symmetry in a physical system is a key ingredient in reducing the dimension of the phase space of a given problem. In particular, mechanics is one of the fields where symmetry and the associated conservation laws have received particular attention starting with the works of Euler and Lagrange. In recent decades, some key achievements are: the theory of symplectic reduction and reduction by stages (see, for example, [1], [4], [14], [18], or [24]), the Marle-Guillemin-Sternberg normal form (the Hamiltonian slice theorem) [6,11], and the energy-momentum method [9,12,17,19].
The present paper considers Lagrangian systems on tangent bundles, with lifted symmetries and configuration-space isotropy. The geometrical framework for studying such systems is given by degenerate parametrisations ("slice coordinates") for neighbourhoods of phase space points with configuration-space isotropy. More precisely, the configuration space in the neighbourhood of the given symmetric point It is easy to show that these actions are free and proper and commute. The twisted product G × K S is the quotient of G × S by the twist action of K. It is a smooth manifold, and the projection is a principal bundle with fibre K. The left multiplication action of G commutes with the twist action and drops to a smooth G action on G× K S, given by γ·[g, s] K = [γg, s] K .
Now consider a G action on a manifold M, and a point z ∈ M, and let K = G z be the isotropy subgroup of z. A tube for the G action at z is a G-equivariant diffeomorphism from some twisted product G × K S to an open neighbourhood of G · z in M, that maps [e, 0] K to z. The space S may be embedded in G × K S as {[e, s] K : s ∈ S} ; the image of the latter by the tube is called a slice.
The slice theorem of Palais [20] (see also [18, 2.3.28]) states that tubes always exist for smooth proper actions of a Lie group G on manifold M . One version of the theorem is the following: given z ∈ M, with isotropy group K = G z , there always exists a K-invariant Riemannian metric on a neighbourhood of z. (This is due to the compactness of K, see [18] for details.) Let N be the orthogonal complement to g · z. Then there exists a K-invariant neighbourhood S of 0 in N such that the map (where exp z is the Riemannian exponential) is a tube at z for the G action. The K-invariant complement N to g · z is sometimes called a linear slice to the G action at z. The twisted product G × K N may be identified with the normal bundle to the orbit G · z. If M is an open subset of a vector space, with G acting linearly, as in the example in Section 5.1, then S can be identified with a neighbourhood of the origin in a linear subspace of M itself, and the tube defined by τ ([g, s] K ) = g · (q 0 + s).
Configuration space slices. Consider a Lagrangian L : T Q → R, invariant under a proper tangent-lifted action of a Lie group G. Let q 0 ∈ Q and K = G q0 . By Palais' slice theorem, the following map is a tube at q 0 for the G action. We consider the following composition, and we regard the composition τ • π K : G × S → Q as a degenerate "parametrisation" of Q in a neighbourhood of q, defining the "slice coordinates" (g, s). This parametrisation is semi-global in the sense that it is global in the group direction and local in the slice direction. The tangent lift of τ • π K gives the parametrisation T (G × S) → T Q. In this paper we will describe the dynamics on T Q, with configurations in the neighbourhood of the group orbit G · q 0 , by pulling them back to T (G × S).
We denote by g and k the Lie algebras of G and K. Throughout the paper we identify T G with G × g using the left trivialisation given by: where L g is left multiplication by g. Similarly, we make the identification Note that T S is trivial, as S is a subset of a vector space. We write elements of T S as (s,ṡ). The left multiplication action of G and the twist action of K on G × S both lift to free, proper, commuting actions on T (G × S).
The tangent bundle. Next we describe the tangent bundle T (G × K S) using a degenerate parametrisation by G × k ⊥ × T S ⊂ T (G × S). Fix a K-invariant complement of k in g, which we denote k ⊥ (such a complement can always be found by averaging over K, since K is compact). Consider the projection π K : G × S → G × K S. Its tangent map is a K-invariant G-equivariant surjection. If we describe points in T (G × K S) as T π K (g, ζ, s,ṡ), then we have two kinds of degeneracy in our coordinates: first, (g, s) is not uniquely determined by π K (g, s) ; second, given a choice of (g, s), the tangent vector (ζ,ṡ) ∈ T (g,s) (G × S) is not uniquely determined because T (g,s) π K has a kernel, We therefore have the splitting: for all (g, s) ∈ (G, S). This defines a connection on the principal bundle π K : G × S → G × K S, with horizontal subspaces k ⊥ ⊕ T s S (with left-trivialisation) at each (g, s) ∈ G × S. We eliminate the second kind of degeneracy by restricting T π K to G × k ⊥ × T S ⊂ T (G × S). Our new parametrisation of T (G × K S) is (2.5) Note that for any (g, s) ∈ G × S, the map T (g,s) π K is an isomorphism from k ⊥ × T s S to T [g,s] K (G × K S) . Composing π K with the tube τ from Equation 2.2 gives a map (g, s) → g · exp q0 s. Differentiating this gives This formalises the observation that T q Q ∼ = g · q ⊕ S near the point at which the slice S is defined. Since G × k ⊥ × T S and T π K are K-invariant, the map T π K descends to the quotient by K, It is easily checked that this map is a G-equivariant diffeomorphism. The K action on G × k ⊥ × T S is exactly the twist action on G × k ⊥ × T S given by the adjoint action on K on k ⊥ and the tangent-lifted action of K on T S. Thus T π K may be written and we see that T π K is actually a tube for T (G × K S) around 0 ∈ [e, 0] K . From now on, for simplicity of notation, we will identify Q with G × K S and q 0 with [e, 0] K . We will also need the diamond operator, defined for all s ∈ S and σ ∈ T * s S by s σ, ξ = σ, ξ · s ∀ξ ∈ k.
3. Equations of motion. Let L : T Q → R be a G−invariant Lagrangian and let q 0 ∈ Q, with isotropy group K = G q0 . From the previous section, there exists a Since our configuration space is G × S and G acts only on the first factor, one can combine Euler-Lagrange reduction and Euler-Poincaré reduction to arrive at the following extended Euler-Poincaré reduction theorem (for details, see [8,Section 7.3

]):
Theorem 3.1 (Euler-Poincaré Reduction). Let G be a Lie group, S a manifold, L : T (G × S) → R a left-invariant Lagrangian, andl : g × T S → R be defined bỹ l(ξ, s,ṡ) :=L(e, ξ, s,ṡ). For any curve g(t) ∈ G, let ξ(t) = g(t) −1ġ (t). Then a curve (g(t), s(t)) satisfies the Euler-Lagrange equations for LagrangianL if and only if the following reduced variational principle holds: for variations (δξ, δs) with δξ of the form δξ =η + ad ξ η, where η(t) is an arbitrary path in g which vanishes at the endpoints, i.e., η(a) = η(b) = 0. This principle is equivalent to: We now apply this theorem in the particular case outlined before the theorem, in whichL = L•T π K . For every pathc(t) in G×S, let S (c) = b aL c(t),ċ(t) dt . Let k(t, s) be a family of paths in K. For every s, the pathc s (t) := k(t, s)c(t) projects down to the same path in G × K S asc(t), i.e., π K •c s = π K •c. By the chain rule, it follows that T π K c s (t),ċ s (t) = T π K c(t),ċ(t) for every s and t, and thus Therefore, defining In particular, for any path η k (t) in k, consider the family of paths k(t, s) := exp s η k (t) . Then .
It follows from Equation 3.7 that, for any path η k (t) in k, variations ofc(t) of the form (δg(t), δs(t)) = −g(t)η k (t), η k · s(t) lead to δS = 0. Thus it suffices to consider only variations in some complement to these variations, for example variations of the form (δg, δs) such that δg(t) = g(t)η ⊥ (t) for some path η ⊥ (t) in k ⊥ . Note that these are variations in the horizontal direction of the connection defined in Equation 2.4, and we have shown that only these variations are important sinceL is degenerate in the vertical direction.
In summary, we have shown: To find critical points of the action functional forL, it suffices to consider only variations of the form (δg, δs) such that In general, if δg(t) = g(t)η(t) and ξ(t) = g −1 (t)ġ(t), for all t, then the proof of Euler-Poincaré reduction shows that δξ =η + ad ξ η. By the previous lemma, it suffices to consider only those δξ of the form Thus we obtain the following consequence of Euler-Poincaré reduction: where η ⊥ (t) is an arbitrary path in g ⊥ which vanishes at the endpoints, i.e., η ⊥ (a) = η ⊥ (b) = 0; and variations of s(t) that vanish at the endpoints.
Using the standard "integration by parts" argument, the variational principle in Theorem 3.3 is equivalent to: for all paths η ⊥ (t) in k ⊥ and all variations δs(t) of s(t). We now break δl δξ into k * and k • components. Just as we write the k and k ⊥ components of ξ as ξ k and ξ ⊥ , respectively, we will write the k * and k • components of δl δξ as δl δξ k and δl δξ ⊥ . So With this notation, we have We will use a line over ad * ξ to indicate projection onto k • , so, in particular, ad * ξ δl δξ := Pr k • ad * ξ δl δξ .
Since η ⊥ and δs are arbitrary, Equation 3.10 is equivalent to the following "Euler-Poincaré-Lagrange" equations: The first of the Equations 3.11 can be rewritten, breaking both the ξ and the δl δξ on the right hand side into k • and k components: The second term on the last line is zero, because ad * ξ k δl/δξ k ∈ k * and the overline represents projection onto k • . For the third term note that, since k ⊥ is K-invariant, ad η k ξ ⊥ ∈ k ⊥ for all η k ∈ k. It follows that for all ν k ∈ k * ad * ξ ⊥ ν k , η k = ν k , ad ξ ⊥ η k = − ν k , ad η k ξ ⊥ = 0.
Hence the overline can be omitted from the third term. Thus the first of Equations 3.11 is equivalent to: We now use the fact thatL = L • T π K in a second way: since T (g,s) π K has kernel k · (g, s), which in left-trivialised coordinates equals we obtainL(g, ξ, s,ṡ) =L(g, ξ − ξ k , s,ṡ + ξ k · s) for all ξ ∈ k, and thus: for all δξ k ∈ T ξ k k. Since δξ k is arbitrary, this is equivalent to δL δξ k = s δL δṡ .
Sincel(ξ, s,ṡ) =L(e, ξ, s,ṡ) =L(g, ξ, s,ṡ), the previous two equations are also valid withL replaced withl. Up to this point we have considered a general base curve (ξ(t), s(t)) in g × S for the variations. However, we can restrict attention to paths in k ⊥ × S, since we are interested in paths c(t) in G × K S. Indeed, as we have seen in the previous section, the projection π K : G × S → G × K S is a principal bundle, with a connection given by the splittings in Equation 2.4. Given any path c(t) in G × K S, and an element (g 0 , s 0 ) ∈ π −1 K (c(0)) (the fibre over c(0)), there exists a unique horizontal lift of c(t) to a pathc(t) such thatc(0) = (g 0 , s 0 ). Note thatc(t) is horizontal if and only ifc(t) = (g(t), s(t)) and g −1 (t)ġ(t) ∈ k ⊥ for all t. In left-trivialised coordinates, if c(t),ċ(t) = (g(t), ξ(t), s(t),ṡ(t)) thenc(t) is a horizontal curve if and only if ξ k (t) = 0 for all t. Note that c(t) satisfies the Euler-Lagrange equations for L if and only if its horizontal liftsc(t) satisfy the Euler-Lagrange equations forL. Thus a curve c(t) satisfies the Euler-Lagrange equations for L if and only if any (and hence all) of its horizontal lifts to G × S satisfies Equations 3.14 with ξ replaced by ξ ⊥ .
We have now eliminated all k components from Equations 3.14. Thus it makes sense to restate these equations in terms of the restriction ofl to k ⊥ × T S, which we denotel : k ⊥ × T S → R.  We have proven the following:

Note that the first equation above is a modified version of the Euler-Poincaré equation, while the second equation is an Euler-Lagrange equation.
Reconstruction. The reconstruction of the curve g(t) along the group is given by the non-autonomous differential equatioṅ Remark 3.6. If the isotropy subgroup K is trivial, then the theorem above reduces to a simple consequence of the Lagrange-Poincaré equations given in [4]. 4. Variational principle. In the previous section, we derived the Euler-Poincaré bundle equations, stated in Theorem 3.5, from the variational principle stated in Theorem 3.3. Note however that the variational principle in Theorem 3.3 is stated in terms of the functionl : g × T S → R, while the Euler-Poincaré bundle equations involve only the function l : k ⊥ × T S → R, which is the restriction ofl to k ⊥ × T S. Can we state a corresponding variational principle on k ⊥ × T S ?
One approach is to replace variations δξ k (an arbitrary variation in the k direction) with equivalent variations in the T s S direction, using the fact, established in the previous section, that and so δξ k = Pr k ad ξ η ⊥ and δξ ⊥ =η ⊥ +Pr k ⊥ ad ξ η ⊥ . We introduce the notation ad ξ η ⊥ k := Pr k ad ξ η ⊥ and ad ξ η ⊥ ⊥ := Pr k ⊥ ad ξ η ⊥ . Thus δξ k = ad ξ η ⊥ k and δξ ⊥ =η ⊥ + ad ξ η ⊥ ⊥ .
It follows that in this case, Note that the k component of δξ no longer appears in the formula. If ξ(t) = ξ k (t) ∈ k, thenl may be replaced by l in the above equation. Thus we have shown that, at a base curve ξ k (t), s(t),ṡ(t) , the condition for variations as described in Theorem 3.3, is equivalent to the condition for variations of the form where η ⊥ (t) is an arbitrary path in g ⊥ that vanishes at the endpoints, and δs is an arbitrary variation of s that vanishes at the endpoints. Thus Theorem 3.3 implies the following variational principle: Letl (ξ, s,ṡ) =L (g, ξ, s,ṡ), and let l be the restriction ofl to k ⊥ × T S. For every curve c(t) in G × S, letc(t) be the horizontal lift of c(t) to G × K S, in the sense described above. Letc(t) = (g(t), s(t)) and ξ(t) = g −1 (t)ġ(t) for all t. Then c(t) satisfies the Euler-Lagrange equations for L if and onlyc(t) satisfies for variations of the form where η ⊥ (t) is an arbitrary path in g ⊥ that vanishes at the endpoints, and δs is an arbitrary variation of s that vanishes at the endpoints.

5.
Example: Simple mechanical systems. In this section we consider the theory above in the case of simple mechanical systems. We also provide an example by considering a three-point-mass system at a collinear configuration. Let L : T Q → R be a Lagrangian characterised by a non-degenerate left Ginvariant Riemannian metric K(·, ·) : T Q×T Q → R and a left G-invariant potential V : Q → R . The Lagrangian reads: Let q 0 be a symmetric point with isotropy group K. We take q 0 to be the base of the slice coordinates. Let S be the orthogonal complement with respect to the given metric to the tangent orbit through q 0 , i.e., For simplicity we confine ourselves to a linear G action, so that points in the configuration space Q can be represented as q = g(q 0 + s) ≡ [g, s] K (see Section 2). Since our analysis is performed for finite-dimensional configuration spaces Q, we can take advantage of the matrix representation of the metric K.
We represent each velocity vector v q ∈ T q Q as v q = (g, ξ ⊥ , s,ṡ) ∈ G × k ⊥ × T S, using the parametrisation in Equation 2.5. A direct derivation of q = g(q 0 + s) gives v q = g ξ ⊥ (q 0 + s) +ṡ and the dynamics of the system takes place in the model Under the change of variable (q, v q ) = (g, ξ ⊥ , s,ṡ) the kinetic term becomes: We define the reduced locked inertia tensor I : S → L (k ⊥ ) * , k ⊥ , the Coriolis term C : S → L T s S, (k ⊥ ) , and the reduced mass m : S → L T s * S, T s S as following: Note that I(s) is a non-degenerate inner product on k ⊥ and it represents the restriction of the usual locked inertia tensor to the Lie algebra k ⊥ . Similarly, m(s) is a non-degenerate inner product on T s S and it represents the restriction of the kinetic metric to the subspace T s S (the terminology comes from the fact that in classical mechanics the kinetic metric is given by the mass tensor of the system).
The mixed map C(s) couples the system and the terminology is inherited from the usual Coriolis forces in mechanics. Our choice of coordinates with the slice based at q 0 ≡ [e, 0] K enforces C(0) = 0, i.e., the system is decoupled at the base.

5.1.
Collinear three-point-mass systems. An example of a simple mechanical system at an isotropic configuration point is provided by an SO(3)-invariant threepoint-mass system in a collinear configuration.
Let m 1 , m 2 , m 3 be three point masses aligned along the Oz axes in a Cartesian system of reference of the Euclidian space E 3 . The point masses m 1 , m 2 , m 3 are interacting though means of a given bonding potential V depending only on the pairwise distances. Example of such systems are given by Eulerian configurations in the classical Newtonian three-body problem (although the collinear configuration need not be a relative equilibrium), or collinear molecules formed by three atoms.
The configuration space we use the reduced Jacobi-Bertrand-Haretu coordinates (see [12]), that is, by the relative vector r 1 determined by m 1 , m 2 , and the vector r 2 describing the position of m 3 relative to the center of mass of the m 1 m 2 , subsystem. The symmetry group is SO(3) and acts diagonally on the configurations space Q = R 3 × R 3 (where we ignore possible collisions) and by cotangent lift on T Q = T R 6 . We denote by v 1 and v 2 the velocity vectors with based at r 1 and r 2 , respectively. The Lagrangian of the system is Let r 0 1 = (0, 0, λ 1 ), r 0 2 = (0, 0, λ 2 ) be the (initial) collinear configuration of the system. The isotropy subgroup of (r 0 1 , r 0 2 ) is the SO(2) group of rotations around the Oz axes. The isotropy subalgebra k consists in instantaneous rotations about the Oz axis which we denote by so(2) z . Its complement k ⊥ is formed by instantaneous rotations about Ox and Oy denoted by so(3) xy . The slice S is given by all vectors (w 1 , w 2 ) ∈ R 3 × R 3 such that In this case S is a four-dimensional subspace of R 6 . A choice to describe S is Using Theorem 3.5, the dynamics near the collinear configuration (r 0 1 , r 0 2 ) is described in the reduced model space so (3)  The diamond operator definition as specialised in the present context is: for all s ∈ S = R 4 and σ ∈ T * S = T * R 4 , s σ, ξ = σ, ξ · s for any ξ ∈ so(2) z . In fact, s σ is the momentum map corresponding to the co-tangent lifted action of the isotropy group SO(2) z on the slice T * S = T * R 4 . Writing the infinitesimal action of so(2) z on S as embedded in R 6 (Equation 5.29) we have that for all ξ = (0, 0, ξ z ) ∈ so(2) z ξ · (s 1 , s 2 , s 3 , −ρs 1 , −ρs 2 , s 4 ) = ((0, 0, ξ z ) × (s 1 , s 2 , s 3 ) , (0, 0, ξ z ) × (−ρs 1 , −ρs 2 , s 4 )) = (−ξ z s 2 , ξ z s 1 , 0 , ξ z ρs 2 , −ξ z ρs 1 , 0) .