MORSE FOR ELASTICA

. In Riemannian manifolds the elastica are critical points of the restriction of total squared geodesic curvature to curves with ﬁxed length which satisfy ﬁrst order boundary conditions. We verify that the Palais-Smale condition holds for this variational problem, and also the related problems where the admissible curves are required to satisfy zeroth order boundary conditions, or ﬁrst order periodicity conditions. We also prove a Morse index theorem for elastica and use the Morse inequalities to give lower bounds for the number of elastica of each index in terms of the Betti numbers of the path space.


1.
Introduction. The natural generalisation to Riemannian manifolds of the classical elastica problem studied by Euler and Bernoulli (see [18] or [28] for a historical survey) is the following: find critical points of the restriction of to the set Ω T of immersed curves which have prescribed initial and final points, initial and final tangent directions, and length . Here x : I = [0, 1] → M is a sufficiently regular curve with length on a complete Riemannian manifold M , k(x) = ∇ T T is the geodesic curvature of x and T =ẋ ẋ the unit tangent vector. We will use the term elastica to refer to critical points of F |Ω T . Among the special cases of elastica we distinguish pinned elastica and closed elastica as those which are critical subject to zeroth order boundary conditions and first order periodicity conditions respectively. These are special cases in the sense that they satisfy the same differential equation with special boundary conditions (see Section 2), but of course not in the sense that they are special cases of the same variational problem (i.e. the pinned and closed constraints are not special cases of the constraints in the original problem). In the absence of the length constraint, the resulting critical curves are known as free elastica. Note that the terms elastica and elastic curve are often used interchangeably, but the latter is somewhat equivocal so we will avoid it.
In modern times elastica have reappeared in several different contexts. In approximation theory they are known as nonlinear splines: a mathematical model for the drafting tool known as a spline (see eg. [17,10,20]). The better known cubic splines are used for ease of computation, not because they are a good approximation to drafting splines. Elastica also appear as a model for curve completion in computer vision [23], and as an important example of an optimal control problem with nonholonomic constraints [11]. As a consequence of the variety of applications, the problem has been approached from several different perspectives. For example Bryant and Griffiths used the theory of exterior differential systems to prove a partial integrability result in homogeneous spaces and study solutions of the Euler-Lagrange equations for elastica in the Euclidean and hyperbolic planes [4]. At about the same time, Langer and Singer obtained similar results using Frenet frames and elliptic functions [14]. Jurdjevic has shown that the Euler-Lagrange equations are completely integrable in surfaces of constant curvature using techniques from geometric control theory [11,12]. This includes in particular elastica on SO(3), by way of the double cover by S 3 . Popiel and Noakes also studied elastica in Lie groups, reducing the problem to the Lie algebra and solving for elastica in SO(3) by quadratures [26].
In [14] Langer and Singer also studied stability properties of the negative gradient flow of F , which they later termed the curve straightening flow. They proved that the only stable closed free elastica in S 2 are non-trivial closed geodesics. This motivated the study of the curve straightening flow on Riemannian manifolds as a method of finding non-trivial closed geodesics. In [15] they showed that this flow is well behaved on closed curves in R 3 with fixed length. They also proved that for almost all initial curves the flow approaches a circle, i.e. the circles are the only stable closed elastica in R 3 . Subsequent work on the curve-straightening flow was carried out by Linnèr [21,22]. Moreover, in [19] Linnèr investigates free elastica in the Euclidean plane and gives some conditions for existence and non-existence.
There is also a considerable mathematical physics literature on free elastica and critical points of other functionals depending on geodesic curvature. We mention [24], [2] and references therein. However, these authors are mainly interested in finding explicit solutions to initial value problems in semi-Riemannian manifolds, usually without constraints on the length or speed.
1.1. The Palais-Smale condition for F . The natural domain for the total squared geodesic curvature F is the set of C 1 immersions with square integrable second covariant derivative along the curve. This set, which we will denote Imm 2 (I, M ), is an open submanifold of the Hilbert manifold H 2 (I, M ) consisting of C 1 curves with square integrable second covariant derivative. The space Ω T of admissible curves can be given the structure of a submanifold of Imm 2 (I, M ) defined by the boundary conditions and the length constraint.
Let (x i ) ⊂ Ω T be a sequence which is minimizing for F , i.e. F (x i ) → inf F . It is possible to prove 1 , as in [12] p. 17, that such a sequence has a weakly convergent subsequence and therefore a limiting curve of class H 2 exists.
There are good reasons for wanting to prove stronger convergence results, such as the Palais-Smale (PS) condition. In general, a PS sequence for a smooth real valued function f on a complete Hilbert manifold X is a sequence (x i ) of points on which f is bounded and |df xi | → 0, and we say f satisfies the PS condition if any PS sequence has a (strongly) convergent subsequence. In particular, for the elastica problem this means that a PS sequence for F |Ω T must have a subsequence which converges in the H 2 metric. If the PS condition holds then the associated negative gradient flow is a positive semi-group and has at least one critical point as a limit point (cf. [25] p. 183). We note that this condition provides a constructive proof of the existence: it ensures that the method of gradient descent will locate critical points. Moreover, the PS condition makes available the minimax and Morse theoretic methods of counting critical points.
In this paper we will verify the PS condition for the elastica and pinned elastica variational problems on any complete Riemannian manifold M . For closed elastica we will do so under the additional assumption that M is compact. The relationship between these and the earlier results of [16] will be discussed at the end of this section. We will also prove a Morse index theorem for elastica and use the Morse inequalities to give lower bounds for the number of elastica in terms of the Betti numbers of the appropriate path space.
It is not possible to prove that F satisfies the PS condition on its natural domain Imm 2 (I, M ) because F is invariant under reparametrization, and the orbits of the action of reparametrization on immersed curves are not compact. It follows that any critical point is contained in a non-compact orbit of critical points at the same level of F , which contradicts the PS condition. Several methods of resolving this kind of problem are discussed in [25] p. 245, with regard to the length functional. One of these methods is to find a second function which 'breaks the symmetry', meaning it is not invariant under reparametrization, but has the same critical points as the original function in the following sense: each critical point of the symmetry breaking function is a critical point of the original function and each orbit of critical points of the original function contains a critical point of the symmetry breaking function. This is the preferred method for the length function; the energy function E = 1 0 ẋ 2 dt is not parametrization invariant and it is well known that the critical points of E (geodesics) are arc-length proportionally parametrized critical points of the length.
An alternative which is also discussed in [25] is to impose a so-called 'gauge fixing condition' to define a smooth submanifold of the domain which intersects each of the orbits only once. This method turns out to be the most appropriate for F . The condition we choose is that the curves should have constant speed 2 v. This leads to a neat simplification of F : on the submanifold Σ v of constant speed curves F coincides with the total squared covariant acceleration: We show in Lemma 2.2 that a curve x which is parametrized proportional to arc length is a critical point of F |Ω T if and only if it is a critical point of J|Σ v with v = . We therefore carry out all our analysis on J|Σ v .
Langer and Singer have proved related results in [16] but with a different objective. They aim to prove that the curve straightening flow on closed curves is well behaved by showing that F satisfies the PS condition. The parametrization invariance of F makes this impossible, so Langer and Singer restrict F to normalised curves: those parametrized proportional to arc length. They are not interested in fixing the length, because the curve straightening flow is intended to be used to find closed non-trivial geodesics whose length may not be known in advance.
The restriction of F to normalized curves still does not satisfy the PS condition because a subset of curves on which F is bounded does not necessarily have bounded length (in the terminology of Elíasson, see Section 3, this means that F is not weakly proper ). Thus we have the counterexamples to the Palais-Smale condition mentioned in [21] §1.7: the sequence x n of geodesics wrapping around the sphere n times, and circles in the plane with radii increasing without bound. The sequence of geodesics wrapping around the sphere is a counterexample to the Palais-Smale condition but it is not an example of a curve straightening trajectory that has no convergent subsequence. It is still an open question whether such an example exists on a compact manifold. The success of Linnér [22] in numerically generating periodic geodesics in sphere-like surfaces seems to suggest that the curve-straightening flow may in fact be convergent in this case.
To circumvent the difficulties outlined above, Langer and Singer consider the modified function instead, with α assumed to be positive so that F α bounds the length. They prove that F α , α > 0 satisfies the PS condition on manifolds of closed, normalized curves on compact Riemannian manifolds. They also remark that their techniques can be used to prove that F |Ω T satisfies the PS condition, thus there is some overlap with our Theorem 5.5. Nevertheless it seems worth providing a detailed treatment, particularly since the techniques used in this paper lend themselves to the development of a Morse index theorem for elastica.

2.
Lagrange multipliers and elastica. We have already mentioned that the natural domain of F is the set Imm 2 (I, M ), which is an open submanifold of the Hilbert manifold H 2 (I, M ) consisting of C 1 curves with square integrable second covariant derivative. Our approach to the geometry of such manifolds of maps aligns closely with the work of Elìasson: see [6,7], or the summaries in [9,27]. Suppose p, q ∈ M , v ∈ T p M and w ∈ T q M . We will make use of the following subsets of H 2 (I, M ) For now we will assume that Ω * and Σ v * are submanifolds of Imm 2 (I, M ) * and H 2 (I, M ) * obtained as the preimages of regular values and v respectively. In section 4 we will prove that this is true for Σ v * under the assumption that it contains no geodesics. To begin with we work with the following version of the Lagrange multiplier theorem which is similar to that in [1] p. 211.
Theorem 2.1. (Lagrange multiplier theorem) Let X be a Banach manifold, E a Banach space and f : X → R, φ : X → E differentiable maps. Suppose e 0 ∈ E is a regular value of φ so that Ω := φ −1 (e 0 ) is a submanifold of X, with T x Ω = ker dφ x split in T x X, and denotef := f |Ω. Then the following are equivalent for x ∈ Ω: For elastica we have the following By the Lagrange multiplier theorem x is a critical point of F |Ω * iff there exists β ∈ R such that x is a critical point of with domain Imm 2 (I, M ) * . From [14] p. 3 the derivative of F β in arc length proportional parametrization is (taking account of the different sign given to the Lagrange multiplier) where the higher derivatives of T are understood as weak derivatives. As in [14], supposing W | t=0,1 = ∇ T W | 0,1 = 0, setting dF β x W = 0 and using the fundamental lemma of calculus of variations gives the Euler-Lagrange equation The fact that weak solutions of this equation are also strong solutions, i.e. the higher derivatives of T are actually continuous, is a consequence of the regularity theory of elliptic operators. Alternatively, one can use a so-called bootstrap argument based on the Du Bois-Reymond lemma to show inductively that weak solutions of (3) are in fact smooth (cf. [13] p. 21). Note that the Euler-Lagrange equation (3) is not always equivalent to d(F |Ω * ) x = 0 or d(F β | Imm 2 (I, M ) * ) x = 0. When we consider F |Ω T the boundary terms in (2) vanish automatically because the tangent space T x Ω T consists of fields W along x of class H 2 which satisfy W | t=0, = 0 and ∇ T W | t=0, = 0. However, for x to be a critical point of F |Ω p,q , we require in addition to (3) that x satisfy the natural boundary conditions ∇ T T | 0, = 0 in order for the boundary terms in (2), and therefore dF β x , to vanish for all W ∈ T x Ω p,q . This is the precise sense in which pinned elastica are a special case of elastica; both satisfy the Euler-Lagrange equation, but pinned elastica necessarily have vanishing acceleration on the boundary, whereas elastica do not.
For F |Ω c , the tangent space T x Ω c consists of those fields W along x which satisfy . It then follows from (3) and derivatives thereof that a critical point satisfies ∇ k T T (0) = ∇ k T T ( ) for any k, i.e. the critical points are C ∞ -periodic.
Applying the Lagrange multiplier theorem to the restriction of total acceleration . Equivalently, changing λ to its Riesz representative in H 1 (I, R), x is a critical point of with Euler Lagrange equation (cf. [26]) Proof. If ẋ = , then equation (3) becomes combining (5) with ẋ = and derivatives thereof forcesΛ(t) = 3 2 d dt ∇ tẋ 2 (cf. [26] for = 1). Integrating and substituting into (5) gives the same equation as above, with β the constant of integration ofΛ. Corollary 1. x ∈ Ω p,q is a pinned elastica parametrized proportional to arc length iff it is a critical point of J|Σ v p,q with v = , and x is a closed elastica parametrized proportional to arc length iff it is a critical point of J|Σ v c with v = .
Proof. Follows from the previous lemma as well as the observation that when x is parametrized proportional to arc length the natural boundary conditions obtained from each of (2) and (4) coincide.
It is tempting now to prove that J λ satisfies the PS condition, since then a sequence (x i ) in Σ v v,w which is a PS-sequence for J λ |H 2 (I, M ) has a subsequence which converges in H 2 (I, M ) v,w to a critical point, and Σ v v,w is closed in H 2 (I, M ). However this doesn't prove thatJ := J|Σ v 1 , or even J λ |Σ v 1 satisfies the PS condition, since a PS sequence for J|Σ v 1 need not be a PS sequence for J λ : in general the condition |dJ xi | → 0 does not imply |dJ λ xi | → 0 because the former has a smaller domain. For multiplicity results, in particular those obtained using Morse theory, we need to prove thatJ itself satisfies the PS condition.
3. Lagrange multipliers and the PS condition. This section will serve as an outline of the method we will use to verify the PS condition for elastica. We begin with an excerpt from [27] reviewing the method of Elíasson (cf. [8], [9]), and then develop the modifications which are necessary to deal with constraints such as fixed speed. We then show how these modifications are related to the method of Lagrange multipliers, and how to use Lagrange multipliers to characterise the nullspace of the Hessian in the presence of such constraints.
Let X, X 0 be Banach manifolds modelled on B, B 0 respectively, and suppose X ⊂ X 0 , B ⊂ B 0 with the latter a continuous linear inclusion. Then X is a weak submanifold of X 0 if for any x 0 in the closure of X there is a chart (θ 0 , U 0 ) for X 0 containing x 0 , such that, setting U = U 0 ∩X, we have θ 0 (U ) ⊂ B and the restriction of θ 0 to U is a chart θ : U → θ(U ) for X. Any chart for X which arises in this way will be called a weak chart 3 at x 0 .
Note that this definition allows the topology of the weak submanifold to be finer than the relative topology.
A Finsler structure on a Banach manifold X is a continuous function v → v on the tangent bundle τ : T X → X such that the restriction to each fibre T x X is a norm, and such that in any local trivialisation Θ : τ −1 U → θ(U ) × B, and for any constant k > 1, we have with η ∈ B and ξ sufficiently close to ξ 0 = θ(x). That is, the fibre norms are locally equivalent. Suppose X is a weak submanifold of X 0 and let B denote the norm for B and | | 0 the norm for B 0 . We call a Finsler structure on X locally bounded with respect to (weak charts from) X 0 if for any x 0 in the closureX and any constant L, there is a local trivialisation Θ over a weak chart θ at x 0 and a constant c such that (ii) locally bounding with respect to X 0 if for any constants K, L and x 0 ∈X there is a weak chart (U, θ) at x 0 and a constant α such that for all ξ ∈ θ(U ) with |ξ| 0 < K and f θ (ξ) := f (θ −1 (ξ)) < L, we have ξ B < α. (iii) locally coercive with respect to X 0 if it is C 1 and for any x 0 ∈X and any constant K, there is a weak chart (U, θ) at x 0 and there exist constants for all ξ, η ∈ θ(U ) with ξ B < K, η B < K. If f is of class C 2 we have an equivalent condition: The assumption of an upper bound for |ξ| 0 is not included in the original definition of locally bounding [8] because it does not need to be assumed if f is weakly proper. However we will find it useful to be able to prove that f is locally bounding independently.
Theorem 3.1. (Elíasson [9]) Let X be a regular Banach manifold and a weak submanifold of X 0 as above, with a locally bounded Finsler structure. If f : X → R is of class C 1 and weakly proper, locally bounding and locally coercive each with respect to X 0 , then f satisfies the Palais-Smale condition.
Suppose now that Ω is a submanifold of X, which is in turn a weak submanifold of X 0 . Note that it is not necessarily true that Ω is also a weak submanifold of X 0 . Of course at any point in Ω there is a chart for X which restricts to a chart for Ω, i.e. satisfies the submanifold property, but in general it is not necessary that this chart is the restriction of a chart for X 0 , viz. a weak chart. For example, consider Σ 1 : the submanifold of H 1 (I, R 2 ) consisting of unit speed curves in the Euclidean plane. The natural charts exp h •ξ → ξ, h ∈ C ∞ (I, R 2 ) for H 1 (I, R 2 ) are weak charts with respect to C 0 (I, R 2 ). Suppose r 1 , r 2 ∈ H 1 (I, R 2 ) are parametrizations of the upper and lower halves of a circle respectively, with unit speed and the same initial and terminal points. Then in the natural chart centred at r 1 , the representative of r 2 is r 2 − r 1 . However, exp r1 1 2 (r 2 − r 1 ) will not have unit speed. This means 1 2 (r 2 − r 1 ) is not in the local image of Σ 1 , and therefore the natural charts for H 1 (I, R 2 ) do not satisfy the submanifold property for Σ 1 .
For this reason the definitions above are not directly applicable and require the following modifications.
Remark 1. In this and subsequent sections it will frequently be the case that we are interested in bounding some quantity by a constant, but the precise value of the constant is not important. It will therefore be convenient to use the symbol C to denote a floating constant, i.e. it may change during a calculation but is nevertheless independent of any variables.
Definition 3.2. Let X be a weak submanifold of X 0 and Ω a submanifold of X (which is not necessarily a weak submanifold of X 0 ), and suppose we have a smooth projection pr T Ω : T X|Ω → T Ω. We will say pr T Ω is locally bounded with respect to (weak charts from) X 0 if for any ω 0 in the X 0 -closure of Ω, and any constant L, there is a weak chart (θ, U ) for X at ω 0 and a constant C such that pr T Ω (ξ)η B ≤ C η B for all ξ ∈ θ(U ∩ Ω) with ξ B < L, and all η ∈ B (we are adopting a standard abuse of notation whereby pr T Ω is used to denote both the map and its local representative). It then follows that | pr T Ω (ξ)| ≤ C. We call f = f |Ω locally coercive with respect to (X, X 0 ) if for any ω 0 in the X 0 -closure of Ω, and any constant α, there is a weak chart (U, θ) at ω 0 and constants C + > 0 and C such that for all ξ, η in θ(U ∩ Ω) with ξ B , η B < α. Note the absence, in contrast with (7), of the square on |ξ − η| 0 .
3. Let f : X → R be a smooth function, where X is a weak submanifold of X 0 with a locally bounded Finsler structure. Suppose also that Ω is a submanifold of X with a smooth projection pr T Ω : T X|Ω → T Ω which is locally bounded with respect to X 0 . Then iff := f |Ω is weakly proper with respect to X 0 , f is locally bounding with respect to X 0 , andf is locally coercive with respect to (X, X 0 ), theñ f satisfies the Palais-Smale condition.
Proof. Let (x i ) ⊂ Ω be a sequence for which f (x i ) is bounded and |df xi | → 0.
Sincef is weakly proper we can find a subsequence converging in X 0 to some x 0 . We choose a weak chart θ : U → B at x 0 , with corresponding trivialisation Θ : τ −1 (U ) → B × B of the tangent bundle, and a subsequence ξ i := θ(x i ). Then ξ i is bounded in B because f is locally bounding. Using the local coercivity off we have where we have also used the assumption that the projection is locally bounded. Moreover, since the Finsler structure for X is locally bounded, Finally, using the assumption |df (x i )| → 0, and the B 0 -convergence of (ξ i ), we have from (10) that (ξ i ) is Cauchy in B and then the corresponding subsequence (x i ) converges in Ω because it is closed in X.
We consider again the setting of Theorem 2.1: where e 0 ∈ E is a regular value, meaning T x φ is surjective and has a split kernel for all x ∈ Ω := φ −1 (e 0 ). Then φ * T E = X × E and we have a short exact sequence of VB morphisms Suppose the above sequence admits a right split, i.e. a VB morphism such that φ * T φ • r = I Ω×E , or fibrewise: dφ x • r x = I E . Equivalently ([1] p. 183) we have a splitting T X|Ω ∼ = T Ω ⊕ Im r and a smooth projection pr T Ω : T X|Ω → T Ω given fibrewise by pr T Ω (x) = 1 − r x dφ x .
Lemma 3.4. The sequence (11) admits a local right split at any x ∈ Ω.
Proof. Since φ is a submersion at x ∈ Ω = φ −1 (e 0 ) there is a chart θ : U → B for X at x such that: Then since φ θ (γ −1 (e), w) = e for any e ∈ V, w ∈ U 2 , we also have So defining r θ : gives a local right split.
If X admits partitions of unity then a split can be constructed from such local splits.
Remark 2. Note that in contrast with Theorem 2.1, where the Lagrange multiplier is treated as an extra variable, λ is now a function X → B * and is defined in advance on all of Ω by a choice of r. This is not necessary in order to write down Euler-Lagrange equations, but it is needed for Theorem 3.7.
Lemma 3.6. Let Ω, X, r be as above with X a weak submanifold of X 0 . Suppose f : X → R is locally coercive with respect to X 0 , and that in a weak chart θ, U for X where (7) holds we also have for any η ∈ B, whenever ξ ∈ θ(U ∩ Ω) with ξ B ≤ C. Then f |Ω is locally coercive with respect to (X, X 0 ).
Proof. Since pr T Ω x = I −r x dφ x , for any η ∈ B we have Df (ξ)η = Df (ξ)(pr T Ω (ξ) + r(ξ)Dφ(ξ))η when ξ B ≤ C. Then since f is locally coercive on X with respect to X 0 , using (13), Recall that given a critical point x of f and vector fields V, W on X, the Hessian Hess x f (V, W ) := W x V f is bilinear and symmetric in V and W , and depends only on the vectors V x , W x . The Morse index of a critical point x of f is the dimension of the maximal subspace on which Hess x f is negative definite. We say f is a Morse function if Hess f is strongly nondegenerate at every critical point, i.e. if the associated self-adjoint operator is an isomorphism.
Suppose f is a Morse function on a complete Riemannian manifold X which satisfies the Palais-Smale condition. Let m i denote the number of critical points of f with index i, and β i the ith Betti number of X. Then the weak Morse inequalities state that β i ≤ m i (see [25] p. 220).
For f to be a Morse function it is necessary that the nullspace of Hess x f be trivial at each critical point x. We observe that V x ∈ null Hess x f iff Hess x is also a critical point of V f . Just as it is necessary to introduce a Lagrange multiplier in order to write the condition df x = 0 in strong form (i.e. as a differential equation), so it is also necessary to introduce a Lagrange multiplier in order to characterise the nullspace of Hess xf .
Theorem 3.7. The following are equivalent statements for x ∈ Ω a critical point off and V x ∈ T x Ω: • V x is in the nullspace of Hess xf , that is, for all W ∈ T x X.
Proof. Suppose x is a critical point off and V x ∈ T x Ω, and let V be an extension of V x to X such that V (Ω) ⊂ T Ω. Then from equation (12) we see that Now V x is in null Hess xf iff x is a critical point of Vf iff x is a critical point of (V f − (λ, V φ))| Ω , which, by the Lagrange multiplier theorem (2.1), is equivalent to for all W ∈ T x X, where µ ∈ B * , and the (W λ, V φ) term vanishes because V φ(x) = 0.

4.
Manifolds of constant speed curves. We will focus here on conditions which ensure that Σ v * is the preimage of a regular value of ν : Imm 2 (I, M ) * → H 1 (I, R) * , where H 1 (I, R) * is a suitable submanifold of H 1 (I, R). According to Lemma 2.2 we will only need to work with Σ v * , and the proofs for Ω * are similar. We recall that in order for x ∈ Imm 2 (I, M ) to be a regular point of ν the requirement is that dν x should be surjective and have split kernel. For Banach spaces the latter is not automatic, however the kernel is a closed subspace so in a Hilbert space it has a closed orthogonal complement, i.e. it splits.
Define the endpoint maps We begin with * as void. The derivative of ν is dν(x)V = 1 ẋ ∇ t V,ẋ , and every x ∈ Imm 2 (I, M ) is a regular point of ν. Indeed suppose w ∈ H 1 (I, R) and let V ∈ H 2 (x * T M ) be any solution of ∇ t V = w ẋ ẋ, then dν(x)V = w, i.e. dν(x) is surjective. In particular, it follows that the constant function v is a regular value and thus Σ v = ν −1 (v) is a submanifold of H 2 (I, M ).
Again every x ∈ Imm 2 (I, M ) p is a regular point because for any w ∈ H 1 (I, R) there is a solution of ∇ t V = w ẋ ẋ with V (0) = 0 and therefore V ∈ T x H 2 (I, M ) p . For ν 0 := ν| Imm 2 (I, M ) p,q the situation is more complicated because an element V ∈ T x H 2 (I, M ) p,q must satisfy V | t=0,1 = 0. For a given w ∈ H 1 (I, R), in order to construct such a V which is also in the pre-image dν(x) −1 w, we look for a solution of where {E i (t)} is a collection of vector fields along x which span the orthogonal complement ofẋ, and the u i ∈ H 1 (I, R) are functions which we are free to choose. Equation (15) represents a linear time dependent control system. Such a system is called controllable on [0, 1] if for any initial state V (0) and any V 1 ∈ T q M there exist controls u i and a corresponding solution V such that V (1) = V 1 . If this system is controllable then x is a regular point of ν 0 . In order to write (15) in a more familiar form we work in an orthonormal parallel frame {e k } along x so that V = V k e k ,ẋ =ẋ k e k , E i = E k i e k with repeated indices summed and V,ẋ, E i ∈ R n . Then we can write (15) asV = w ẋ ẋ + Bu (16) where u ∈ H 1 (I, R n−1 ) and B is the n × (n − 1) matrix with the coordinates of E i in the ith column. We address the question of controllability as follows. First consider the linear time dependent control systeṁ b = Bu (17) Suppose (17) is controllable and a is a solution ofȧ = w ẋ ẋ. Then given V 0 , V 1 there exists u and a corresponding b such that b(0) = V 0 − a(0), b(1) = V 1 − a(1), so that V = a + b is a solution to (16) with V(0) = V 0 and V(1) = V 1 . Thus controllability of (16) is equivalent to controllability of (17). A necessary and sufficient condition for (17) to be controllable on [0, 1] is that the matrix should be non-singular, in which case a particular control which drives the solution to b(1) = b 1 is given by u = B T W −1 (b 1 − b 0 ), (see e.g. [3] p. 76). If W is singular then there exists a non-zero y ∈ R n such that Then y T B(t) = 0 almost everywhere on I. This is only possible if there exists a real valued function α such that y = α(t)ẋ, and then since y is constantαẋ + αẍ = 0, i.e.αẋ + α∇ tẋ = 0 Since we have assumed α = 0 it then follows that x is a regular point of ν 0 if it is not a reparametrized geodesic.
In particular if ν(x) ≡ v then ∇ tẋ ,ẋ = 1 2 d dt ẋ 2 = 0 and then (18) holds iff x is a geodesic. Thus if there are no geodesics joining p, q with constant speed v (and therefore length L(x) = 1 0 vdt = v) then v is a regular value of ν 0 and Σ v p,q is a submanifold of H 2 (I, M ) p,q .
Next we characterise regular points of the restriction ν 1 := ν| Imm 2 (I, M ) v,w → H 1 (I, R) v , w . The codomain is now a submanifold of H 1 (I, R) with tangent space H 1 (I, R) 0,0 = {w ∈ H 1 (I, R) : w| t=0,1 = 0}, and an element V of the tangent space of the domain must satisfy V | t=0,1 = 0 and ∇ t V | t=0,1 = 0. Therefore instead of (15) we look for solutions of where β : I → R is any smooth function which satisfies β(0) = 0 = β(1) and otherwise β(t) = 0. Then since w(0) = 0 = w(1), any solution of (19) automatically satisfies ∇ t V | 0,1 = 0. Moreover the system (19) is controllable by precisely the same argument as above, and therefore x is a regular point of ν 1 , provided there is no solution to (18). In particular, it follows that if there are no geodesics in Σ v v,w then it is a submanifold.
Finally, let us consider the restriction ν c := ν| where the codomain is the submanifold of H 1 (I, R) consisting of periodic functions and the tangent space In this case we again look for a solution of (19) but now withẋ and w periodic, and then ∇ t V is automatically periodic. Moreover, if the system is controllable then we can set V (0) = V (1) and so x is a regular point. Again it follows that if there are no geodesics in Σ v c then it is a submanifold. We summarize the required results from above in the following Lemma. 5. The Palais-Smale condition for elastica. Our goal in this section is to prove that J|Σ v * satisfies the PS condition using Theorem 3.3. We will assume henceforth that v, (p, q), (v, w) where relevant, are such that Σ v * contains no geodesics and is therefore a submanifold by Lemma 4.1.
For Σ v v,w , the short exact sequence corresponding to (11) is 20) and we will begin by constructing a right split r for the above sequence. For x ∈ Σ v v,w and w ∈ H 1 (I, R) we will define r x w as follows. We have already observed that a solution V of (19) will satisfy dν x V = w. Setting r x w := V where V is a solution of (19) with V | t=0,1 = 0 will then satisfy the desired property: dν x r x = I. However we haven't specified the frame {E i } forẋ ⊥ or the controls u i and so r x is not yet well-defined.

PHILIP SCHRADER
First we will show how to construct a particular frame forẋ ⊥ for any where δ is the Kronecker delta function. Hence (21) defines an orthonormal frame { 1 vẋ , E i } along x. We will use this adapted frame in (19) and as in (16) we (temporarily) work in an orthonormal parallel frame along x and write (19) aṡ with V,ẋ, E ∈ R n . We will also assume that β is normalised to Since the E i are orthonormal and β is normalized we have 1 0 BB T dt = n − 1. So we let η = 1 n−1 a(1) and then u = 1 n−1 B T a(1), i.e. u i = β n−1 (E i · a(1))E i . In covariant terms this means we define where a and b are the solutions of ∇ t a = w vẋ , a(0) = 0 (24) and where by P t we mean parallel translation along x for time t beginning at p = x(0). In order to apply Theorem 3.3 we will need to prove that the projection induced by r is locally bounded. This will require some estimates for r x w 2 .
First we estimate a 2 . From (24) we have ∇ t a 2 = w 2 . Then since a(0) = 0, using the fundamental theorem of calculus and the Cauchy-Schwarz and Hölder inequalities gives from which we observe a 0 ≤ 2 w 0 and also |a| 0 ≤ 2 w 0 . Differentiating (24) gives and therefore ∇ 2 t a 2 = ( w v ) 2 ∇ tẋ 2 +ẇ 2 . Now overall we have From (25), the Cauchy-Schwarz inequality, and recalling that parallel translation gives isometries where the last step uses the inequality |a| 0 ≤ 2 w 0 proved above. Then since b(0) = 0 the same argument used for a gives b 0 ≤ C w 0 . Differentiating (25) gives Combining the preceding estimates for b shows that b 2 ≤ C w 0 + C w 0 ∇ tẋ 0 , which together with (28) yields The next task is to prove that the projection pr T Σ v v,w = I −r x dν x , henceforth abbreviated to pr, is locally bounded with respect to (H 2 , C 1 ). For this we will need to infer bounds on the local expression for pr in a trivialisation induced by a weak chart, from bounds obtained in tangent spaces, such as (30) above. In order to do so we require the following auxiliary lemmas.
Proof. From [7] Theorem 11 (or see [27] for a summary) the local expressions forẋ and ∇ tẋ with respect to the induced trivialisation Θ h are ∂ h ξ = ∇ t ξ + Q 1 (ξ) and (∇ t ∂) h ξ = ∇ 2 t ξ + Q 2 (ξ) where Q 1 , Q 2 are polynomial differential operators of order 0 and 1 respectively. Using these local expressions, and the fact that the Finsler structure on H 0 (H 2 (I, M ) * T M ) is locally bounded (cf. [27] Lemma 2) : and similarly ∇ tẋ Proof. Similar to the proof of [27] Lemma 2, we have: using the fact that G(ξ) is positive definite, the assumption that |ξ| 0 is bounded, and the local formula ( [27] eq. (5)) for (∇ i t ) h . Proposition 1. The projection pr : T H 2 (I, M ) v,w → T Σ v v,w , obtained from the right split r (23) as pr = 1 − r x dν x , is locally bounded with respect to C 1 (I, M ).
(1 + ∇ tẋ 0 ) and using (30): Let θ h , U h be the natural chart for H 2 (I, M ) centred at h. Then for any ξ ∈ θ h (U h ) with ξ 2 ≤ C, writing x = θ −1 h (ξ), we have by Lemma 5.1 that ẋ 1 ≤ c and so from (31) r Now we will write pr(ξ)η = Θ h (x, pr x V ), where Θ h is the local trivialisation corresponding to θ h , and η = Θ h (x, V ), then by Lemma 5.2 and (32) we have where we have also used the local boundedness of the Finsler structure on T H 2 (I, M ). Since the above inequality has been shown to hold for any ξ ∈ θ h (U h ) such that ξ 2 ≤ C, we have shown that pr is locally bounded.
Proof. The derivative of J at x ∈ H 2 (I, M ) is From (27) and (29) and therefore, recalling from (21) Now using the estimate for |a| 0 from (26), Using the bounds obtained for a 0 , b 0 we have and then from (33) Now suppose we work in a natural chart (θ h , U h ) centred at h. Then for any ξ ∈ φ h U h with ξ 2 ≤ C we have from Lemma 5.1 that x := φ −1 h ξ satisfies ẋ 1 ≤ C. Moreover, x(I) is contained in a compact subset of M because the length and x(0) = p are fixed. Thus, from (34) we have |dJ x r x dν x V | ≤ C|V | 1 , and then locally, for any η ∈ H 2 (h * T M ), h (ξ, η)| 1 ≤ C|η| 1 because the Finsler structure |.| 1 is locally bounded by a very similar argument to the proof of Lemma 2 in [27]. Since J is locally coercive with respect to C 1 by Theorem 3 in [27], the result now follows from Lemma 3.6.
Proof. Σ v p,q is equicontinuous by [27] Lemma 5, and since each x ∈ Σ v p,q has length v and a fixed initial point there exists a closed and bounded K ⊂ M such that x(I) ⊂ K for all x ∈ Σ v p,q . K is compact by the Hopf-Rinow theorem and therefore Σ v p,q is pointwise relatively compact (i.e. given a sequence (x i ) ⊂ Σ v p,q and fixed t 1 , (x i (t 1 )) has a convergent subsequence). Hence by the Arzelà-Ascoli theorem Σ v p,q is a compact subset of C 0 (I, M ), which contains Σ v v,w as a closed subset. For Σ v c the initial point is not fixed and so we assume that M is compact in this case. -By Proposition 1, pr is locally bounded with respect to C 1 -J|Σ v v,w is weakly proper with respect to C 1 by Lemma 5.4. -J is locally bounding with respect to C 1 by [27] Theorem 3 -J|Σ v v,w is locally coercive with respect to (H 2 , C 1 ) by Proposition 2.
We now consider the pinned elastica, i.e. J|Σ v p,q . In this case it is not possible to use exactly the same right split r because (21) required a fixed initial adapted basis for T p M but we are now allowing the direction ofẋ(0) to vary. Moreover, by the hairy ball theorem a global smoothlyẋ(0)-dependent choice of adapted basis for T p M may be impossible. Fortunately, as we will see below, a global definition will not be needed. We define r 0 in a C 1 neighbourhood of x 0 ∈ Σ v p,q as follows. Supposeẋ 0 (0) = v 0 and let U be a neighbourhood of v 0 in the sphere of radius v in T p M such that the orthonormal frame bundle is trivial over U. Fix a smooth section f of the orthonormal frame bundle over U. Then for any x ∈ Σ v p,q witḣ x(0) ∈ U solve (cf. (21)) to obtain an adapted orthonormal frame {ẋ v , F i } along x adapted toẋ. Then as before we define r 0 x by (23)-(25) (although β is actually no longer needed), but now using F i instead of E i . Theorem 5.6. J|Σ v p,q satisfies the PS condition. Proof. Let (x i ) ⊂ Σ v p,q be a PS sequence for J|Σ v p,q . Then since J|Σ v p,q is weakly proper with respect to C 1 (I, M ) by Lemma 5.4, there is a subsequence, still denoted (x i ), such that (x i ) converges in C 1 to x 0 ∈ C 1 (I, M ). We may therefore choose a natural chart θ h , U centred at h ∈ C ∞ (I, M ) and containing x 0 , and a subsequence (x i ) ⊂ U with ξ i := θ(x i ). If necessary we may then further restrict attention (and take a further subsequence) to a subet U ⊂ U such that for any x ∈ U ,ẋ(0) is contained in a neighbourhood of v 0 :=ẋ 0 (0) in the sphere of radius v in T p M which has trivial orthonormal frame bundle. We then define r 0 on U as described above. The estimates (26) − (30) are also valid with this definition of r 0 (on U ). Moreover, the proofs of Propositions 1 and 2 also carry through to prove that on U the corresponding projection pr T Σ v p,q = 1−r 0 is locally bounded with respect to C 1 , and J|U is almost locally coercive. As in the proof of Theorem 3.3 it follows that ξ i is Cauchy and converges in H 2 (h * T M ), and therefore x i converges in Σ v p,q . Theorem 5.7. J|Σ v c satisfies the PS condition, provided M is compact. Proof. For the same reasons as those given above for Σ v p,q , we can only define r c locally. We mimic the construction of the adapted orthonormal frame F i above. Then again we define r c x by (23)-(25) using F i , and the periodicity of w andẋ ensures that r c x w is C 1 -periodic as required. We may then follow the same argument as in the proof of Theorem 5.6, because J|Σ v c is weakly proper when M is compact (Lemma 5.4). For ν we have V ν = 1 ẋ ∇ t V,ẋ , and v,w , notice that the only terms in (35) and (36) which depend on the values of V away from x are those involving ∇ W V . When we calculate W V J − λW V ν these terms group together to form dJ x ∇ W V , which is zero since x is also a critical point ofJ. It will be convenient to represent the Lagrange multipliers λ, γ as elements of H 1 (I, R) and write Λ := λ −λ, Γ := γ −γ (weakly). After repeated integration by parts and several applications of Bianchi identities we find that is the same large collection of curvature terms that appears in [5] eq. (9). Thus by Lemma 3.7 and the fundamental lemma of calculus of variations, we have that V x ∈ T x Σ v v,w is in the nullspace of Hess xJ iff which we call the Jacobi equation for elastica. From Lemma 2.2 the value of Λ is known. Similiarly, if we take the inner product of equation (38) withẋ and use the constraints ẋ = 1, ∇ t V,ẋ = 0, derivatives thereof, and the Euler-Lagrange equation (5) to simplify we find (after several manipulations) It then follows that the nullspace of Hess xJ at a critical point x ofJ, being the intersection of T x Σ v v,w with the space of solutions of the system (38),(39), is finite dimensional. Lemma 6.1. If x is a critical point ofJ then Hess xJ is strongly nondegenerate iff the associated self-adjoint operator hess xJ : T x Σ v v,w → T x Σ v v,w has trivial kernel.
Proof. We have seen above that ker hess xJ is finite dimensional. Since it is selfadjoint we have ker hess xJ = coker hess xJ , and therefore if ker hess xJ is trivial then hess xJ is an isomorphism. Note that (i) does not appear in the statement of this theorem in [29] but it is assumed earlier in the paper.
We let Then (iv) is satisfied and (iii) has just been proved. For (ii), suppose there exists V ∈ N t ∩ N k , k > t, then V (τ ) = 0 for all τ ∈ (t, k) and V satisfies the Jacobi equation for elastica. But then by local uniqueness of solutions of the Jacobi equation and the compactness of I, V = 0 on the entire unit interval. As for (i) we proceed as follows. At a critical point, Hess xJ is equal to the restriction of Hess x J to T x Σ v v,w × T x Σ v v,w . Now since J is the integral of a strongly elliptic polynomial differential operator (see [27]), we have the following inequality, obtained in the proof of Theorem 3 in [27]: where α > 0, for all V ∈ T x Σ v v,w . If supp V ⊂ (0, ε) then by the Hölder inequality Then with ε sufficiently small we have that Hess xJ is positive definite on (0, ε). Now choose t i ∈ I, i = 0, 1 . . . N such that Hess xJ is positive definite on the subspace Y := {w ∈ T x Σ v v,w : w(t i ) = 0 = ∇ t w(t i )}. The map P : H 2 (x * T M ) → × i (T x(ti) M ) 2 defined by P i (V ) := (V (t i ), ∇ t V (t i )) is surjective so ker P has finite codimension. Then Y = ker P ∩ T x Σ v v,w has finite codimension as a subspace of T x Σ v v,w , and (i) holds. We have now proved the following theorem. NowJ is a Morse function iff 0 and 1 are not conjugate along any elastica satisfying the given first order boundary conditions, in which case we say that the boundary conditions themselves are non-conjugate. Proof. This is an application of the Morse inequalities (see eg. [25] p. 220).
Concluding remarks. In section 5 we have cat(Σ v * ) as a lower bound for the total number of critical points. Typically we would compare the homotopy type (and therefore category) of this path space with that of the based loop space. However in the case of elastica it is not clear that any such general statements can be made, since the based loop space may contain homotopy classes of curves which all have length greater than v. It might be interesting to study the topology of Σ v * . At the beginning of section 6 it was explained that J|Σ v p,q and J|Σ v c are not Morse functions. However, we have not excluded the possibility that they are Morse-Bott functions; it may be that the critical sets are nondegenerate critical manifolds. Finally, it is possible that Theorem 3.3, or some variant thereof, will be useful for other constrained variational problems.