CONSTRAINED LAGRANGIANS AND DEGENERATE HAMILTONIANS ON MANIFOLDS: AN INDEX THEOREM

PAOLO PICCIONE AND DANIEL V. TAUSK

Abstract. A (possibly time-dependent) constrained Lagrangian L on a manifold M produce via the Legendre transform, under a certain hyper-regularity assumption, a degenerate Hamiltonian H on the cotangent bundle T M^{∗}. By constrained Lagrangian we mean a smooth function L defined on an open subset of IR × Δ, where Δ is a smooth distribution on M; examples of the theory come from sub-Riemannian geometry and from the so called Vakonomic mechanics. Non abnormal stationary points γ : [a, b] → M of the Lagrangian action functional L associated to L correspond to solutions Γ : [a, b] → T M^{∗} of the Hamiltonian H. Given such a critical point γ, using the Hamiltonian formalism, we introduce the notion of focal points along the corresponding solution Γ and we prove that, if L is convex, then the number of focal points along Γ equals the Morse index of L at γ.

1. Introduction

Sub-Riemannian geometry is the geometry of manifolds endowed with a smooth positive definite metric tensor defined on a smooth distribution. In this setup, the metric tensor defines an action functional in the set of all (absolutely continuous) curves that are everywhere tangent to the given distribution, and its stationary points are the normal sub-Riemannian geodesics. The existence of a different, although not disjoint, class of length minimizers is well known, it consists of the so-called abnormal minimizers. The normal minimizers are solutions of a Hamiltonian defined on the cotangent bundle; such Hamiltonian is degenerate, and its solutions do not satisfy the Euler–Lagrange equations corresponding to a Lagrangian defined on the tangent bundle.

In a recent paper (see [19]) the authors have introduced a general notion of Legendre transform for maps defined on arbitrary vector bundles; under a suitable hyper-regularity assumption, to each such map there corresponds a map on the dual bundle. In particular, when applied to the case of a Lagrangian map L defined on an open subset of IR × Δ, where Δ is a smooth distribution on a manifold M, this construction yields a Hamiltonian map defined on an open subset of IR×Δ^{∗}. Such Hamiltonian induces in a canonical way a Hamiltonian map H on an open subset of IR × T M^{∗}; the main result of [19] is that critical points of the Lagrangian action functional correspond to solutions of this Hamiltonian H.

While working on this problem, the authors came across an article by Professor I. Kupka and Professor W. Oliva where a similar problem was resolved in the case of constrained Lagrangians of the form L(m, v) = \frac{1}{2}g(v, v) + V (m), where m ∈ M, v ∈ T_{m}M, g a positive definite metric tensor and V a smooth map on M. The critical points of the corresponding action functional are the trajectories of the so called Vakonomic mechanics, in which case M is the configuration space of a constrained mechanical system, Δ represents the constraint, the term \frac{1}{2}g(v, v) represent the kinetic energy and V is a potential.

Date: October 2000.

2000 Mathematics Subject Classification. 37J05, 37J50, 37J60, 53C17, 70H03, 70H05.

The first author is partially sponsored by CNPq (Processo n. 301410/95), the second author is sponsored by FAPESP (Processo n. 98/12530-2).

Besides sub-Riemannian geometry and the Vakonomic mechanics, interest in more general Lagrangians comes from Control theory.

In this paper we study the second variation of the Lagrangian action functional at its critical points. Using the Hamiltonian formalism we define the notions of focal point along a solution and of focal index of the solution. We use a suitable notion of convexity of the Lagrangian function L that is the analogue of the positivity assumption of the sub-Riemannian metric tensor. We prove that, if L is convex, the index of the second variation of the action functional is finite. On the other hand, in some pathological situation focal points along a solution may accumulate; this is not the case, for instance, when the data are real-analytic.

The main result of the paper is that, given a convex constrained hyper-regular Lagrangian L, if the number of focal points along the solution of the corresponding degenerate Hamiltonian is finite, then the Morse index equals the number of focal points counted with multiplicity.

2. Constrained Lagrangians and degenerate Hamiltonians

Let us recall from [19] some facts about the Lagrangian and the Hamiltonian formalism for constrained variational problems.

Let ξ_{0} be a real finite-dimensional vector space and let ξ_{0}^{∗} denote its dual. Given a smooth map Z : U_{0} ⊂ ξ_{0} → IR defined on the open subset U_{0}, assume that dZ : U_{0} → ξ_{0}^{∗} is a diffeomorphism onto an open subset V_{0} ⊂ ξ_{0}^{∗}. Then, the Legendre transform of Z is the smooth map Z^{∗} : V_{0} → IR defined by:

Z^{∗} = E_{Z} ∘ (dZ)^{−1},

where E_{Z} : U_{0} → IR is the map:

E_{Z}(v) = dZ(v)v − Z(v), v ∈ U_{0}.

A simple computation shows that, identifying ξ_{0} and its bidual ξ_{0}^{∗∗}, then dZ^{∗} = (dZ)^{−1} and Z^{∗∗} = Z, i.e., the Legendre transform is involutive.

We now consider the following setup: let M be a smooth manifold and Δ ⊂ T M be a smooth distribution on M, i.e., a smooth vector sub-bundle of the tangent bundle T M. Let ω denote the canonical symplectic form on the cotangent bundle T M^{∗}, let Δ^{o} ⊂ T M^{∗} denote the annihilator of Δ and denote by ω_{0} the (possibly degenerate) closed 2-form on Δ^{o} given by the restriction of ω; a characteristic of Δ is a C^{1} curve η : [a, b] → Δ^{o} with η′(t) ∈ Ker(ω_{0}(η(t))) for all t.

A (time-dependent) constrained Lagrangian in M is a smooth map L : U → IR defined on an open subset U ⊂ IR × Δ; given t ∈ IR and m ∈ M, we denote by U_{(t,m)} the open set in Δ_{m} given by:

U_{(t,m)} = {v ∈ Δ_{m} : (t, v) ∈ U}

and by L_{(t,m)} : U_{(t,m)} → IR the map L_{(t,m)}(v) = L(t, v).

Let P be a smooth submanifold of M and q ∈ M be fixed; denote by Ω_{P,q} the Banach manifold of all C^{1} curves γ : [a, b] → M with γ(a) ∈ P and γ(b) = q. We consider the action functional L corresponding to L given by:

L(γ) = ∫_{a}^{b} L(t, γ̇(t)) dt

defined on the space:

Ω_{P,q}(Δ; U) = {γ ∈ Ω_{P,q} : γ̇(t) ∈ Δ, (t, γ̇(t)) ∈ U, ∀ t}.

It is well known (see for instance [19]) that the set Ω_{P,q}(Δ; U) is a smooth Banach submanifold of Ω_{P,q} around each one of its points that is not abnormal. Recall that a curve γ ∈ Ω_{P,q}(Δ; U) is abnormal (with respect to P and Δ) if there exists a non zero characteristic η of Δ that projects onto γ and such that η(a) ∈ T_{γ(a)}P^{o}; here ^{o} denotes the annihilator of a subspace. The Lagrangian action functional L is smooth around the curves γ that are not abnormal.

Let γ ∈ Ω_{P,q}(Δ; U) be a curve that is not abnormal; we want to determine conditions under which γ is a critical point of L.

Definition 2.1. The fiber derivative 𝔽L : U → Δ^{∗} of the constrained Lagrangian L : U ⊂ IR × Δ → IR is the map defined by 𝔽L(t, v) = d(L_{(t,m)})(v), where v ∈ T_{m}M. The constrained Lagrangian L is hyper-regular if the map

U ∋ (t, v) ⟼ (t, 𝔽L(t, v)) ∈ IR × Δ^{∗}

is a diffeomorphism onto an open set V. If L is hyper-regular, its Legendre transform L^{∗} : V → IR is the map:

L^{∗}(t, α) = (L_{(t,m)})^{∗}(α), (t, α) ∈ V, α ∈ Δ_{m}^{∗}.

The degenerate Hamiltonian corresponding to the hyper-regular constrained Lagrangian L is the map H : Ṽ ⊂ IR × T M^{∗} → IR defined by H(t, p) = L^{∗}(t, p|_{Δ_{m}}), where p ∈ T_{m}M^{∗} and Ṽ is the open set:

Ṽ = {(t, p) ∈ IR × T M^{∗} : (t, p|_{Δ_{m}}) ∈ V, p ∈ T_{m}M^{∗}}.

We will call Hamiltonian in M any smooth real valued function H defined on an open subset of IR × T M^{∗}; recall that a solution of the Hamiltonian H is an integral curve Γ : [a, b] → T M^{∗} of the symplectic gradient H⃗ of H, defined by

ω(H⃗(t, p), ·) = dH_{t}(p),

where H_{t} = H(t, ·).

Let us recall the following result from [19] relating the critical points of an hyper-regular constrained Lagrangian and the solutions of the corresponding degenerate Hamiltonian:

Proposition 2.2. Let L : U ⊂ IR × Δ → IR be an hyper-regular time-dependent constrained Lagrangian and let H be its corresponding degenerate Hamiltonian. Let γ ∈ Ω_{P,q}(Δ; U) be a curve that is not abnormal. Then, γ is a critical point of L if and only if γ is the projection on M of a solution Γ : [a, b] → T M^{∗} of H such that Γ(a) ∈ T_{γ(a)}P^{o}. □

We will now introduce a formalism for studying the second variation of the Lagrangian action functional at its critical points.

Definition 2.3. Let L : U ⊂ IR × Δ → IR be a constrained Lagrangian on M; we say that L is convex if for all (t, m) ∈ IR × M the map L_{(t,m)} is convex^{1} in U_{(t,m)} ⊂ Δ_{m}.

If γ ∈ Ω_{P,q}(Δ; U) is a critical point of the action functional L, then the Morse index i_{Morse}(γ) is defined to be the index of the Hessian d^{2}L_{γ} at the critical point γ, which is the (possibly infinite) dimension of a maximal subspace of the tangent space T_{γ}Ω_{P,q}(Δ; U) on which the symmetric bilinear form d^{2}L_{γ} is negative definite.

1recall that a smooth function on an open set of a vector space is called convex if its Hessian is positive semi-definite at every point.

An important class of examples of hyper-regular convex constrained Lagrangians comes from sub-Riemannian geometry, where one takes L(v) = \frac{1}{2}g(v, v), g being a positive definite metric tensor on Δ. The corresponding degenerate Hamiltonian is given by H(p) = \frac{1}{2}g^{−1}(p|_{Δ}, p|_{Δ}), and the solutions of H (or better, their projections on M) are called normal sub-Riemannian geodesics. We also obtain as a special case of our theory the so-called Vakonomic mechanics, where the constrained Lagrangian L takes the form:

L(t, v) = \frac{1}{2}g(v, v) + V (m), v ∈ T_{m}M,

where \frac{1}{2}g(v, v) is a positive definite metric tensor on Δ (the kinetic energy) and V : M → IR is a smooth function (the potential energy).

Given p ∈ T M^{∗}, recall that the vertical subspace Ver_{p} ⊂ T_{p}T M^{∗} is defined as the tangent space at p to the fiber of T M^{∗} containing p, i.e., Ver_{p} = Ker(dπ_{p}) where π : T M^{∗} → M is the canonical projection.

Definition 2.4. Let H be a (time-dependent) Hamiltonian function in M and let F : A ⊂ IR × IR × T M^{∗} → T M^{∗} denote its flow, i.e., given t_{0} ∈ IR and p ∈ T M^{∗}, the map t ↦ F (t, t_{0}, p) is the maximal integral curve of H⃗ with F (t_{0}, t_{0}, p) = p. We write F_{t,t_{0}} = F (t, t_{0}, ·).

Let Γ : [a, b] → T M^{∗} be a solution of H, so that Γ(t) = F_{t,a}(Γ(a)) for all t ∈ [a, b]. A vector field ρ along Γ is called a solution of the linearization of H⃗ if

ρ(t) = dF_{t,a}(Γ(a))ρ(a), ∀ t ∈ [a, b].

Given a smooth submanifold P of M with Γ(a) ∈ T P^{o}, we say that ρ is a P -solution of the linearization of H⃗ if in addition it satisfies ρ(a) ∈ T_{Γ(a)}(T P^{o}).

A point Γ(t), t ∈ ]a, b], is said to be P -focal along Γ if there exists a non zero P - solution ρ of the linearization of H⃗ with ρ(t) ∈ Ver_{Γ(t)}. The multiplicity mul(t) of the P -focal point Γ(t) is the dimension of the space of such ρ’s. If the number of P -focal points along Γ is finite, we define the P -focal index i_{foc}(Γ; P) of Γ by:

i_{foc}(Γ; P) = ∑_{t∈]a,b[} mul(t),

where we set mul(t) = 0 if Γ(t) is not P -focal.

The well known Morse index theorem in Riemannian geometry (see for instance [6]) or in the theory of convex Hamiltonian systems (see for instance [7]) relates the Morse index of a critical point of the action functional with the number of its conjugate points. The main result of this paper is the following generalization of the Morse index theorem to the case of constrained Lagrangians:

Theorem 2.5 (Index Theorem for constrained convex Lagrangians). Let L : U ⊂ IR × Δ → IR be a convex, hyper-regular time-dependent constrained Lagrangian and let H be the corresponding degenerate Hamiltonian. Let γ ∈ Ω_{P,q}(Δ; U) be a non abnormal curve that is a critical point of the Lagrangian action functional L and let Γ : [a, b] → T M^{∗} be the corresponding solution of H.

Then, the Morse index i_{Morse}(γ) is finite; if the number of P -focal points along Γ is finite the following equality holds:

(2.1)  i_{Morse}(γ) = i_{foc}(Γ, P).

The proof of Theorem 2.5 will be given in the rest of the paper; we conclude this Section with a few remarks on the notion of P -focal points.

Remark 2.6. Let L be a hyper-regular constrained Lagrangian on M and let H be the corresponding degenerate Hamiltonian. Let Γ : [a, b] → T M^{∗} be a solution of the Hamiltonian H with Γ(a) ∈ T P^{o} and let γ be its projection on M. A simple computation in local coordinates shows that γ is abnormal if and only if there exists a non zero P -solution ρ of the linearization of H⃗ which is everywhere vertical, i.e., ρ(t) ∈ Ver_{Γ(t)} for all t. According to Definition 2.4, it follows that every point along a solution Γ which projects onto an abnormal curve γ is P -focal.

Remark 2.7. Suppose that M, Δ and L are real-analytic; let Γ be a solution of H with Γ(a) ∈ T P^{o} whose projection γ on M is not abnormal. Then, we will prove later (see Corollary 4.3) that the number of P -focal points along Γ is finite.

It is also possible to prove that, under the assumption that the distribution Δ is strongly bracket generating, then there are only a finite number of P -focal points along any solution of H. In particular, Theorem 2.5 gives a generalization of the Morse Index Theorem for normal sub-Riemannian geodesics of Kishimoto [9].

Reduction to a differential system in IR^{n} ⊕ IR^{n}^{∗}. We will now describe how to reduce the study of P -focal points along the solution of a Hamiltonian and the second variation of the Lagrangian action functional on a manifold M to the study of suitable systems of ODE’s in the space IR^{n} ⊕ IR^{n}^{∗}. The space IR^{n} ⊕ IR^{n}^{∗} will be considered as a symplectic vector space, endowed with the canonical symplectic form

(2.2)  ω((v, α), (w, β)) = β(v) − α(w).

This construction was introduced in [18], where the reader will find additional details.

Let us consider a fixed hyper-regular time-dependent constrained Lagrangian L : U ⊂ IR × Δ → IR on the manifold M, and let H : Ṽ ⊂ IR × T M^{∗} → IR be the corresponding degenerate Hamiltonian. Let Γ : [a, b] → T M^{∗} be a solution of H with Γ(a) ∈ T P^{o} and denote by γ its projection on M. A symplectic trivialization of T T M^{∗} along Γ is a smooth family {ψ_{t}}_{t∈[a,b]} of symplectomorphisms ψ_{t} : T_{Γ(t)}T M^{∗} → IR^{n} ⊕ IR^{n}^{∗} which map the vertical space Ver_{Γ(t)} onto {0} ⊕ IR^{n}^{∗}.

Given a symplectic trivialization of T T M^{∗} along Γ, we define for each t ∈ [a, b] an isomorphism Φ_{t} of IR^{n} ⊕ IR^{n}^{∗} by:

Φ_{t} = ψ_{t} ∘ dF_{t,a}(Γ(a)) ∘ ψ_{a}^{−1}.

It is well known that the flow of a Hamiltonian preserves the symplectic form, hence Φ_{t} is a symplectomorphism for all t. Set:

(2.3)  X(t) = Φ′_{t} ∘ Φ_{t}^{−1}, t ∈ [a, b].

Consider the following linear homogeneous system of ODE’s in IR^{n} ⊕ IR^{n}^{∗}:

(2.4)  (\begin{matrix}v \\ α \end{matrix})′ = X (\begin{matrix}v \\ α \end{matrix});

it is easy to see that a vector field ρ along Γ is a solution of the linearization of H⃗ if and only if (v(t), α(t)) = ψ_{t}(ρ(t)) is a solution of 2.4. Set:

(2.5)  ℓ_{0} = ψ_{a}(T_{Γ(a)}T P^{o}) ⊂ IR^{n} ⊕ IR^{n}^{∗};

then ρ is a P -solution of the linearization of H⃗ if in addition the solution (v, α) of (2.4) satisfies the initial condition

(2.6)  (v(a), α(a)) ∈ ℓ_{0}.

We observe that X is actually a smooth curve in the Lie algebra sp(2n, IR) of the symplectic group Sp(2n, IR), and ℓ_{0} is a Lagrangian subspace of IR^{n} ⊕ IR^{n}^{∗}. The study of systems of the form (2.4) with initial condition of the form (2.6) is motivated by the above construction; Section 3 is devoted to the study of these systems. We will only be concerned with the study of systems arising from convex Lagrangians; the convexity property corresponds to a suitable positive semi-definiteness condition on a n × n block of the matrix function X.

3. Positive semi-definite symplectic systems

We will consider the symplectic space IR^{n}⊕IR^{n}^{∗} endowed with the canonical symplectic form (2.2). Let us denote by Sp(2n, IR) the symplectic group of (IR^{n} ⊕IR^{n}^{∗}, ω), which is the Lie group of all symplectomorphisms of (IR^{n} ⊕ IR^{n}^{∗}, ω), and by sp(2n, IR) its Lie algebra. Recall that elements in sp(2n, IR) can be identified with 2n × 2n real matrices of the form

(3.1)  X = (\begin{matrix}A & B \\ C & −A^{∗} \end{matrix}) ,

where A, B, C are n × n real matrices, with B and C symmetric, and where A^{∗} denotes the transpose of A.

Given real vector spaces V and W, we denote by Lin(V, W) the space of all linear maps from V to W, and by Lin(V) the space Lin(V, V); similarly, by B(V, W) we will denote the set of bilinear maps 𝔟 : V × W → IR and by B(V) the space B(V, V). By B_{sym}(V) we will denote the subspace of B(V) consisting of the symmetric bilinear forms. We will use throughout the article the canonical identification between the spaces B(V, W) and Lin(V, W^{∗}), where W^{∗} denotes the dual of W. Such identification is obtained by associating to each 𝔟 ∈ B(V, W) the linear map V ∋ v ↦ 𝔟(v, ·) ∈ W^{∗}. With this identification in mind, in (3.1) we have A ∈ Lin(IR^{n}), A^{∗} ∈ Lin(IR^{n}^{∗}), B ∈ Lin(IR^{n}^{∗}, IR^{n}) ≃ B(IR^{n}^{∗}) and C ∈ Lin(IR^{n}, IR^{n}^{∗}) ≃ B(IR^{n}); clearly, B and C are symmetric bilinear forms.

Definition 3.1. A symplectic differential system in IR^{n} ⊕ IR^{n}^{∗} defined on [a, b] is a linear system of differential equations of the form:

(3.2)  \{ \begin{matrix}v′ = Av + Bα, \\ α′ = Cv − A^{∗}α, \end{matrix} v : [a, b] → IR^{n}, α : [a, b] → IR^{n}^{∗},

where X = (\begin{matrix}A & B \\ C & −A^{∗} \end{matrix}) : [a, b] → sp(2n, IR) is a smooth curve in the Lie algebra

sp(2n, IR). The symplectic differential system (3.2) is said to be positive semi-definite if the symmetric bilinear form B(t) is positive semi-definite for all t, and if its rank rk(B(t)) = k ≤ n is constant in [a, b]; in this case, we say that k is the rank of the positive semi-definite symplectic differential system.

Given a symplectic differential system, we denote by Φ its fundamental matrix, which is the smooth curve Φ : [a, b] → Sp(2n, IR) in the symplectic group satisfying

Φ′ = XΦ, and Φ(a) = Id.

Moreover, we will look at the distribution {D_{t}}_{t∈[a,b]} of subspaces of IR^{n} given by the images of B(t):

(3.3)  D_{t} = Im(B(t)),

for all t ∈ [a, b]; clearly, the symmetry of B(t) implies Ker(B(t)) = D_{t}^{o}, where D_{t}^{o} is the annihilator of D_{t} in IR^{n}^{∗}. If the symplectic differential system is positive semi-definite of rank k, then t → D_{t} is a smooth map in the Grassmannian of all k-dimensional subspaces of IR^{n}.

Remark 3.2. In the case that the symplectic system X is obtained from a solution Γ : [a, b] → T M^{∗} of a Hamiltonian H as described at the end of Section 2, the bilinear form B(t) corresponds to the Hessian of the restriction of H to the fiber of T M^{∗} containing Γ(t), i.e., the Hessian of H with respect to the momenta. More precisely (see [18, Section 3.3]), B(t) is the push-forward of the bilinear form d^{2}(H_{t}|_{T_{γ(t)}M^{∗}})(Γ(t)) by the isomorphism given by the restriction of ψ_{t} to the vertical space Ver_{Γ(t)} ≅ T_{γ(t)}M^{∗}.

Moreover, if H is a degenerate Hamiltonian corresponding via Legendre transform to a hyper-regular constrained Lagrangian L : U ⊂ IR × Δ → IR, then ψ_{t} induces an isomorphism from T_{γ(t)}M to IR^{n} that takes Δ_{γ(t)} onto D_{t}.

Let us now choose a Lagrangian subspace ℓ_{0} of (IR^{n} ⊕ IR^{n}^{∗}, ω), i.e., an n-dimensional subspace of IR^{n}⊕IR^{n}^{∗} such that ω|_{ℓ_{0}×ℓ_{0}} = 0. It is easy to see that to each such Lagrangian subspace there corresponds a unique pair (P, S), where P ⊂ IR^{n} is a subspace and S : P × P → IR is a symmetric bilinear form, in such a way that:

(3.4)  ℓ_{0} = {(v, α) ∈ IR^{n} ⊕ IR^{n}^{∗} : v ∈ P, α|_{P} + S(v, ·) = 0 ∈ P^{∗}}.

We will be interested in solutions of the symplectic system (3.2) satisfying the Lagrangian initial condition:

(3.5)  (v(a), α(a)) ∈ ℓ_{0} ⟺ v(a) ∈ P, α(a)|_{P} + S(v(a), ·) = 0.

A smooth curve (v, α) : [a, b] → IR^{n} ⊕ IR^{n}^{∗} will be called an X-solution if it satisfies the symplectic differential system (3.2); and X-solution will be called an (X, ℓ_{0})-solution if in addition it satisfies the initial condition (3.5). We will usually identify the system (3.2) with its coefficient matrix X; for instance, we say that (v, α) is a solution of X meaning that (v, α) is a solution of (3.2), or that (v, α) is a solution of (X, ℓ_{0}) if (v, α) is a solution of X that satisfies (3.5).

Let us denote by 𝕍 the set of all v’s such that (v, α) is an (X, ℓ_{0})-solution for some α:

(3.6) 𝕍 = {v : [a, b] → IR^{n} : there exists α such that (v, α) is an (X, ℓ_{0})-solution};

moreover, for all t ∈ [a, b], we denote by 𝕍[t] the space:

(3.7)  𝕍[t] = {v(t) : v ∈ 𝕍}.

The space of (X, ℓ_{0})-solutions (v, α) is obviously n-dimensional; the space 𝕍 may have dimension smaller than n, and this motivates the following:

Definition 3.3. Let us consider a positive semi-definite symplectic differential system of the form (3.2) with a Lagrangian initial condition (3.5). We say that (X, ℓ_{0}) is abnormal if there exists a non zero (X, ℓ_{0})-solution (v, α) with v ≡ 0 in [a, b]. Equivalently, (X, ℓ_{0}) is abnormal if there exists a non zero solution α of the following:

(3.8)  α′ = −A^{∗}α, α(a) ∈ P^{o}, α(t) ∈ D_{t}^{o} for all t,

where P ⊂ IR^{n} is the space defined by ℓ_{0} as in (3.4) and D_{t} ⊂ IR^{n} is the subspace defined in (3.3).

We say that (X, ℓ_{0}) is special if (X|_{[a,s]}, ℓ_{0}) is not abnormal for all s ∈ ]a, b]. Given a positive semi-definite symplectic system with Lagrangian initial condition (X, ℓ_{0}), an instant t_{0} ∈ ]a, b] is said to be (X, ℓ_{0})-focal if there exists a non zero (X, ℓ_{0})-solution (v, α) such that v(t_{0}) = 0. The multiplicity mul(t_{0}) of a (X, ℓ_{0})-focal instant t_{0} is defined to be the dimension of the space of such (v, α)’s:

(3.9)  mul(t_{0}) = dim({(v, α) solution of (X, ℓ_{0}) : v(t_{0}) = 0}).

Observe that if (X, ℓ_{0}) is abnormal, then every instant t_{0} ∈ ]a, b] is focal.

We give some sufficient conditions for a positive semi-definite symplectic system to be special:

Lemma 3.4. If P + D_{a} = IR^{n}, then (X, ℓ_{0}) is special.

Proof. Suppose that α : [a, s] → IR^{n}^{∗} is a solution of (3.8). Since α(a)|_{P} = α(a)|_{D_{a}} = 0 and IR^{n} = P +D_{a}, it follows that α(a) = 0. This implies α ≡ 0 and (X, ℓ_{0}) is special. □

Remark 3.5. Observe that if X : [a, b] → sp(2n, IR) is a real-analytic function, then if (X, ℓ_{0}) is not abnormal then it is special.

We now pass to the description of the last ingredient of our theory, which is the so-called index form of a positive semi-definite symplectic system. To this aim, given a smooth map X : [a, b] → sp(2n, IR) of the form (3.1), we consider the family {B_{0}(t)}_{t∈[a,b]} of symmetric bilinear forms B_{0}(t) ∈ B_{sym}(D_{t}^{∗}), where D_{t} is defined in (3.3), and B_{0}(t) is defined by the following commutative diagram:

(3.10)  \xymatrix{IR^{n}^{∗} \ar[d] \ar[r]^{B(t)} & D_{t} ⊂ IR^{n} \\ D_{t}^{∗} \ar[ur]_{B_{0}(t)}}

where the unlabeled vertical arrow denotes the restriction map. By definition, if X : [a, b] → sp(2n, IR) is the coefficient matrix of a positive semi-definite symplectic system, then B_{0}(t) is positive definite for all t ∈ [a, b].

Let L^{2}([a, b], IR^{n}) denote the Hilbert space of IR^{n}-valued Lebesgue square-integrable functions on [a, b]; by H^{1}([a, b], IR^{n}) we denote the Sobolev space of IR^{n}-valued absolutely continuous maps on [a, b] whose derivative belongs to L^{2}([a, b], IR^{n}).

Definition 3.6. Let (X, ℓ_{0}) be a positive semi-definite symplectic system in IR^{n}⊕IR^{n}^{∗}; define A, B, C, P and S as in (3.1)and in (3.4). Consider the family of subspaces {D_{t}}_{t∈[a,b]} of IR^{n} defined in (3.3) and the family of symmetric bilinear forms B_{0}(t) ∈ B_{sym}(D_{t}) defined in (3.10). Let us consider the Hilbert space:

H = {v ∈ H^{1}([a, b], IR^{n}) : v(a) ∈ P, v(b) = 0};

endowed with the following inner product:

(3.11)  ⟨v, w⟩ = ∫_{a}^{b} v′(t) · w′(t) dt,

where v′(t) · w′(t) denotes the Euclidean inner product of v′(t) and w′(t). The index form I of (X, ℓ_{0}) is the bounded symmetric bilinear form given by:

(3.12) I(v, w) = ∫_{a}^{b} [B_{0}^{−1}(v′ − Av, w′ − Aw) + C(v, w)] dt − S(v(a), w(a)),

defined on the closed subspace K of H:

(3.13)  K = {v ∈ H : v′(t) − A(t)v(t) ∈ D_{t} a.e.}.

The symmetry of I follows easily from the symmetry of the bilinear forms B_{0}, C and S.

We will prove below (Lemma 3.10) that the kernel of I is related to the solutions of (X, ℓ_{0}).

We will first discuss the notion of isomorphism in the class of symplectic differential systems, which will allow us to reduce the proof of our main results to the case of symplectic systems of a simpler form.

Definition 3.7. Let X, X̃ : [a, b] → sp(2n, IR) be the coefficient matrices of two symplectic differential systems in IR^{n} ⊕ IR^{n}^{∗} and let Φ, Φ̃ : [a, b] → Sp(2n, IR) denote the corresponding fundamental matrices. Let L_{0} denote the Lagrangian subspace {0} ⊕ IR^{n}^{∗} of (IR^{n} ⊕IR^{n}^{∗}, ω). An isomorphism ϕ : X ≅ X̃ is a smooth curve ϕ : [a, b] → Sp(2n, IR) in the symplectic group satisfying ϕ(t)(L_{0}) = L_{0} and either one of the following equivalent conditions:

(a) X̃(t) = ϕ(t)X(t)ϕ(t)^{−1} + ϕ′(t)ϕ(t)^{−1}, for all t ∈ [a, b];

(b) Φ̃(t) = ϕ(t)Φ(t)ϕ(a)^{−1}, for all t ∈ [a, b];

(c) t ↦ (v(t), α(t)) is an X-solution if and only if t ↦ ϕ(t)(v(t), α(t)) is an X̃-

solution. Given an isomorphism ϕ : X ≅ X̃ and Lagrangian initial conditions ℓ_{0} and ℓ̃_{0} for X and X̃ respectively, we say that ϕ is an isomorphism of the pairs (X, ℓ_{0}) and (X̃, ℓ̃_{0}) if in addition ϕ satisfies ϕ(a)(ℓ_{0}) = ℓ̃_{0}; in this case we write ϕ : (X, ℓ_{0}) ≅ (X̃, ℓ̃_{0}).

It is easily seen that any symplectomorphism ϕ of IR^{n} ⊕ IR^{n}^{∗} that preserves the Lagrangian L_{0} is given in block matrix form by:

(3.14)  ϕ = (\begin{matrix}Z & 0 \\ Z^{∗}^{−1}W & Z^{∗}^{−1} \end{matrix}) ,

where Z : IR^{n} → IR^{n} is an isomorphism and W is a symmetric bilinear form in IR^{n}. Hence, an isomorphism ϕ between symplectic differential systems is obtained by giving a smooth curve t ↦ Z(t) of isomorphisms of IR^{n} and a smooth curve t ↦ W (t) of symmetric bilinear forms in IR^{n}. If A, B, C and Ã, B̃, C̃ denote the n × n blocks of isomorphic symplectic differential systems X and X̃ (see (3.1)), and ϕ : X ≅ X̃ is an isomorphism given by (3.14), a straightforward computation shows that the following relations hold:

\begin{matrix}Ã = ZAZ^{−1} − ZBW Z^{−1} + Z′Z^{−1}, \\ B̃ = ZBZ^{∗}, \\ C̃ = Z^{∗}^{−1}(W A + C − W BW + A^{∗}W + W′)Z^{−1}. \end{matrix} ; (3.15)

The notions of positive semi-definiteness, abnormality, index form and focal instants are invariant by isomorphisms of symplectic differential systems with Lagrangian initial conditions:

Proposition 3.8. Let (X, ℓ_{0}) and (X̃, ℓ̃_{0}) be isomorphic symplectic differential systems with Lagrangian initial conditions; let ϕ : (X, ℓ_{0}) ≅ (X̃, ℓ̃_{0}) be an isomorphism given by maps Z and W as in (3.14). Then,

• X is positive semi-definite iff X̃ is positive semi-definite;

• X is abnormal (resp., special) iff X̃ is abnormal (resp., special). If X is positive semi-definite, denote by I, K and Ĩ, K̃ the index form and the space (3.13) associated respectively to (X, ℓ_{0}) and (X̃, ℓ̃_{0}). Then, • v ∈ K iff Zv ∈ K̃;

• Ĩ(Zv, Zw) = I(v, w) for all v, w ∈ K;

• (X, ℓ_{0}) and (X̃, ℓ̃_{0}) have the same focal instants with the same multiplicities.

Proof. It is obtained by straightforward calculations using (3.15). □

We now prove that every positive semi-definite symplectic differential system is isomorphic to a system where the coefficient B is constant:

Proposition 3.9. Let X : [a, b] → sp(2n, IR) be a positive semi-definite symplectic differ-

ential system of rank k. Then, X is isomorphic to a symplectic system X̃ = (\begin{matrix}Ã & B̃ \\ C̃ & −Ã^{∗} \end{matrix})

with B̃(t) constant of the form:

(3.16)  B̃ = (\begin{matrix}0 & 0 \\ 0 & I_{k} \end{matrix}) ,

where I_{k} is the identity k × k matrix.

Proof. Since B(t) has constant rank, there exists a smooth curve Z of isomorphisms of IR^{n} such that ZBZ^{∗} is given by the righthand side of (3.16). Then, define ϕ : X ≅ X̃ as in formula (3.14) setting W ≡ 0. □

We are now ready to prove the following:

Lemma 3.10. The kernel of the index form is given by:

Ker(I) = 𝕍 ∩ H,

where 𝕍 is defined in (3.6).

Proof. By Propositions 3.8 and 3.9 we can assume that B is the constant bilinear form in the right hand side of the equality (3.16) and that D_{t} = {0}^{n−k} ⊕ IR^{k}. Then, I takes the form

(3.17)  I(v, w) = ∫_{a}^{b} [(v′ − Av) · (w′ − Aw) + C(v, w)] dt − S(v(a), w(a)).

We consider the extension of I to H × H which is defined again by (3.17). We write K as the kernel of a bounded linear operator F : H → L^{2}([a, b], IR^{n−k}) defined by:

(3.18)  F (v) = π(v′ − Av),

where π : IR^{n} → IR^{n−k} is the projection onto the first n − k coordinates. F has closed image; for, consider the restriction of F to the subspace of H consisting of those v’s whose last k coordinates vanish. Using the compact inclusion of H^{1} in L^{2}, it is easy to see that such restriction is a Fredholm operator, hence it has finite codimensional closed image. Since the image of F contains such finite codimensional closed space, it follows that the image of F is closed. Then, we have the following characterization of vectors in the kernel of I in K:

v ∈ Ker(I|_{K×K}) ⟺ I(v, ·) ∈ K^{o} = Ker(F)^{o} = Im(F^{∗}) ⊂ H^{∗}, ∀ v ∈ K,

where^{o} denotes the annihilator of a subspace of H and F^{∗} is the transpose of F. It follows that v ∈ K is in the kernel of I|_{K×K} if and only if there exists λ ∈ L^{2}([a, b], IR^{n−k}) such that: (3.19)

∫_{a}^{b} [(v′−Av)·(w′−Aw)+C(v, w)−λ·(w′−Aw)] dt−S(v(a), w(a)) = 0, ∀ w ∈ H, where λ is considered as an IR^{n}-valued map whose last k coordinates are zero. By standard bootstrap arguments, the above equality implies that the map v′ − Av − λ is absolutely continuous, and using integration by parts in (3.19) we get the following relations:

(3.20)  (v′ − Av − λ)′ = Cv − A^{∗}(v′ − Av − λ),

and

(3.21)  [v′(a) − A(a)v(a) − λ(a)]|_{P} + S(v(a), ·) = 0 ∈ P^{∗}.

Now, set α = v′ − Av − λ and observe that Bα = B(v′ − Av) = v′ − Av because v ∈ K. The conclusion follows from (3.20) and (3.21). □

4. The index theorem

Recall that if V is a real finite dimensional vector space endowed with an inner product ⟨·, ·⟩ then, given a self-adjoint linear endomorphism T : V → V the quantity

max_{∥x∥=1}⟨T (x), x⟩

equals the largest eigenvalue of T. Using this fact and the spectral theorem for compact self-adjoint operators on real Hilbert spaces, it is easy to prove the following:

Lemma 4.1. Let H be an infinite dimensional real Hilbert space endowed with an inner product ⟨·, ·⟩. Let K : H → H be a compact self-adjoint operator on H and set T = Id + K. For each integer k ≥ 1 define:

(4.1)  λ_{k}(T) = inf_{V ⊂H subspace dim(V)=k} max_{\begin{matrix}x∈V \\ ∥x∥=1 \end{matrix}}⟨T (x), x⟩.

Then:

• If T has an infinite number of eigenvalues smaller than 1 then (λ_{k}(T))_{k≥1} is the non decreasing sequence of such eigenvalues repeated according to multiplicity.

• If T has only a finite (possibly zero) number of eigenvalues smaller than 1 then (λ_{1}(T), …, λ_{n}(T)) is the sequence of such eigenvalues repeated according to multiplicity and λ_{k}(T) = 1 for k > n. □

In the notation of the statement of Lemma 4.1, let 𝔟 : H × H → IR be the symmetric bilinear form given by 𝔟 = ⟨T ·, ·⟩. The index of 𝔟, i.e., the supremum of the dimensions of the subspaces of H where 𝔟 is negative definite, is obviously given by:

(4.2)  index of 𝔟 = cardinality of {k ≥ 1 : λ_{k}(T) < 0} < +∞.

Observe also that the kernel of 𝔟 (which coincides with the kernel of T) has dimension equal to the cardinality of {k ≥ 1 : λ_{k}(T) = 0}.

Let X be a positive semi-definite symplectic differential system with Lagrangian initial conditions determined by a Lagrangian subspace ℓ_{0} ⊂ IR^{n} ⊕ IR^{n}^{∗}. Using Propositions 3.8 and 3.9 we see that, in order to prove the index theorem, there is no loss of generality in assuming that B is constant and given by the righthand side of (3.16). In that case, D_{t} is equal to the subspace {0}^{n−k} ⊕ IR^{k} ⊂ IR^{n} and the bilinear form B_{0}(t)^{−1} coincides with the restriction to D_{t} of the Euclidean inner product for all t.

Let I : K×K → IR be the index form corresponding to (X, ℓ_{0}); we consider the Hilbert space K endowed with the inner product ⟨·, ·⟩ defined in (3.11). Observe that, by our choice of B, I takes the form (3.17). Let T : K → K be the self-adjoint linear endomorphism of K which represents I, i.e., I = ⟨T ·, ·⟩. Comparing (3.17) with (3.11) and using the compact inclusion of the Sobolev space H^{1} in C^{0} it is easy to see that T is a compact perturbation of the identity of K, i.e.:

T = Id + K,

where K is a compact self-adjoint endomorphism of K.

Given t ∈ ]a, b], we consider the symplectic differential system X|_{[a,t]} obtained from X by restriction to the interval [a, t]; we define objects H_{t}, K_{t}, I_{t} and T_{t} associated to (X|_{[a,t]}, ℓ_{0}) in analogy respectively with the objects H, K, I and T associated to X. Similarly, we define maps F_{t} : H_{t} → L^{2}([a, t], IR^{n−k}) in analogy with (3.18); recall that K_{t} = Ker(F_{t}) for all t ∈ ]a, b].

We identify each v ∈ H_{t} with the map ṽ : [a, b] → IR^{n} defined by:

ṽ(s) = \begin{cases}v(s), & s ∈ [a, t], \\ 0, & s ∈ [t, b]; \end{cases}

in this way H_{t} is identified with a closed subspace of H. Moreover, observe that K_{t} = K ∩ H_{t} and that I_{t} is the restriction of I, i.e.:

(4.3)  ⟨T_{t}(x), y⟩ = ⟨T (x), y⟩, x, y ∈ K_{t}.

For each t ∈ ]a, b] and each integer k ≥ 1 set λ_{k}(t) = λ_{k}(T_{t}); it follows directly from (4.3) that t ↦ λ_{k}(t) is a non increasing function for each k.

We will now show that, for t sufficiently close to a, λ_{k}(t) is positive for all k:

Lemma 4.2. For t sufficiently close to a, the index form I_{t} is positive definite on K_{t}.

Proof. We consider an extension of I_{t} to H_{t} as in the proof of Lemma 3.10. Such extension is the index form of a symplectic differential system in IR^{n} whose upper right n×n block is positive definite. It follows from the Index Theorem^{2} in [18] that this extension is positive definite in H_{t}, and so I_{t} is positive definite in K_{t}. □

Corollary 4.3. Suppose that X is real-analytic and that (X, ℓ_{0}) is special. Then there are only a finite number of focal instants.

Proof. It follows from Lemma 3.10 and Lemma 4.2 that there are no focal instants near t = a. Moreover, if {(v_{1}, α_{1}), …, (v_{n}, α_{n})} is a basis of the space of all (X, ℓ_{0})-solutions, then it is easy to see that the focal instants corresponds to the zeroes in ]a, b] of the real analytic function det(v_{1}(t), …, v_{n}(t)). The conclusion follows. □

We will show later on that if X is special then the functions t ↦ λ_{k}(t) are continuous (Lemma 4.8); assuming this fact for a moment, we can now prove the following:

Theorem 4.4 (Index Theorem for positive semi-definite symplectic systems). Let X be a positive semi-definite symplectic differential systems with Lagrangian initial conditions determined by a Lagrangian subspace ℓ_{0} ⊂ IR^{n} ⊕ IR^{n}^{∗}. The index of the index form I corresponding to (X, ℓ_{0}) is finite; if (X, ℓ_{0}) has only a finite number of focal instants then this index equals the focal index of (X, ℓ_{0}):

index ofI = i_{foc}(X, ℓ_{0}).

2When the upper right n × n block of the coefficient matrix of a symplectic system is positive definite, the index theorem is a classical result in the literature, see for instance [7].

Proof. It follows from (4.2) that the index of I equals the number of k’s such that λ_{k}(b) < 0. Since T is a compact perturbation of the identity, the number of such k’s is finite, thus I has finite index. Observe that if there are only a finite number of (X, ℓ_{0})-focal instants then (X, ℓ_{0}) is special; hence, by Lemma 4.8 below it follows that each map t ↦ λ_{k}(t) is continuous.

It follows from Lemma 3.10 that an instant t ∈ ]a, b] is focal iff λ_{k}(t) = 0 for some k; moreover, the number of such k’s is precisely the multiplicity of such focal instant. Since t ↦ λ_{k}(t) is non increasing and there are only a finite number of (X, ℓ_{0})-focal instants, there can be at most one t ∈ ]a, b] with λ_{k}(t) = 0. Since t ↦ λ_{k}(t) is continuous and it is positive near t = a (Lemma 4.2), there exists t ∈ ]a, b[ with λ_{k}(t) = 0 iff λ_{k}(b) < 0. The conclusion follows. □

Proof of Theorem 2.5. After a reduction to a symplectic differential system (see the last part of Section 2), the results is an easy consequence of Theorem 4.4. For, observe that a symplectic trivialization {ψ_{t}}_{t∈[a,b]} induces an isomorphism between the tangent space T_{γ}Ω_{P,q}(Δ; U) and the dense subspace of K consisting of the C^{1} maps. The only thing that needs to be checked is that such isomorphism carries the Hessian d^{2}L_{γ} to the index form I. This is done using the method of Lagrange multipliers as in [19] and the techniques of [18]. □

We have left open the question of proving the continuity for the maps t ↦ λ_{k}(t); to this aim we need to introduce some technical definitions. Given Hilbert spaces H and H′, we denote by Lin(H, H′) the Banach space of bounded linear maps from H to H′.

Definition 4.5. Let H be a Hilbert space and {H_{t}}_{t∈I} be a family of closed subspaces of H, where I ⊂ IR is an interval. We say that {H_{t}} is a continuous family of subspaces if for each t_{0} ∈ I there exists a continuous map t ↦ ϕ_{t} ∈ Lin(H, H) defined in a neighborhood of t_{0} in I such that ϕ_{t} is an isomorphism of H for all t and ϕ_{t}(H_{t}) is independent of t.

We have the following criterion for establishing the continuity of a family of subspaces:

Lemma 4.6. Let H, H′ be Hilbert spaces and let t ↦ F_{t} ∈ Lin(H, H′) be a continuous map defined on an interval I ⊂ IR such that F_{t} is surjective for all t. Then, H_{t} = Ker(F_{t}) is a continuous family of subspaces of H.

Proof. See for instance [8, Lemma 2.9]. □

Using Lemma 4.6, we now want to prove that {K_{t}}_{t∈]a,b]} is a continuous family subspaces in an appropriate sense. Observe indeed that they are not subspaces of a fixed Hilbert space, but rather K_{t} ⊂ H_{t}. To overcome this problem, the following strategy is used: for each t ∈ ]a, b] consider the isomorphism ψ_{t} : H_{t} → H which is obtained by affine reparameterization of a map defined on [a, t] to a map defined on [a, b]. Then, we push-forward to H all the objects considered in H_{t} using ψ_{t}. For instance, by continuity of the family K_{t} we mean more precisely the continuity of the family ψ_{t}(K_{t}) of subspaces of H (details of a similar construction may be found in [8, Section 3]). To prove this continuity, using Lemma 4.6 the only thing left to prove is the surjectivity of the maps F_{t}:

Lemma 4.7. If (X, ℓ_{0}) is not abnormal, then F is surjective.

Proof. It was observed in the proof of Lemma 3.10 that the image of F is closed in the space L^{2}([a, b], IR^{n−k}). We need to show that it is also dense; this is equivalent to showing that the transpose F^{∗} is injective, where we consider F as a densely defined unbounded operator from L^{2}([a, b], IR^{n} to L^{2}([a, b], IR^{n−k}. It is easy to compute the transpose F^{∗}, given by:

(4.4)  F^{∗}(u) = −u′ − A^{∗}u,

defined in the dense subspace of L^{2}([a, b], IR^{n−k} consisting of those u ∈ H^{1}([a, b], IR^{n−k}) with u(a) ∈ P^{⊥}. In (4.4) we are identifying u with an IR^{n}-valued map whose last k coordinates are zero. If α ∈ H^{1}([a, b], IR^{n}^{∗}) is the map whose coordinates in the canonical basis are the same as those of u, then u ∈ Ker(F^{∗}) if and only if (0, α) is an (X, ℓ_{0})- solution. This concludes the proof. □

Lemma 4.8. In the notations as above, if (X, ℓ_{0}) is special, then for each k the map t ↦ λ_{k}(t) is continuous in ]a, b].

Proof. Using Lemma 4.6 and Lemma 4.7 we have that T_{t} depends continuously on t in an appropriate sense (see remarks above). Clearly, formula (4.1)defines a continuous function Lin(H, H) ∋ T ↦ λ_{k}(T) for each k. This concludes the proof. □

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Departamento de Matemática, Universidade de São Paulo, Brazil

E-mail address: piccione@ime.usp.br
URL: http://www.ime.usp.br/~piccione

Departamento de Matemática, Universidade de São Paulo, Brazil

E-mail address: tausk@ime.usp.br
URL: http://www.ime.usp.br/~tausk