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Coisotropic Variational Problems

In this article we study constrained variational problems in one independent variable defined on the space of integral curves of a Frenet system in a homogeneous space G/H. We prove that if the Lagrangian is G-invariant and coisotropic then the extremal curves can be found by quadratures. Our proof is constructive and relies on the reduction theory for coisotropic optimal control problems. This gives a unified explanation of the integrability of several classical variational problems such as the total squared curvature functional, the projective, conformal and pseudo-conformal arc-length functionals, the Delaunay and the Poincar{\'e} variational problems.


Introduction
The present paper is an outcome of our attempt to understand the general mechanisms underlying the integrability of constrained variational problems for curves of constant type in homogeneous spaces [10,11,17,18]. The Pfaffian differential systems arising from curves of constant type lead to the notion of generalized Frenet system for curves of a homogeneous space G/H. Roughly speaking, a generalized Frenet system of order k on G/H is a G-invariant submanifold S ⊂ J k (R, G/H) of the jet space J k (R, G/H), which may be linearized by a left-invariant affine sub-bundle of T (G).
From the geometrical viewpoint the integral curves of such systems are canonical lifts of curves of constant type on G/H. The most elementary example is the classical Frenet-Serret differential system for generic curves in Euclidean space. We then consider a G-invariant Lagrangian and we investigate the corresponding Euler-Lagrange system. The general construction of the momentum space and of the Euler-Lagrange system of a constrained variational problem in one independent variable is due to Griffiths (we refer to [5,12,16] as the standard references on the subject and to [6] as the original source of inspiration of the approach developed by Griffiths). We adhere to the terminology introduced in [12,5] and say that a Lagrangian L is non-degenerate if the geometric description of the Marsden-Weinstein reduced spaces in terms of the phase portraits of the system. This procedure is constructive and applies to several concrete examples (see refs. [5,7,12,16,20,21,22,23]).
The paper is organized as follows. In the next section we recall the basic definitions and properties of linear control systems on Lie groups and Frenet systems of curves in homogeneous spaces.
In Section 3, we examine variational problems defined by invariant Lagrangians for linear control systems on Lie groups. From a geometrical viewpoint we deal with k-th order variational problems for curves of constant type in a homogeneous space that depend on the generalized curvatures.
Since all the derived systems have constant rank, the extremal curves of the variational problem are the projections of the integral curves of the Euler-Lagrange system. Therefore, we focus our attention on the momentum space and on the Euler-Lagrange system. First we investigate the geometry of the momentum space Y of a regular invariant Lagrangian of a linear control system of a Lie group G. We show that Y is of the form G×F , where F is an immersed submanifold of g×g * (we call F the phase space of the variational problem). Next we study non-degenerate Lagrangians. We prove that if L is non-degenerate, then the phase space F can be realized as a submanifold of g * . We define the linearized phase portraits and the Legendre transform and analyze the structure of the characteristic vector field of a non-degenerate Lagrangian. In Section 4 we study coisotropic Lagrangians. We prove that the integral curves of the characteristic vector field passing through a point of the bifurcation set are orbits of one-parameter subgroups of the symmetry group G.
Therefore, from this point on, we focus our attention on the regular part Y r of the momentum space. We show that Y r is of the form G × F r , where F r is an open subset of the phase space. We prove that F r intersects the coadjoint orbits, O(µ), of G transversally and that P r (µ) = F r ∩ O(µ) are smooth curves (referred to as the phase portraits). Subsequently we introduce the moment map J : Y r → g * and prove that the Marsden-Weinstein reduction J −1 (µ)/G µ can be naturally identified with the phase portrait P r (µ). We also show that every µ ∈ J(Y r ) is a regular element of g * which implies that the isotropy subgroups G µ are Abelian, for every µ ∈ J(Y r ). We then examine more closely the phase flow φ and the characteristic vector field ξ. We prove that if the Lie algebra g possesses a non-degenerate Ad-invariant inner product then the differential equation fulfilled by the phase flow can be written in Lax form. From the Noether conservation theorem we know that the characteristic vector field ξ is tangent to the fibers J −1 (µ) of the moment map.
We define a canonical connection form θ µ on the Marsden-Weinstein fibrations J −1 r (µ) → P(µ) whose horizontal curves are the integral curves of the characteristic vector field. Since the base is 1-dimensional and the structure group G µ is Abelian, the horizontal curves can be found by a single quadrature. This shows that the extremal curves of an invariant coisotropic Lagrangian are integrable by quadratures. As a byproduct, we prove that if the canonical connection θ µ is complete, which is generically the case when µ is a regular value of the moment map, then the connected components of J −1 r (µ) are Euclidean cylinders and the characteristic vector field ξ can be linearized on J −1 r (µ). We would like to stress that the connection form θ µ can be constructed explicitly from the data of the problem, so that the integration process can be performed in a completely explicit way.
Finally, in two appendices, we summarise the background material that we use from the theory of Pfaffian differential systems and constrained variational problems in one independent variable.
Throughout the paper, we demonstrate how our general results apply to the specific example of isotropic curves in R (2,1) . We show how to derive the Frenet system for such curves, and show that the variational problem is coisotropic if we take the Lagrangian to be a linear function of the curvature. We prove that the phase portraits may be parametrised in terms of elliptic functions, and construct the sections of the Marsden-Weinstein fibration required to reduce the integration to quadratures. Other concrete geometrical examples where the general scheme described in this paper are implemented may be found in [5,6,20,21,22,23]. In all of these cases the generic phase portrait is an elliptic curve, so that the extremal curves can be integrated in terms of elliptic functions and elliptic integrals.
2. Linear control systems on Lie groups and Frenet systems in homogeneous spaces 2.1. Linear control systems on Lie groups. Let G be a Lie group with Lie algebra g. The natural pairing between g and g * will be denoted by (η, V ) ∈ g * × g → η; V ∈ R. We let Θ ∈ g * ⊗ g be the Maurer-Cartan form of G. If we fix a basis (e 0 , . . . , e n ) of g, then Θ = θ J ⊗ e J , where (θ 0 , . . . , θ n ) is the basis of g * dual to (e 0 , . . . , e n ).
Definition 2.1. Let A ⊂ g denote an affine subspace of g of the form P + a = {P + A : A ∈ a}, where P ∈ g and a ∈ Gr h (g) with P / ∈ a. The set of such affine subspaces of g will be denoted P h (g). We call M := G × A the configuration space of the affine subspace A, and denote by π G : M → G and π A : M → A the natural projections onto the two factors.
We now fix a left-invariant form ω ∈ g * such that ω; P = 1 and ω ∈ a ⊥ (i.e. ω; A = 0, for all A ∈ a). We may fix a basis (e 0 , . . . , e n ) of g such that P = e 0 , a = span(e 1 , . . . , e h ) and we let θ 0 , . . . , θ n be the components of the Maurer-Cartan form with respect to (e 0 , . . . , e n ). Such a basis may be chosen so that θ 0 = ω. Using the projection π G , we may pull-back the differential 1-forms ω, θ 1 , . . . , θ n to M , to define a set of 1-forms on M which, by the standard abuse of notation, we again denote by ω, θ 1 , . . . , θ n ∈ Ω 1 (M ) 1 . Let k 1 , . . . , k h denote the affine coordinates on A defined by the affine frame (P, e 1 , . . . , e h ). We then define the 1-forms We then define the Pfaffian differential system (A, ω) on M to be the Pfaffian differential ideal generated by the 1-forms {η j : j = 1, . . . n} with the independence condition given by ω 2 .
Definition 2.2. (A, ω) is the linear control system associated to A ∈ P h (g).
Note that the ideal A has constant rank, being generated by a rank n sub-bundle, Z ⊂ T * (M ). The sub-bundle Z is of the form G × Z, where 1 We will generally follow the usual practice in the method of moving frames and omit the pull-back signs to simplify notation. This should cause no confusion as we will clearly specify the manifolds that we are working on. 2 More invariantly, if we fix ω ∈ g * with ω ∈ a ⊥ and ω; P = 1, then we define the g-valued 1-form Θ ∈ Ω 1 (M, g) . A is then the differential ideal generated by { µ; Θ : µ ∈ g * }.
and where the embedding of Z as a sub-bundle of T * (M ) is given by We may use the left-invariant trivialization to identify T (G) and G × g. The tangent space to M at (g, Q) is then identified with g ⊕ a. With this identification at hand, the integral elements of (A, ω) at (g, Q) are the 1-dimensional subspaces of g ⊕ a of the form (Q, v), where v ∈ a.
A smooth curve γ = (α, β) : (a, b) → M , where (a, b) ⊆ R, is a parametrized integral curve of the control system (A, ω) if and only if α : (a, b) → G is a solution of the linear system Thus, as a control system, the points of the affine space A play the role of the inputs. Note that if we assign a smooth map β : (−ǫ, ǫ) → A and a point g 0 ∈ G, then there exists a unique integral curve of the control system, γ = (α, β), satisfying the initial condition α(0) = g 0 .
Consider the linear subspaces a k ⊂ g defined recursively by The smallest integer N such that a N = a N +1 is called the derived length of A. Note that Z s := A × a ⊥ s is contained in Z, for s = 1, . . . , N . We set Z s = M × Z s , s = 1, . . . , N and we consider the sequence of sub-bundles If we denote by A s the Pfaffian differential ideal generated by Z s , then is the derived flag of the control system (see ref. [4,12] for more details about derived flags). We have thus proved the following: Proposition 2.3. All the derived systems of a linear control system on a Lie group G have constant rank.

2.2.
Frenet systems in homogeneous spaces. Let H ⊂ G be a closed Lie subgroup and consider the homogeneous space G/H. The left-action of G on G/H induces an action of G on the jet space J k (R, G/H), called the k-th prolongation of the action of G on G/H. Definition 2.4. A differential relation (in one independent variable) of order k on G/H is a submanifold S of J k (R, G/H) such that dt| S is nowhere vanishing. We define the Pfaffian differential system (I, dt) on S, given by restriction to S of the canonical contact system on J k (R, Note that the integral curves of the Pfaffian differential system (I, dt) are the k-order jets j k (γ) of curves γ : (a, b) → G/H that satisfy the differential relation j k (γ)| t ∈ S, for all t ∈ (a, b). with the properties that • If γ : (a, b) → G/H is a smooth curve of type S then Γ = Φ • j k (γ) is an integral curve of the control system (A, ω); • If Γ : (a, b) → M is an integral curve of (A, ω) such that Im(Γ) ⊂ Φ(S), then γ = π G/H •Γ : The method of moving frames [11,12] gives an algorithmic procedure for the construction of the Frenet systems for curves of constant type in homogeneous spaces (see [10,17,18]). We refer the reader to [12] for the explicit construction of the Frenet system of generic curves in the affine space R 3 , to [7,23] for the Frenet system of generic curves in RP 2 , to [20,26] for the Frenet system of generic curves in the conformal 3-sphere and to [22] for the Frenet systems of generic Legendrian curves in the strongly pseudoconvex real hyperquadric Q 3 of CP 2 .
Definition 2.6. Let F := π G • Φ : S → G and K := π A • Φ : S → A denote the two components of the map Φ. We call F the Frenet map and K the curvature map.
Let γ : (a, b) → G/H be a curve of type S, then Γ = Φ • j k (γ) : (a, b) → M is called the canonical lift of γ. The maps are called the Frenet frame field and the curvature function of γ, respectively.
Definition 2.7. The generalized arc-length of a curve γ : (a, b) → G/H of type S is the smooth function s γ : (a, b) → R, unique up to a constant, such that ds γ = Γ * (ω), where Γ : (a, b) → M is the canonical lift of γ. Each curve γ ⊂ G/H of type S may be parametrized in such a way that ds γ = dt. In this case, we say that the curve γ is normalized .
Proof. We set U Φ = π A (Φ(S)). Thus, U Φ is an open subset of A such that Φ(S) ⊆ G × U Φ . Take any (g 0 , Q 0 ) ∈ G × U Φ . Since Q 0 ∈ U Φ then there exists g 1 ∈ G such that (g 1 , Q 0 ) ∈ Φ(S). Now let Γ : (−ǫ, ǫ) → M be an integral curve of the control system (A, ω) such that Γ(0) = (g 1 , Q 0 ). Since Φ(S) is an open set we may, by restricting the value of ǫ if necessary, assume that Im(Γ) ⊂ Φ(S). Then, the projection of Γ onto G/H is a curve γ : (−ǫ, ǫ) → G/H of type S such that Γ = Φ[j k (γ)]. Using the G-invariance of S it follows that g 0 g −1 1 γ is another curve of type S. Thus, from the equivariance of Φ it follows that g 0 g −1 1 Γ(0) = (g 0 , Q 0 ) belongs to Φ(S). This shows that The elements of the open subset U Φ may therefore be considered as the "geometrical inputs" of the control system (A, ω). In particular, the curvature function K gives a complete set of local differential invariants for curves of type S. More precisely, if γ, γ : (a, b) → G/H are normalized curves of type S with K γ = K γ , then γ and γ are congruent to one other, in the sense that there exists a g ∈ G such that gγ(t) = γ(t), for all t ∈ (a, b). Moreover given any smooth map K : (a, b) → U Φ ⊂ A there exists a normalized curve γ : (a, b) → G/H of type S, unique up to congruence, such that K γ = K.
If we fix an affine frame (P, e 1 , . . . , e h ) of A and if we let k 1 , . . . , k h be the corresponding coordinates, we may identify the configuration space M with G × R h . Thus, we may write K γ = (k 1 γ , . . . , k h γ ), where k 1 γ , . . . , k h γ are smooth functions that depend on the k-jet of γ. These functions can be viewed as the generalized curvatures of γ.
2.3. Isotropic curves in R (2,1) . An example that will illustrate our considerations concerns variational principles for isotropic curves in 3-dimensional Minkowski space. Let R (2,1) denote Minkowski 3-space endowed with the Lorentzian inner product We fix the spatial orientation by requiring that the standard basis (e 1 , e 2 , e 3 ) is positively oriented, and we fix the time orientation defined by the positive light cone Let G be the restricted Poincaré group E(2, 1) i.e. the group of isometries of R (2,1) that preserve the given orientations. The group G may conveniently be described as the space of pairs g = (q, A) where q ∈ R (2,1) and A = ( We let g denote the Lie algebra of G, consisting of all matrices of the form We now define the Maurer-Cartan form Ω ∈ Ω 1 (G, g), which takes the form Differentiating these relations, we obtain the structure equations Recall that the Maurer-Cartan forms ω 1 , ω 2 , ω 3 , ω 1 1 , ω 2 1 , ω 1 2 are linearly independent and generate the space, g * , of left-invariant 1-forms on G. Definition 2.9. A null (or isotropic) curve in R (2,1) is a smooth parametrized curve α : (a, b) ⊂ R → R (2,1) such that α ′ (t) ∈ L + for all t ∈ (a, b). We shall assume that α is without flex points, in the sense that The linear differential form ω α := α ′′ (t) 1/2 dt is nowhere vanishing, and is invariant under changes of parameter and the action of the group G. Without loss of generality we may assume that α is normalized, in the sense that α ′′ (t) 1/2 = 1, ∀t ∈ (a, b) .
(This condition fixes the parameter t up to an additive constant.) The curvature of α is defined by At each point of the curve we may define the frame g(t) = (α(t), A(t)) ∈ G given by This frame defines a canonical lift of the curve α to the group G, referred to as the Frenet frame field along α. An application of the method of moving frames shows that the Frenet frame field is the unique lift of α to G with the property that We illustrate the construction of the Frenet system for isotropic curves in R (2,1) , viewed as a homogeneous space of the group G. Let be the standard coordinates on the jet space J 3 (R, R (2,1) ) ∼ = R × R (2,1) × R (2,1) × R (2,1) × R (2,1) . The differential relation S ⊂ J 3 (R, R (2,1) ) is defined by Holonomic sections of S are third order jets j 3 (α) of normalized isotropic curves α : (a, b) → R (2,1) . We define κ : S → R by κ j 3 (α) is then the curvature of the isotropic curve α. The affine space A = P + a ⊂ g is then the straight line It is convenient to identify the configuration space M = G × A with G × R by means of the map With this identification at hand, the linear control system (A, ω) is generated by the linear differential forms We now consider a smooth curve Γ : (a, b) → M and let be the two components of Γ. Then Γ is an integral curve of the control system (A, ω) if and only if g : (a, b) → G is the Frenet field along the isotropic curve α : (a, b) → R (2,1) , and k is the curvature of the curve α. The mapping Φ : Remark 2.10. Using the structure equations for E(2, 1) we find that where π = dκ + κ 2 η 4 .

2.4.
Coadjoint action of E(2, 1). For later convenience, we now discuss the coadjoint action of E(2, 1) on e(2, 1) * , the dual of its Lie algebra. Our discussion follows the discussion of the coadjoint representation of E(3) given in Guillemin and Sternberg [14].
Using the Maurer-Cartan forms, we identify g * with R (2,1) ⊕ R (2,1) by means of the map The coadjoint action of G on g * then takes the form for all g = 1 0 We now define the map the components which, C 1 and C 2 , generate the space of Casimir functions. We recall the following standard material: • Let G be a Lie group, and g * the dual of the Lie algebra of G. Let µ ∈ g * . The isotropy group of G at µ is the closed subgroup of G defined by • The Lie algebra of G µ is • The rank of the group G is defined as • An element µ ∈ g * is regular if dim(g µ ) = rank(G), otherwise µ is a singular element of g * . The set of regular elements of g * will be denoted by g * r , while g * s will denote the set of singular elements.
• By a theorem of Dixmier (cf. [8,9]), the isotropy group G µ and the isotropy Lie algebra g µ of a regular element µ ∈ g * r are Abelian.
An invariant Lagrangian L gives rise to a variational problem defined on the integral curves of the linear control system (A, ω). From this point of view the Lagrangian L is considered as a cost function. It is then an optimal control problem to minimize the cost among the integral curves of (A, ω). If (A, ω) comes from a Frenet system (S, A, Φ) on the homogeneous space G/H then the Lagrangian L defines a geometric action functional L : S → R acting on the space of the normalized curves of type S: Note that the geometric action functional L depends only on the generalized curvatures of γ.
Example 3.2. The simplest invariant variational problem for a Frenet systems is the arc-length functional , which is defined by a constant Lagrangian (see [7,20,22,23,12] for more details about the arc-length functionals for generic curves in the conformal and pseudoconformal threedimensional sphere, in the real projective plane and in the affine plane). Another typical example of an invariant Lagrangian is the Kirchhoff variational problem for the Frenet system of generic curves in R 3 , defined by the action functional The extremal curves are the canonical lifts of the Kirchhoff elastic rods of R 3 . When a = 0, we get the total squared curvature functional. Other examples of invariant variational problems for curves in R 3 have been considered in ref. [19].
Given an invariant Lagrangian L : A → R, we construct the corresponding affine sub-bundle The Liouville one-form ψ is given by where π : G × Z → G denotes the projection onto the first factor. where (θ 0 , . . . , θ n ) is the dual basis of g * . We use the following index range: i, j = 1, . . . , h, gives an explicit identification between G × R h × R n and Z. With this identification at hand, the tautological 1-form can be written as:  .
Proof. First we claim that the momentum space, Y , is G-invariant. To show this, for any g ∈ G, we consider the submanifold g · Y ⊂ Z. The G-invariance of the exterior differential forms ψ, Ψ and ω implies that left translation L g : Z → Z sends integral elements of (C(Ψ), ω) into integral elements of (C(Ψ), ω). Hence, for every point p ∈ g ·Y , there exists an integral element of (C(Ψ), ω) tangent to g · Y . Since the momentum space Y is maximal with respect to this property, it follows that g · Y ⊆ Y . Thus the group G acts on Y . Since this action is free and proper, the quotient space F = Y /G exists as a manifold. The natural projection π : Y → F is constant along the fibers of the map (g, Q, η) ∈ Y → (Q, η) ∈ A × g * . Thus it induces a smooth one-to-one immersion j : F → A × g * . We conclude the proof by observing that the map (id, j) : Definition 3.7. We call F the phase space of the system. Note that a point p ∈ F is of the form p = (Q, η), where Q ∈ A, η ∈ g * . We define the maps We refer to Λ as the Legendre transform and H as the Examples of invariant non-degenerate variational problems include the total squared curvature functional in two and three-dimensional space forms [12,5], the Kirchhoff variational problem in R 3 , the Poincaré and the Delaunay functionals [12,16,21], the projective, the conformal and the pseudoconformal arc-length functionals (cf. [7,20,22]).
Given a non-degenerate variational problem, it follows that ω ∧ (Ψ Y ) m defines a volume form on Y , and that Ψ Y is of maximal rank on Y . Therefore there exists a unique vector field ξ ∈ X(Y ) Definition 3.9. ξ is the characteristic vector field of the non-degenerate variational problem (A, ω, L).
If (A, ω, L) is non-degenerate then the Euler-Lagrange system is simply the Cartan system of the canonical 2-form restricted to the momentum space: E = C(Ψ Y ). Therefore, for such variational problems, the integral curves of the Euler-Lagrange system are the integral curves of the characteristic vector field ξ (see ref. [12]). We therefore have Proposition 3.11. If L is non-degenerate then the Legendre transform Λ : F → g * is an immersion.
Proof. Let ξ be the characteristic vector field of the momentum space. The Liouville form ψ, the canonical two-form Ψ and the independence condition ω are G-invariants, therefore the character- implies that there exist smooth maps A ξ : F → g and Φ ξ : F → a ⊕ g * with the property that then it is simple to check that every such vector lies in the kernel of the canonical 2-form Ψ. Since this null-distribution is generated by ξ, we therefore have Since A ξ is non-vanishing, however, this holds if and only if ker [dΛ| p ] = {0}.
From now on we will assume that the Legendre transform Λ is a one-to-one immersion, so that the phase space, F , can be considered as a submanifold (not necessarily embedded) of g * .
Consequently, we will think of the momentum space as an immersed submanifold of G × g * . The notation introduced in the preceding paragraphs can then be simplified as follows: • the Legendre map Λ is the inclusion of F into g * ; • the tangent space F (η) of F at η ∈ F is a linear subspace of g * and R(η) = F (η); • the Liouville form and the canonical two-form on Y are the restrictions to Y of the Liouville form and the standard symplectic form on T * (G); • the characteristic vector field ξ can be written as where A ξ : F → g and Φ ξ : F → g * are smooth functions such that Φ ξ (η) ∈ F (η), for all η ∈ F .
From now on, we will adhere to these simplifications.
With this notation at hand, we may use the left-invariant trivialization of T (G) to identify the tangent space We then have the explicit isomorphism where A ∈ g = T id (G) and A| g = (L g ) * A ∈ T g (G). With this identification, the Liouville form ψ becomes the cross-section of T * (Y ) defined by Then, from the standard formula it follows that the canonical 2-form Ψ = dψ ∈ Ω 2 (Y ) takes the form for all η ∈ F and for all A + v, B + w ∈ g ⊕ F (η). The following result shows that the characteristic vector field ξ may be written in terms of the Hamiltonian H: Theorem 3.13. The characteristic vector field ξ is given by Proof. Given a point η ∈ F , we set It then follows from equations (3) and (4) that a tangent vector A+v ∈ g⊕F (η) to the momentum space Y at the point (g, η) belongs to the kernel of Ψ if and only if We now let (g 0 , η 0 ) ∈ Y and let Γ : (−ǫ, ǫ) → Y be the integral curve of the characteristic vector field ξ with initial condition Γ(0) = (g 0 , η 0 ). We write Γ(t) = (g(t), η(t)), where g : (−ǫ, ǫ) → G and η : (−ǫ, ǫ) → F are smooth maps such that is an integral curve of the linear control system (A, ω). We then have Therefore, we conclude that Since ξ belongs to the kernel of Ψ, we conclude from equation (7) that This yields the required result. 3 It is a general fact that if π : Y → M is the momentum space of a regular variational problem (I, ω, L) on the configuration space M and if Γ : (a, b) → Y is an integral curve of the Euler-Lagrange system, then γ = π • Γ is an integral curve of the Pfaffian differential system (I, ω) on M (see [12]).
Remark 3.16. The phase flow φ ξ : D → F satisfies the Euler equation If there exists a G-equivariant isomorphism g ∼ = g * , then we can identify g and g * . (For example, if G is semisimple, then take the pairing defined by the Killing form of g.) Using this identification, the Euler equation can be written in the Lax form Definition 3.17. We denote by F s = {η ∈ F : Φ ξ (η) = 0} the set of all fixed points of the phase flow and by F r the complement of F s . We call F s and F r the singular and the regular parts of the phase space, respectively. We call Σ = G × F s ⊂ Y the bifurcation set and refer to Y r = Y \ Σ as the regular part of the momentum space. The intersection P r (µ) = F r ∩ O(µ) ⊂ P(µ) is called the regular part of the phase portrait P(µ). The connected component P(µ) of P r (µ) containing µ is referred to as the reduced phase portrait of µ.
The following result, which may be verified by applying the uniqueness theorems for ordinary differential equations, characterises integral curves of the characteristic vector field that intersect the bifurcation set: Proposition 3.18. Let p = (g, η) ∈ Σ be a point of the bifurcation set. The integral curve Γ ξ (−, p) : R → Y of the characteristic vector field ξ passing through p is the orbit of the oneparameter subgroup generated by H(η): This result implies that if (A, ω) comes from a Frenet system of curves in G/H, then the curve γ ⊂ G/H of type S that corresponds to Γ ξ (−, p), where p ∈ Σ, has constant curvature (i.e. K γ = constant).
Since this result completely characterises the behaviour of integral curves that intersect the bifurcation set Σ, we shall henceforth restrict our attention to the regular parts of the phase space and momentum space. Therefore, to simplify the notation, F , Y and P(µ) will be used to denote the regular parts of the phase space, the momentum space and the phase portraits, respectively. (2,1) . We now return to our example of isotropic curves in R (2,1) considered in Section 2.3. Let m be a non-zero constant and consider the variational problem on the space V of parametrized integral curves Γ : t ∈ (a, b) → (g(t), k(t)) ∈ G × R of the Pfaffian system (A, ω) defined by the action functional

The Poincaré variational problem for isotropic curves in R
Geometrically, this amounts to an analogue of the Poincaré variational problem where we minimize the arc-length functional (defined by the integral of the canonical line-element of the null curve) amongst normalized null curves α ⊂ R (2,1) subject to the additional constraint that the integral of the curvature k along the curve be held constant.
Theorem 3.19. The momentum space, Y , is the 9-dimensional sub-manifold of Z defined by the equations and the Euler-Lagrange system (E, ω) is the Pfaffian differential system on Y with independence condition ω generated by the linear differential forms (η 1 , . . . , η 5 , σ 1 , σ 2 , σ 3 ) where Proof. We let V 1 ⊂ T ( Z) be the sub-variety of 1-dimensional integral elements of the Cartan system (C(Ψ), ω) and denote by Z 1 ⊂ Z the projection of V 1 under the bundle map T ( Z) → Z. From equations (9a) and (9b) we then deduce that Z 1 is the submanifold defined by λ 1 = m. Denote by C(Ψ) 1 the restriction to Z 1 of the Cartan system. Then C(Ψ) 1 is generated by the linear differential forms (η 1 , . . . , η 5 , λ 2 ω, β 2 , . . . , β 5 ). We then consider the sub-variety V 2 ⊂ T ( Z 1 ) consisting of integral elements of (C(Ψ) 1 , ω) and let Z 2 ⊂ Z 1 denote the projection of V 2 . We therefore have that Z 2 is the sub-manifold of Z 1 defined by λ 2 = 0. Denote by C(Ψ) 2 the restriction to Z 2 of C(Ψ) 1 .
Corollary 3.20. The momentum space Y associated with the Poincaré variational problem for isotropic curves in R (2,1) is the rank-3, affine sub-bundle Y = G × F ⊂ T * (G) ∼ = G × g * where F ⊂ g * is defined by The variational problem is non-degenerate, and the characteristic vector field takes the form Proof. It follows from the preceding theorem that the restriction of the Liouville to the momentum space takes the form The form of Y and F follow directly from this equation. The dimension of Y equals 9, and a straightforward calculation shows that which is nowhere vanishing. Hence the variational problem is non-degenerate. The form of the characteristic vector field follows from a direct calculation.
Remark 3.21. Since the variational problem is non-degenerate, the Euler-Lagrange system E coincides with the Cartan system of Ψ. The characteristic line-distribution Ξ ⊂ T (Y ) of Ψ is transverse to the independence condition ω, and is generated by the characteristic vector field ξ. Without loss of generality, we may choose a parametrization of our integral curve such that Γ * (ω) = dt.
In this case, we may write Γ : t ∈ (a, b) → (g(t), k(t), λ 4 (t), λ 5 (t)) ∈ Y = G × R 3 . From these relations, and the explicit form of the differential forms η i and σ i , we deduce the following result.
and g(t) ∈ G is a solution of Remark 3.24. Although, in the present case, the Lie group G is not semi-simple, it is naturally embedded in SL(4, R). Using the Killing form on sl(4, R) we deduce that the Euler equation (12) may be written in Lax form: Proposition 3.25. If Γ : (a, b) → Y is an integral curve of the Euler-Lagrange system, with Γ * (ω) = dt, then the curvature k(t) satisfies the third order ordinary differential equation Conversely, any non-constant solution k : (a, b) → R of this equation determines a parametrized integral curve of the Euler-Lagrange system, unique up to the action of E(2, 1).
The explicit form of the Casimir operators implies the following result.
Proposition 3.27. If Γ : (a, b) → Y , parametrized such that Γ * (ω) = dt, is an integral curve of the Euler-Lagrange system then the curvature k(t) satisfies the first order ordinary differential Remark 3.28. Letting h(t) := 1 4 k − 1 3m , we deduce that h satisfies dh dt Hence the curvature k and the functions λ 4 , λ 5 corresponding to any solution of the Euler-Lagrange system may be expressed in terms of Weierstrass elliptic functions with invariants g 2 , g 3 .
Remark 3.29. A short calculation using the explicit form of η = ψ Y given in (11) and the coadjoint action of g on g * , which can be derived from (2), shows that in the present case the linearised phase-portrait Π(η) = F (η) ∩ O(η) is one-dimensional, and is spanned by the vector ad * [H(η)]η = Hence the regular parts of the phase portraits are one-dimensional in the current problem. We now introduce a more general class of variational problems for which this is the case.

Coisotropic variational problems
Consider a smooth manifold M equipped with an exterior differential 2-form Ψ. The kernel of Ψ x will be denoted by N (Ψ) x . Suppose that a Lie group G acts on M . Denote by A ♯ the fundamental vector field on M corresponding to A ∈ g and, for each Definition 4.1. The action of G on M is coisotropic with respect to Ψ if Remark 4.2. The notion of a coisotropic action arises naturally when studying collective complete integrability of Hamiltonian systems (see [13,14,25]).

Definition 4.3. An invariant Lagrangian L :
A → R is said to be coisotropic if it is non-degenerate and if the action of G on the regular part of the momentum space Y is coisotropic with respect to Ψ Y , the restriction of the canonical two-form Ψ to Y .
(Recall that we are now using the notation Y , F and P(µ) to denote the regular parts of the momentum space, phase space and phase portraits, respectively.) Proof. Using the left-invariant trivialisation, we find that the polar space of g ♯ (Y ) (g,η) is given by First, assume that L is coisotropic, i.e.
Let V ∈ Π(η). Then V ∈ F (η) and there exists an A ∈ g such that V = −ad * (A)η. Then A + V belongs to g(η) ⊥ and hence V must be a real multiple of ad * [H(η)]η.
Proposition 4.6. Let L : A → R be an invariant coisotropic Lagrangian and let Y = G × F be the corresponding momentum space. Suppose that F is non empty, we then have: • dim(Y ) = dim(G) + rank(G) + 1; • the regular part of the phase space, F , intersects the coadjoint orbits transversally; • The regular parts P(µ) of the phase portraits are smooth and 1-dimensional; • Every η ∈ F is a regular element of g * . In particular, the isotropy group G η and the isotropy algebra g η are Abelian.
Thus F intersects the coadjoint orbits transversally. Since dim(F (η)) ∩ O(η)) = 1, it follows that P(η) = F ∩ O(η) is a smooth curve such that T η [P(η)] = Π(η). Moreover, from the transversality condition, it follows that F cannot be contained in the set g * s of the singular element of g * . Thus, F ∩ g * r is non-empty. Therefore, there exists an η ∈ F such that k = k(η) = rank(G). This gives the required result.
Remark 4.7. The regular part of the phase space, F , is foliated by the nowhere vanishing vector field Φ ξ and the leaves are the phase portraits. Furthermore, if X ⊂ F is a local section of such a foliation then X is also a local section of the coadjoint representation.
Proof. From Proposition 4.6 we know that each η ∈ F is an element of g * r , and hence J(g, η) ∈ g * r for all (g, η) ∈ Y . The differential of the moment map is given by the formula This implies that for all η ∈ F . Since F and O(η) intersect transversally, this implies that J is a submersion. Therefore J −1 (µ) is a sub-manifold of Y and the tangent space T (g,η) [J −1 (µ)] is naturally isomorphic to ker[dJ| (g,η) ]. From the formula (16) and the fact that the Lagrangian is coisotropic, we deduce that ker dJ| (g,η) = span[ξ| (g,η) ] + g η , for all η ∈ F , as required. Note that this relation implies that the characteristic vector field ξ belongs to ker [dJ], and therefore that ξ| (g,η) is tangent to J −1 (µ). We shall denote the restriction of ξ to the fibre J −1 (µ) by ξ µ . The isotropy group G µ acts on J −1 (µ) by (g, η) → (hg, η) for each h ∈ G µ . This action is clearly free and proper, so the quotient space Y µ := J −1 (µ)/G µ exists as a one-dimensional manifold. The map gives J −1 (µ) the structure of a principal fibre-bundle with structure group G µ . Moreover, the vector field ξ µ is horizontal with respect to the fibration J −1 (µ) → Y µ . We also consider the fibration of J −1 (µ) over P(µ) defined by π µ : (g, η) ∈ J −1 (µ) → η ∈ P(µ).
The structure group is again the isotropy subgroup G µ . Furthermore, π µ is constant along the fibers of the fibration π µ , and therefore descends to a diffeomorphism of Y µ onto P(µ).
Definition 4.10. We adopt standard terminology, referring to Y µ as the Marsden-Weinstein reduction of Y at µ, and to π µ : J −1 (µ) → Y µ as the Marsden-Weinstein fibration at µ. We may consider π µ : J −1 (µ) → Y µ as a principal G µ bundle over Y µ , where the right-action of G µ on Definition 4.11. Let µ ∈ J(Y ). The restriction of the Marsden-Weinstein fibration J −1 (µ) to the reduced phase portrait P(µ) is said to be the reduced Marsden-Weinstein fibration. We shall denote this fibration by π µ : P µ → P(µ).
Definition 4.14. On P µ we consider the g µ -valued 1-form θ µ defined by This defines a connection on the reduced Marsden-Weinstein fibration P µ → P(µ). We call θ µ the canonical connection of the reduced Marsden-Weinstein fibration P µ → P(µ).

4.1.
Isotropic curves in R (2,1) . In the case of our problem for isotropic curves in R (2,1) , we have defined a map where Given the form (10) of the characteristic vector field ξ, we see that the regular part of the phase space F is given by the complement of the set of points with To show that the action of E(2, 1) on the regular part of the momentum space, Y , is coisotropic, we consider a general vector field on Y : This vector field lies in g(Y ) ⊥ (g,k,λ4,λ5) if and only if and If (k, λ 4 , λ 5 ) lies in the regular part of F then conditions (19) and (20) are linearly independent. Therefore in this case dim g(Y ) ⊥ = 3.
It is easily checked that the characteristic vector field ξ belongs to g(Y ) ⊥ , as do the following vector fields: Hence the action is coisotropic.
The momentum map and the basic invariants: From the form of the coadjoint action of E(2, 1) given earlier, we deduce that the moment map takes the form J(g; y) = (Ap(y), Av(y) − (Ap(y)) × Q) , The basic invariants, which correspond to constants of motion of the system, are the Casimir operators If we choose µ = (m 1 , m 2 ) ∈ g * r , where m 1 , m 1 ∈ R (2,1) , then J −1 (µ) is the set If we perform the substitution then the first equation becomes the cubic relation where we have defined the "modified Casimirs" .
Parametrization of the phase portraits: We define the discriminant of the cubic polynomial appearing in equation (23) D(m 1 , m 2 ) = 27g 2 3 − g 3 2 . There are two non-degenerate cases that we must consider.
In the degenerate cases where the discriminant vanishes D = 0, the cubic is singular, and our curve is rational. In this case, the ℘ and ℘ 3 functions degenerate into elementary functions.

Integrability by quadratures
Proposition 5.1. The integral curves of the characteristic vector field ξ with momentum µ ∈ J(Y ) are the horizontal curves of the canonical connection θ µ on P µ .
Since the structure group of the Marsden-Weinstein fibrations is Abelian and the base manifolds are 1-dimensional, the horizontal curves of the canonical connection can be found by a single quadrature. The explicit integration of the horizontal curves requires four steps: • Step one: take a smooth parametrization η : (a, b) → P(µ) of the phase portrait.
Conclusion: the image of the curve Γ : (a, b) → Y defined by is contained in P µ and Γ is horizontal for the canonical connection θ µ . Any horizontal curve of P µ arises in this way.
Remark 5.2. In the case where the symmetry group G is a classical matrix group, with G ⊂ Aut(V ) for V a finite-dimensional vector space, and the Lagrangian is a polynomial function, then we have: • The phase space F is an algebraic subset of g * .
• The generic coadjoint orbit is defined by polynomial equations F j (η) = 0, j = 1, . . . , k, where k is the rank of g and where F 1 , . . . , F k is a basis of the Ad * -invariant polynomial functions g * → R. • The phase portraits are real-algebraic curves.
In the simplest cases the phase portraits are rational or elliptic curves, so they can be easily parametrized by means of elementary or elliptic functions (see the examples considered in refs. [5,12,16,20,21,22,23]). The third step in the construction above can be treated in a rather easy way if g is semi-simple and if the momentum µ is a regular semisimple element of g * . In this case the construction of a cross-section of the Marsden-Weinstein fibration is a linear-algebra problem involving the structure of the Cartan subalgebras of g. Definition 5.3. Consider µ ∈ J(Y ). We say that µ is a complete momentum if ǫ − (η) = −∞ and ǫ + (η) = ∞ for some (and hence for all) η ∈ P(µ).
Proposition 5.4. If µ ∈ J(Y ) is a complete momentum, then the connected components of the reduced Marsden-Weinstein fibration P µ → P(µ) are Euclidean cylinders and ξ µ is a linear vector field.
Remark 5.5. If P(µ) is compact and the isotropy subgroup G µ is compact, then the connected components of the reduced Marsden-Weinstein fibration P µ are (k + 1)-dimensional tori.
Finally, we show how the above integration procedure may be carried out in our example.
Positive type: For an orbit of positive type, where C 1 = p 2 > 0, we may assume, up to the action of G on O(µ), that µ = (m 1 , m 2 ) is in the standard form: where C 2 := p, v .
Since p 2 > 0, we may define a the R (2,1) -valued map with the property that A 2 2 = 1. We can now complete A 2 to a frame field (A 1 , A 2 , A 3 ) by adding any R (2,1) -valued functions A 1 , A 3 with the property that A i , A j = g ij , i, j = 1, 2, 3, and we fix the orientation of this basis by the requirements that More explicitly, we can define the vector S = λ 4 e 1 + λ 5 − 1 2 (1 − mk) e 2 − λ 4 e 3 , with the property that (p, S) = 0. We then define A = (A 1 , A 2 , A 3 ) : P µ → SO(2, 1) by Defining the map Q : P µ → R (2,1) by we let g := (Q, A) : P µ → P(µ).
Negative type: For orbits of negative type we have C 1 := p 2 < 0. We treat the case where the vector p is future-directed, although the past-directed case may be treated similarly. The standard form of elements in this case is µ = (m 1 , m 2 ), where To define a suitable basis, we fix a vector S ∈ R (2,1) with S 2 = 1 and p, S = 0. (For example, the vector S = e 2 + p 2 /p 3 e 1 .) We then define a basis Letting A := (A 1 , A 2 , A 3 ) ∈ SO(2, 1), and we then define g := (Q, A). It follows that η = Ad * (g)µ, and hence that the map (p, v) → g(p, v) −1 , (p, v) is a cross-section of the Marsden-Weinstein fibration.
Null type: Finally, orbits of negative type have C 1 := p 2 = 0 with p = 0. Again we treat the case where the vector p is future-directed, the past-directed case being similar. In this case, we may use the action of E(2, 1) to reduce µ = (m 1 , m 2 ) to the standard form We now define the null basis vector A 1 := p, and extend to a basis (A 1 , A 2 , A 3 ) by defining, for example, We let and then define g = (Q, A). Again it follows that η = Ad * (g)µ and therefore that the map (p, v) → g(p, v) −1 , (p, v) is a cross-section of the Marsden-Weinstein fibration.
The explicit parametrizations of the orbits given in Section 4.1 have the property that η * (σ µ ) = dt, and hence v µ = 1. From the explicit forms of the cross-sections, g, it is straightforward to check for each type of orbit that g(t) −1 g(t) ′ + g(t) −1 H[η(t)]g(t) lies in g µ for all t in the relevant range, as required. We may then, by direct integration, compute the gauge-transformation (24).
The integral curves of the Euler-Lagrange system are then given by (25).
Appendix A. Pfaffian differential systems with one independent variable Definition A.1. Let M be a smooth manifold. A Pfaffian differential system (I, ω) with one independent variable on M consists of a Pfaffian differential ideal I ⊂ Ω * (M ) and a non-vanishing 1-form ω ∈ Ω 1 (M ) such that ω / ≡ 0 (mod I).
Definition A.2. An integral element of (I, ω) is a pair (x, E) consisting of a point x ∈ M and a 1-dimensional linear subspace E ⊂ T x (M ) such that η| E = 0, ∀η ∈ I and ω| E = 0. We denote by V (I, ω) the set of integral elements of (I, ω). We say that I has constant rank if it is generated by the cross-sections of a sub-bundle Z of T * (M ).
We denote the set of integral curves of the system by V(I, ω).
Definition A.4. We say that the Pfaffian system in one independent variable (I, ω) is reducible if there exists a non-empty submanifold M * ⊆ M such that We call M * the reduced space. We define on M * the reduced Pfaffian system, denoted by (I * , ω), which is obtained by restricting the original system (I, ω) to M * .
A basic result is the following, a proof of which may be found in [12].
There is an algorithmic procedure for constructing the reduction of a Pfaffian system [12]. To construct the reduced space M * , we consider the projection M 1 ⊆ M of V (I, ω) to M . If M 1 is a non-empty submanifold of M , we then define (I 1 , ω 1 ) to be the restriction of (I, ω) to M 1 . We then construct V (I 1 , ω 1 ), the set of integral elements of (I 1 , ω 1 ). Repeating this construction, we inductively define The canonical contact system I on J k (R, M ) is defined to be the Pfaffian differential ideal generated by the forms η i a = dx i a − x i a+1 dt, i = 0, . . . , n, a = 0, . . . , k − 1 (where x i 0 = x i ). The independence condition of the system is given by the 1-form dt. The integral curves Γ : (a, b) → J k (R, M ) of I such that Γ * (ω) = dt are the canonical lifts j k (γ) of maps γ : (a, b) → M .
Appendix B. Constrained variational problems in one independent variable Definition B.1. Let (I, ω) be a Pfaffian differential system on a smooth manifold M and let L : M → R be a smooth function. The triple (I, ω, L) is said to be a constrained variational problem in one independent variable. The function L is referred to as the Lagrangian of the variational problem.
The Lagrangian L gives rise to the action functional L : V(I, ω) → R defined (perhaps not everywhere) on the space of the integral curves of (I, ω) by L(γ) = γ γ * (L ω) .
Definition B.2. By an extremal curve of (I, ω, L) we mean an integral curve γ that is a critical point of the functional L when one considers compactly supported variations of γ through integral curves of the system. Let us suppose that the Pfaffian ideal I is generated by a sub-bundle Z ⊂ T * (M ). We then let Z ⊂ T * (M ) be the affine sub-bundle Lω + Z. We denote by ψ ∈ Ω 1 ( Z) the restriction to Z of the tautological one-form on T * (M ), and call ψ the Liouville form of the variational problem. We let Ψ be the 2-form dψ and we consider on Z the Cartan system C(Ψ) together with the independence condition ω. Definition B.3. We say that (I, ω, L) is a regular variational problem if the Cartan system (C(Ψ), ω) is reducible. The reduced space Y ⊂ Z of (C(Ψ), ω) is called the momentum space of the variational problem. The restriction of the Cartan system (C(Ψ), ω) to Y is called the Euler-Lagrange system of the variational problem, and denoted (E, ω).
The importance of the Euler-Lagrange system comes from the following theorem (cf. [12,3] This theorem allow us to find critical points of the variational problem from the integral curves the Euler-Lagrange system. However, not all the extremals arise this way for a general variational problem. It is known that if all the derived systems of Z have constant rank, then all the extremals are projections of the integral curves of the Euler-Lagrange system [3]. Other results in this direction have been proved by L. Hsu [16]. Definition B.5. A variational problem (I, ω, L) is said to be non-degenerate if the Cartan system (C(Ψ), ω) is regular, in the sense of Definition A.6.