Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems

We consider the question of linear stability of a periodic solution z(t) with finite spatio-temporal symmetry group of a reversible-equivariant Hamiltonian system obtained as a minimizer of the action functional. Our main theorem states that z(t) is unstable if a subspace W associated with the boundary conditions of the minimizing problem is a Lagrangian subspace with no focal points on the time interval defined by the boundary conditions and the second variation restricted to the subspace W at the minimizer has positive directions. We show that the conditions of our theorem are always met for a class of minimizing periodic orbits with the standard mechanical reversing symmetry. Comparison theorems for Lagrangian subspaces and the use of time-reversing symmetries are essential tools in constructing stable and unstable subspaces for z(t). In particular, our results are complementary to the recent paper of Hu and Sun [14].


Introduction
The discovery and proof of existence of the collision-less Figure-Eight periodic orbit of Chenciner-Montgomery-Moore [7] in the planar Newtonian three body problem has renewed interest in the search for periodic solutions of the Newtonian N -body problem.Since then, many new periodic solutions of the N -body problem with spatio-temporal symmetries have been found by various authors, see for instance [5,6,8,11].In this paper we address the question of linear instability of periodic solutions with spatio-temporal symmetries in reversible-equivariant Hamiltonian systems obtained as minimizers of a Lagrangian action functional.
Our main result is Theorem 5.1 which states that a periodic solution z(t) obtained via minimization of an action functional is unstable if a Lagrangian subspace W associated with the boundary conditions of the minimizing problem has no focal points on a fundamental domain of the spatio-temporal symmetry of z(t) and the second variation restricted to W has at least one positive direction.This second condition is expressed by assuming that the dimension of the kernel of the second variation of the minimization problem is strictly less than half the dimension of phase space.If the second variation of the variational problem at the minimizer vanishes only for the direction corresponding to the flow of the periodic solution, then z(t) is hyperbolic on its energy surface.Following [11], we call a periodic solution of cyclic type if it has a discrete spatio-temporal symmetry group, but no time-reversing symmetry.Otherwise, it is called of dihedral type.Our result applies to both cases.A special case of the cyclic type, occurs when the symmetry group is generated by the identity.This is the classical problem of periodic boundary condtions in the calculus of variations, and our result gives sufficient conditions that the minimizer is unstable, which generalizes an old theorem of Poincaré for minimizing closed geodesics on orientable surfaces.
This result is obtained by generalizing the approach of Offin [20] who shows that the spatial isosceles three-body configuration is unstable, and in fact, hyperbolic given some nondegeneracy condition on the periodic solution.The argument in [20] is to build a Lagrangian subspace of variations associated with the boundary conditions of the minimization problem and show that this Lagrangian subspace has no focal points for all times.
The proof of our main theorem is done by constructing stable and unstable subspaces for z(t).A result of Contreras and Iturriaga [9] shows that the nonexistence of conjugate points for z(t) guarantees the existence of flow invariant Lagrangian subbundles containing the stable and unstable subspaces of z(t).We show that z(t) has no conjugate points by proving the nonexistence of focal points for all positive times for a Lagrangian subspace associated to the boundary conditions of the minimization problem used to obtain z(t).We then use a theorem of Hartman [13] linking the nonexistence of conjugate points to the nonexistence of focal points for a Lagrangian subspace to obtain the Lagrangian subbundles.The existence of (nonempty) stable and unstable subspaces for z(t) inside the Lagrangian subbundles is a consequence of the assumption of nondegeneracy of the variational problem and the symmetry properties of the Hamiltonian system.
Recently, Hu and Sun [14] have addressed the linear stability of periodic solutions with spatiotemporal symmetries obtained as critical points of an action functional.Their main results give criteria for stability in terms of the computation of a Maslov index and their proof relies on a generalization of Bott's iteration formula [3].In particular, they show that a cyclic periodic solution is unstable if the Maslov index is odd.Bolotin and Treschev [2] use the Hill's determinant to obtain instability criteria for action minimizing periodic orbits.In particular, they provide formulae for computing the Morse index of a critical point after symplectic reduction via a continuous symmetry group action.
The paper is structured as follows.The next section introduces the standard set-up of G-reversible equivariant Hamiltonian systems.Section 3 discusses periodic solutions with spatio-temporal symmetry group obtained via minimization of an action functional.In Section 4, we study a subspace W linked to the boundary conditions of the variational problem and give a characterization which guarantees that W is a Lagrangian subspace.We show that for a class of periodic orbits with timereversing symmetry R = diag(I, −I), W is a Lagrangian subspace.Section 5 contains the main result, Theorem 5.1, including corollaries and we also show that the theorem holds for a the class of periodic orbits with time-reversing symmetry R = diag(I, −I) mentioned above.The section ends with a comparison of our main result with the result of Hu and Sun mentioned above.Section 6 expands on the concept of Lagrangian subspace, introduces a useful comparison theorem and discusses the relationship between focal and conjugate points of a Lagrangian distribution and we show the existence of the flow-invariant Lagrangian subbundles E and F. In the final section, we characterize E and F using iterates of the Lagrangian subspace W by the relative monodromy matrix and we end with the proof of Theorem 5.1.

Preliminaries
Let X be an open subset of R n and G be a subgroup of the Euclidean group E(n) acting linearly and faithfully on X .That is, if x ∈ X and g ∈ G, there is an orthogonal representation ρ : G → O(n) and a vector u ∈ R n such that g.x = ρ(g)x + u.
The cotangent bundle of X denoted by M = T * X is a smooth symplectic manifold and we let (q, p) = (q 1 , . . ., q n , p 1 , . . ., p n ) be global coordinates on M. The canonical symplectic form is given by The action of G on X lifts semisymplectically on M as follows.There exists a homomorphism χ : G → Z 2 = {+1, −1} with Γ := ker χ such that g.(q, p) = (ρ(g)q + u, χ(g)[ρ(g −1 ) + u] T .p) Note that since Γ is a normal subgroup of G of index 2, there exists R ∈ G\Γ such that G = Γ, R .In particular, if R is of order 2 we can write G = Γ Z 2 (R).For this paper, we make the following assumption.
Assumption 1 R is an involution.
Note that any mechanical system with Hamiltonian function of the form Kinetic+Potential has timereversing symmetry R = diag(I, −I) which is an involution.Moreover, R commutes with the action of Γ.Note that elements g of G are symplectic or antisymplectic symmetries respectively depending on whether χ(g) = +1 or χ(g) = −1.The symplectic form satisfies Let H(q, p) denote a smooth G-invariant Hamiltonian function on M, then a smooth G reversibleequivariant Hamiltonian vector field X H on M is defined by That is, the following relations are satisfied: X H γ = γX H for all γ ∈ Γ and X H R = −RX H . Let φ t be the flow map of X H , by G-reversible equivariance, for all γ ∈ Γ, φ t satisfies Let dφ t be the linearized flow, by time-reversibility of X H , differentiating φ t R = Rφ −t we obtain We refer to a Hamiltonian function H as being convex if the Hessian of H with respect to p is positive definite and H is superlinear if In particular, all Hamiltonian functions of the type Kinetic+Potential are convex and superlinear.
Assumption 2 We assume that H is a convex, superlinear Hamiltonian function.
Without loss of generality and to simplify the discussion, we make the next assumption Assumption 3 The configuration space X ⊂ R n is a smooth G-manifold where G does not contain a subgroup G acting properly and freely on X .
If such a subgroup G exists, we can use symplectic reduction to obtain a smooth reduced phase space M µ where µ is a regular value of the momentum map J : M → (g ) * with g the Lie algebra of G .Therefore, no elements of G/G act properly and freely on M µ .Given assumption 3, the action on phase space is given by orthogonal matrices.For g ∈ G and (q, p) ∈ M, we have g(q, p) = (ρ(g)q, χ(g)ρ(g)p).
Suppose z(t) is a periodic solution of X H with minimal period T > 0. We recall basic facts about spatio-temporal symmetry groups of periodic solutions, see Golubitsky et al [12] for more details.Because X H is an autonomous system of differential equations, the group S 1 R/T Z acts on z(t) by phase shifts.Let Σ be the subgroup of Γ which leaves z(t) setwise invariant: σ{z(t)} = {z(t)} for all σ ∈ Σ.Let ∆ be the subgroup of Σ leaving z(t) pointwise invariant: δz(t) = z(t) for all δ ∈ ∆.Since Σ leaves {z(t)} invariant, it acts as phase shift on z(t) and so there exists a homomorphism θ : Σ → S 1 .Since ∆ is a normal subgroup of Σ, then Σ/∆ is isomorphic to a subgroup of S 1 .The pair (Σ, ∆) is called the spatio-temporal symmetry group of z(t).Another way of describing the spatio-temporal symmetry group of A T -periodic solution z(t) such that Σ/∆ is isomorphic to the cyclic group Z m for some m ∈ Z is called a discrete rotating wave.In particular, there exists a generator σ of Σ/∆ such that If z(t) is not time-reversible, then the time-reversing symmetry R applied to z(−t) yields a distinct periodic solution u(t) with spatio-temporal symmetry It is typical to use the automorphism of G given by κ(σ) = RσR.Both periodic solutions z(t) and u(t) are important in the analysis of stability.
In the discussion that follows, we focus on the case of a discrete rotating wave z(t) with trivial spatial symmetry group: ∆ = 1.We treat the case ∆ = 1 as a consequence of the trivial ∆ case.

Periodic solutions via variational methods
Let L = L(c(t), ċ(t)) be the Lagrangian obtained from the Hamiltonian system X H via the Legendre transformation.Let π denote the canonical projection π : M → X .
Let Λ = H 1 ([0, T ], X ) be the space of absolutely continuous closed loops with L 2 derivatives from [0, T ] to X .The following action of Σ ⊂ G on H 1 ([0, T ], X ) has been used recently by Ferrario and Terracini [11] in the context of the N -body problem.For all σ ∈ Σ and for all t ∈ [0, T ] We denote by Λ Σ the space of loops in Λ fixed by (3).Note that Λ Σ is a closed linear subspace of Λ.
Consider an action of Σ on [0, T ] given by σ.t = (t + θ(σ)) mod T where θ : Σ → S 1 R/T Z is a homomorphism, similar to the one described in the discussion on spatio-temporal symmetries.Then Λ Σ can also be described as the set of Σ-equivariant closed loops x : [0, T ] → X : x(σ.t) = ρ(σ)x(t).

Consider the action functional
and the following two different but related variational problems: min and min satisfying the symmetric boundary condition where σ ∈ Γ is of order m.The next result is proved in the case of the N -body problem in [11] and establishes the equivalence between the above two variational problems for obtaining periodic solutions with spatio-temporal symmetry.Note that the result in [11] also contains the case of a discrete rotating wave with time-reversing symmetry.We present a straightforward generalization.
X ) be a minimizer of variational problem ( 5) with boundary conditions (6).Then there exists a periodic solution z(t) with c(t) = π(z(t)) of X H with spatio-temporal symmetry (2) and closed equivariant loop c(t) in Λ Σ .Therefore, the minimization problem (4) and the minimization problem (5) with boundary condition (6) are equivalent.
Proof: (i) From the spatio-temporal symmetry condition (2) we obtain σz(0 X ) be a solution of the variational problem (5) with boundary conditions (6).Let p(t) denote the momentum corresponding to c(t) via the Legendre transform.Then, the first variation formula reads From this calculation, it follows that σ T p(T /m) = p(0) and since σ is orthogonal then p(T /m) = σ p(0).Now, setting z(t) = (c(t), p(t)) we find z(T /m) = σz(0).By equivariance of the flow we have σz(t) = z(t + T /m).
We now show the equivalence of the minimization problems.If c(t) = π(z(t)) provides a minimizing solution for variational problem (4), then σc(0) = c(T /m) and min On the other hand, if c(t) provides a minimizing solution of (5), then It follows that A L (c) = A L (c) and that c or c solves both variational problems.
For the remainder of this paper, we study T -periodic solutions z(t) of X H obtained from variational problem (5) and satisfying the boundary conditions (6) and u(t) the image of z(t) by R. Note that since π(z(t)) = c(t), then π(u(t)) = c(−t) and so c(−t) satisfies the variational problem min satisfying the symmetric boundary condition The following are standard results in variational calculus, see [17].Let c(t) be a minimizer of ( 5) satisfying (6).Then, the first variation δA L (c(t)) vanishes for variations Moreover, the second variation δ 2 A L (c(t)) is a bilinear form defined on T X × T X .For variations ξ satisfying the boundary condition σ.ξ(0) = ξ(T /m), the second variation is a semi-definite symmetric bilinear form.That is, where η is the momentum variation associated to ξ via the linearized Legendre transform.Similarly, for the variational problem (7) with boundary conditions (8), where ξ is a variation satisfying ξ(0) = κ(σ)ξ(T /m) and η is the momentum variation associated to ξ via the linearized Legendre transform.
We make the following assumption on the variational problem ( 5) with boundary conditions (6).

Lagrangian subspaces
We now introduce some formalism from symplectic geometry which is useful in the discussion on stability of periodic solutions.A Lagrangian subspace λ is the largest subspace of a symplectic space E on which the symplectic form vanishes.That is, and λ has maximal dimension.This implies dim λ = 1 2 dim E. Because dφ t preserves the symplectic form, for any Lagrangian subspace λ in T M, dφ t λ is also a Lagrangian subspace for all t ∈ R.An important Lagrangian subspace distribution is the vertical space V ⊂ T M defined by dπ(v) = 0 for all v ∈ V .Note that the vertical space distribution is not invariant with respect to the linearized flow, that is, dφ t V is typically not a vertical space for all t ∈ R. Two points θ 1 , θ 2 ∈ M are said to be conjugate if θ 2 = φ t 0 (θ 1 ) for some t 0 = 0 and where S = diag(σ, σ) and T denotes transpose.Proof: Let P = S T dφ T /m .We call P the relative monodromy matrix.Rewrite W as Recall that c(t) = π(z(t)) and consider the mapping F : T z(0) M → T c(0) X defined by Then F (0) = 0 and ∂F ∂η (0) = −dπP 0 I = 0 since c(t) has no conjugate points in the interval [0, T /m].By the implicit function theorem F (ξ, B 0 ξ) ≡ 0 for all σξ(0) = ξ(T /m) and for some linear map B 0 .Thus, the mapping F defines the subspace W of dimension n given by (12).
In order to characterize the bilinear form b more precisely, we describe the action of P on elements of W .Let H(z(t)) and V (z(t)) be respectively the horizontal and vertical spaces in T z(t) M. For some > 0, dφ t W ∩V = {0} for all t ∈ (− , ) and so there exists a linear mapping B(t) : H(z(t)) → V (z(t)) such that B(0) = B 0 and all elements of dφ t W are of the form (ξ(t), B(t)ξ(t)) for all t ∈ (− , ).
Because dφ T /m W is an n-dimensional subspace and by definition of W we have dφ T /m W ∩V = {0}, then there exists a linear map C 0 such that all elements of dφ T /m W are of the form (ξ 1 , C 0 ξ 1 ) T .Again, choosing > 0 small enough, we have dφ τ +T /m W ∩ V = {0} for all τ ∈ (− , ) and there exists a linear mapping C(τ ) : There is a simple case where b is symmetric.
) then using ( 13) is obviously symmetric.In fact, we have the more general statement.
A then the previous equality holds and so b is symmetric.Suppose A where A is a symmetric matrix.Before we look at the next example, we recall results from Roberts [21] and some consequences.

Lemma 4.4 ([21]
) Let dφ t be the linearized flow of the G-reversible equivariant vector field X H near the R-reversible periodic solution z(t) with spatio-temporal symmetry Sz(t) = z(t + T /m).Then, The proof is done by taking the left and right hand side of the identities and showing they satisfy the same linear initial value problem.From the second equation in (14), setting t = T /m and rearranging the terms we obtain SR(dφ T /m ) −1 = dφ T /m SR.
We now check that (dφ To do this, let Q = RS T , and set M (t) = dφ t (z(0)) to be the fundamental matrix solution.Then, let Using arguments as in [21], one can show that Z(s) and U (s) satisfy with initial condition ξ(0) = I.For Z(s) use the fact that Qz(T /2m − s) = z(T /2m + s) and For s = T /2m, equation ( 15) is Example 4.5 (R = diag(I, −I)-symmetric periodic orbit) Consider a periodic orbit z(t) of an n-degree of freedom Hamiltonian system.Suppose that z(t) is R-symmetric where R = diag(I, −I) with spatio-temporal symmetry S: Rz(t) = z(−t) and also there exists We consider the case where σ is an involution and σ = I, this means m = 2.Because Fix(SR) = diag(σ, −σ) with σ an involution, then dim Fix(SR) = n and Fix(SR) is a Lagrangian subspace.We define the Lagrangian subspace λ := (dφ T /4 ) −1 Fix(SR).Then, and by inspection, it is straightforward to see that λ = W as defined by ( 12) and so W is a Lagrangian subspace.In this case we have σ T C 0 σ = −B 0 .If σ = I, then λ = Fix(R) and P = dφ T and we also have P λ = Rλ because dφ t R = Rdφ −t for all t.

Statements of the main results
We can now state explicitly the main results of this paper.The first one gives sufficient conditions for a periodic orbit with spatio-temporal symmetry to be unstable and we state this result for ∆ = 1.
The case ∆ = 1 is obtained as a corollary.
Theorem 5.1 Suppose that the Hamiltonian H and the group G satisfy assumptions 1 through 3.
Let z(t) be a T -periodic solution of the G-equivariant Hamiltonian system X H with spatio-temporal symmetry (2) where H(z(t)) = e is a regular value of H. Assume z(t) is obtained as a minimizer of the action functional (5) satisfying the boundary condition (6).Suppose W is a Lagrangian subspace with no focal points in the interval [0, T /m] and that assumption 4 is satisfied.Then, z(t) is an unstable periodic solution.Moreover, if n = 1 then z(t) is hyperbolic and therefore isolated on its energy level.
The proof is found in Section 7. We now lift the restriction that ∆ = 1 and an immediate corollary is obtained for periodic solutions with nontrivial and compact spatial symmetry group ∆.Indeed, in this case, Fix(∆) is a proper symplectic submanifold of M, invariant for the flow of X H and so there exists a Hamiltonian function H = H | Fix(∆) on Fix(∆) defining a Hamiltonian vector field X H on Fix(∆), see Marsden [16] for details.
Corollary 5.2 Let z(t) be a T -periodic solution such as in Theorem 5.1 with spatio-temporal symmetry group (Σ, ∆) and ∆ nontrivial and compact.If the minimizing problem satisfies the same conditions as Theorem 5.1, then z(t) is unstable in Fix(∆) and hyperbolic if n = 1.
Proof: We apply Theorem 5.1 to z(t) as a T -periodic solution of X H with trivial spatial symmetry group.
Corollary 5.3 Under the assumptions of Theorem 5.1, for any two times t 0 < t 1 , π(z(t)) locally minimizes the action between π(z(t 0 )) and π(z(t 1 )) subject to fixed boundary conditions at t 0 , t 1 .
Proof: In the theory leading to the proof of Theorem 5.1, the assumption that W has no focal points in [0, T /m] leads to the conclusion that W has no conjugate points for all t ∈ R (see Theorem 6.7) and so π(z(t)) locally minimizes the action subject to variations ξ satisfying ξ(t 0 ) = ξ(t 1 ) = 0.
Finally, we conclude this section with an application of Theorem 5.1 to the context of Example 4.5 and which shows that there is a large class of periodic orbits which are unstable.(1) If σ = I, assumption 4 is satisfied.Then z(t) is unstable and locally minimizing over any compact time interval with fixed boundary condition.

Proof:
The subspace λ = W is Lagrangian as shown in Example 4.5.We begin with the case σ = I.The case σ = I is a consequence of this case.If σ = I, then Fix(SR) ∩ V = {0} and so by definition of W , dφ T /2 W does not intersect the vertical.We now show by contradiction that W has no focal points in [0, T ].Suppose that W has a focal point at T /2 − t 0 ∈ (0, T /2).Let ζ ∈ W and where the next to last equality holds by (15) with m = 1.By the form of R, this implies that dφ t ζ crosses the vertical at t = T /2 − t 0 and T /2 + t 0 , but this means z(T /2 − t 0 ) and z(T /2 + t 0 ) are conjugate points, contradicting the minimizing property of π(z(t)).The assumptions of Theorem 5.1 are satisfied, so z(t) is unstable and Corollary 5.3 applies.We now turn to the case σ = −I.We begin by showing that W has no focal points in the intervals (0, T /4) and (T /4, T /2).As above, we claim that W has a focal point at T /4 − t 0 in the interval (0, T /4) if and only if W has a focal point at T /4 + t 0 in the interval (T /4, T /2).We now check this assertion.Let ζ ∈ W and ∈ Fix(SR) such that ζ = dφ −1 T /4 .Then where again the next to last equality holds by (15) with m = 2.Because SR = diag(σ, −σ) where σ is an orthogonal matrix, the claim is proved.As above, this means z(T /4 − t 0 ) and z(T /4 + t 0 ) are conjugate points which contradicts the minimizing property of π(z(t)).
Remark 5.5 (1) In the case σ = I, the orbit of z(t) could in principle develop conjugate points along dφ t and so Corollary 5.3 does not apply.
(2) The case σ = −I occurs for instance in the case of the Hip-Hop orbit of the Newtonian four-body problem.The instability of this case is proved in [15].

Comparison to result of Hu and Sun
In a recent paper, Hu and Sun [14] prove an instability result for cyclic periodic orbits of Hamiltonian systems.Let L(t) be a continuous family of Lagrangian subspaces and α be a fixed Lagrangian subspace.They define the Maslov index of L(t) with respect to α as where [ : ] denotes the intersection number and 0 < << 1.This definition satisfies the axiomatic definition of Maslov index, see Cappell et al. [4] for instance.Note that this Maslov index depends on the choice of α.Consider the symplectic vector space (R 2n × R 2n , ω = ω ⊕ −ω) and we consider Lagrangian subspaces in this space.In particular, Gr A is a Lagrangian subspace for any 2n-dimensional symplectic matrix A. Note that this result does not require π(z(t)) to be a minimizer of the variational problem (4).In fact, in [14] is shown the relationship between the Morse index m − (π(z(t))) of a critical point of ( 4) and the Maslov index: The Hip-Hop orbit of the spatial four-body problem shows the limitation of Theorem 5.6.After reducing by the standard symmetries and also via an order four symmetry, the problem has three degrees of freedom.The Hip-Hop is a minimizer of the Lagrangian action functional with Italian symmetry x(t) = σx(t + T /2), where σ = −I 3 .Therefore, m − (π(z(t))) = 0 and dim ker(σ − I) = 0 which implies µ(Gr (S), Gr (M (t))) = 0 by formula ( 16) and so Theorem 5.6 does not apply.In Lewis et al. [15], the Hip-Hop orbit is shown to be unstable using a method which inspired the approach taken in this paper.

Lagrangian subspaces, focal points and conjugate points
Denote by Λ n the topological space of all Lagrangian subspaces in R 2n called the Lagrangian Grassmannian; it is a submanifold of the Grassmannian manifold of R 2n .For a fixed Lagrangian subspace If λ, β ∈ Λ 0 (α), then we may represent β as summands in α and λ.More generally, if λ and α are nonintersecting Lagrangian subspaces and C : λ → α is a linear map, then β = graph C = { +C | λ ∈ Λ} is a Lagrangian subspace if and only if ω(λ, Cλ) is a symmetric quadratic form.This guarantees that a parametrization for the Lagrangian subspaces in the open set Λ 0 can be obtained in terms of symmetric forms on a subspace β ∈ Λ 0 (α).We can think of the map β → ω(λ, Cλ), where β = graph C, as a coordinate mapping on the chart Λ 0 (α).Note that the dimension of ker ω(λ, Cλ) equals the dimension of β ∩ λ.For β, λ ∈ Λ 0 (α) and β = graph C, the symmetric form on λ associated with β is denoted The introduction of the symmetric form Q is useful for establishing comparison theorems for Lagrangian subspace distributions along periodic solutions arising as minimizers of a variational principle.In particular, Offin develops such comparison theorems in [18] and [19] to prove, respectively, the instability of periodic solutions of Hamiltonian systems with an involutory time-reversing symmetry and the instability of closed geodesics with an involutory time-reversing symmetry on a Riemannian manifold.We now state the comparison theorem which is used for the proof of our main theorem.
Theorem 6.1 (Offin [19]) Let dφ t (x) be the linearized flow of the Hamiltonian vector field X H (x) on R 2n and let λ, λ * denote Lagrangian planes of R 2n with V the vertical plane.Assume that Let z(t) = φ t (z(0)) be a T -periodic solution with spatio-temporal symmetry Sz(t) = z(t + T /m) where S = diag(σ, σ), lying on the energy surface H −1 (e).The linearized flow dφ t in a neighborhood of z(t) satisfies the variational equation The matrix dφ t (z(0)) is the fundamental matrix solution of system (17); that is, dφ 0 (z(0)) = I.We let M (t) := dφ t (z(0)).System ( 17) is a linear Hamiltonian system (with periodic coefficients) and so M (t) is a symplectic matrix for all t ∈ R. M (t) preserves the symplectic form By time-reversibility of the vector field X H , one easily verifies that where u(t) = Rz(−t).We use frequently the first identity in (14) from Lemma 4.4, but written for the matrix M (t): For the fundamental matrix N (t), the equivalent formula is which can be verified directly by checking that both sides of the equation solve (18) with dX H (u(t + T /m)).Note that M (t + T /m) = SM (t)P and we define P = κ(S)N (T /m), therefore N (t + T /m) = κ(S T )N (t)P .
Reverting back to M (t), we obtain, From ( 19), we see that for any k ∈ Z where for k < 0, S k = (S T ) −k .For the remainder of this section we assume If z(t) is R-symmetric then u(t) = z(t) and π(z(t)) has dihedral symmetry.Otherwise it is a distinct periodic orbit with spatio-temporal symmetry u(t) = (RSR)u(t + T /m) as described in the introduction and so π(z(t)) is cyclic.We now establish a few results in this last case.

Cyclic case
Suppose that u(t) = z(t) and consider the subspace since dπR = rdπ.Therefore, We apply Theorem 6.1 to the Lagrangian subspaces W and W and obtain the following result.
Since S is nonsingular on the configuration components, then Iterating this procedure, we conclude that M (t)W ∩ V = {0} for all t ∈ [0, ∞).To show that N (t)W ∩ V = {0} for all t ≥ 0, we begin by noting that where the last equality holds by (19) and M (−t + T /M )W for t ∈ [0, T /m] does not intersect the vertical by Assumption 5. Thus, N (t)W has no focal points for t ∈ [0, T /m].Also, note that π(u(t)) is a minimizer of the action functional in the interval [0, T /m] and so for ξ ∈ dπ(W ) and η the corresponding momentum.Thus, the second variation evaluated at π(u(t)) is nonnegative for elements in dπ(W ); that is, where the equality follows from (10) and the boundary conditions associated with W .Because N (t)W ∩ V = {0} for all t ∈ [0, T /m] then W ∩ V = {0} and P W ∩ V = {0} so we can write ζ = (ξ , η ) ∈ W as and this means by the second variation calculation.By the comparison theorem then N (t)P W ∩ V = {0} for 0 ≤ t ≤ T /m which means for all 0 ≤ t ≤ T /m leading to N (t)W ∩ V = {0} for all 0 ≤ t ≤ 2T /m.Theorem 6.4 M (t)W has no focal point for all t ∈ R.
Proof: From Lemma 6.3, we know that M (t)W has no focal point for all t ≥ 0. Consider M (−t)W with t ≥ 0. Then But, N (t + T /m)W has no focal point for all t ≥ 0 and so the same holds for M (−t)W .
Using the results above, it is now possible to handle the cases of π(z(t)) cyclic and dihedral simultaneously.This is done starting in the next section.

Nonexistence of conjugate points
The proof of our main theorem relies in part on results of Contreras and Iturriaga [9].They study the existence of flow invariant Lagrangian subbundles for orbits which are not necessarily periodic.Their main results are the following.Theorem 6.5 (Contreras and Iturriaga [9]) Suppose that the orbit of θ ∈ M does not contain conjugate points and H(θ) = e is a regular value of H. Then there exists two dφ t -invariant Lagrangian subbundles E, F ⊂ T M along the orbit of θ given by The relationship between the nonexistence of conjugate and focal points is given by a theorem of Hartman which we now state in a version adapted from [9].Theorem 6.6 (Hartman [13])

Proof:
Suppose π(z(t)) is cyclic.From Theorem 6.4, M (t)W ∩ V = {0} for all t ∈ R and so by Theorem 6.6, z(t) has no conjugate points for all t ≥ 0 and for all t ≤ 0. If π(z(t)) is dihedral, the interval t < 0 is done using the Lagrangian subspace RW .We have M (−t)(RW ) = RM (t)W so M (t)(RW ) has no focal points for t < 0. Therefore, z(t) has no conjugate points for all t ∈ R.
Because z(t) has no conjugate points for all t ∈ R, Theorem 6.5 guarantees the existence of dφ tinvariant Lagrangian subbundles E and F for z(t).In [9], it is also shown that Note that in principle, we can have E s = E u = 0 so that we cannot conclude directly that z(t) is unstable, let alone hyperbolic.If the orbit of θ in Theorem 6.5 is periodic there is an immediate corollary to Theorem 6.5.Proposition 6.8 (Contreras and Iturriaga [9]) Let z(t) be a T -periodic solution of ż = X H (z) without conjugate points.Then z(t) is hyperbolic (on its energy level) if and only if E(θ) ∩ F(θ) = {X H (θ)} for some θ ∈ {z(t) | t ∈ [0, T ]}.In this case, E and F are its stable and unstable subspaces.Now, in our setup, if n = 1 in assumption 4, this guarantees that z(t) is the only periodic solution with spatio-temporal symmetry (2) in the energy level H −1 (e).However, we don't know that z(t) is isolated on its energy level, in principle, it may be possible that periodic solutions with other symmetry groups (even with no symmetry at all) could accumulate in a neighborhood of z(t).The conclusions of Theorem 5.1 imply that we obtain the same result as in Proposition 6.8, but with different conditions.
7 Proof of Theorem 5.1 We now state and prove a final lemma before we turn to the actual proof of our main theorem.Lemma 7.1 If π(z(t)) is cyclic then for all integers k ≥ 0, P k W and P −k (RW ) have no focal points for all t ∈ R. If π(z(t)) is dihedral then for all integers k ≥ 0, P k W has no focal point for all t ≥ 0 and P −k (RW ) has no focal point for all t ≤ 0.
Proof: Consider the cyclic case so that M (t)W ∩ V = 0 for all t ∈ R. We do a proof by induction.For k = 1 we have M (t)(P W ) = S T (SM (t)P )W = S T M (t + T /m)W.
.Since M (t+T /m)W ∩V = 0 for all t ∈ R, the same is true for M (t)(P W ). Suppose that M (t)P k−1 W ∩ V = 0 for all t ∈ R, then and by the induction hypothesis, M (t)(P k W ) ∩ V = 0 for all t ∈ R. A similar proof by induction shows the result for P −k (RW ) by using the facts that W = P −1 RW , S T M (t)P −1 = M (t − T /m) and again that M (t)W has no focal points for all t ∈ R. The proof for the dihedral case is identical, but for the fact that M (t)W has no focal point for t ≥ 0 and M (t)(RW ) has no focal point for t ≤ 0.
We recall a few results before attending to the final proof.Denote by E 1 the +1 eigenspace of P .It is shown in [9] that if the φ t orbit of z(t) does not contain any conjugate point, then B(z(0)) ⊆ E∩F.
All bounded solutions are linear combinations of solutions of the form dφ t (z(0))E λ with |λ| = 1.We now show that B(z(0)) = E 1 .Suppose that P has eigenvalues λ, λ on the unit circle with λ = ±1 and consider the eigenspace E λ of λ.The orbits of initial conditions in E λ by dφ t are bounded solutions and so E λ ⊂ E ∩ F. But, E λ must be zero since E ∩ F is isotropic and E λ is a symplectic subspace.
We claim that P cannot have an eigenvalue −1, the proof is below.Therefore, the only eigenvalue of P on the unit circle is +1.Hence B(z(0)) = E 1 .
Since E 1 ⊂ ker δ 2 A L (c(t)) and n < n, then dim E 1 ≤ n < n.If B(z(0)) = E ∩ F, then dim E ∩ F < n and so E ∩ F is a proper subspace of both E and F which implies that E s and E u are nonzero Thus, z(t) is unstable.If B(z(0)) E ∩ F then E ∩ F contains unbounded solutions and so z(t) is unstable.If n = 1 then the centre subspace is only given by the flow direction; E c = dX H (z(0)) = E ∩ F, and so z(t) is hyperbolic and isolated on its energy level.
We now show the claim.Suppose that P = S T M (T /m) has a −1 eigenvalue and let E −1 be the eigenspace of −1.Because the P orbit of E −1 is bounded and E −1 is isotropic then E −1 ⊂ E ∩ F. We construct a Lagrangian subspace α containing E −1 such that α ∩ V = {0}.Then, ω(α, V ) has constant sign.Recall that P k = (S T ) k M (kT /m) and from S = diag(σ, σ), then S : T z(0) M → T Sz(0) M preserves the vertical space V ; that is, SV z(0) ⊂ V Sz(0) .For simplicity of notation, we drop the subscript of V in the various tangent spaces.Thus, we have ω(P k α, V ) = ω(M (kT /m)α, S k V ) = ω(M (kT /m)α, V ).
Suppose that M (k n T /m)α ∩ V = {0} for an increasing sequence {k n } ∞ n=1 ⊂ N, then M (t)α has an infinite number of focal points.The comparison theorem of Arnol'd [1] states that if a Lagrangian subspace has n + 1 moments of verticality in some interval (where n is the number of degrees of freedom), then any other Lagrangian subspace must have at least one moment of verticality.This means M (t)W for t ≥ 0 must have at least one focal point which leads to a contradiction.
Suppose instead that M (kT /m)α has a finite number of focal points.We now show that M (t)α has also an infinite number of focal points for t ≥ 0. Without loss of generality, we can assume that M (kT /m)α has no focal points because we can always begin looking for the infinite number of focal points of M (t)α after the last intersection of M (kT /m)α with V .

Theorem 5 . 4
Consider a T -periodic orbit z(t) with spatio-temporal symmetry given as in Example 4.5 with σ = −I and such that π(z(t)) minimizes an action functional A in the fundamental domain T /m.