HETEROCLINIC ORBITS FOR A CLASS OF HAMILTONIAN SYSTEMS ON RIEMANNIAN MANIFOLDS

Let M be a smooth Riemannian manifold with the metric (gij) of dimension n, and let H = 1 2 g(q)pipj + V (t, q) be a smooth Hamiltonian on M, where ( gij ) is the inverse matrix of (gij). Under suitable assumptions we prove the existence of heteroclinic orbits of the induced Hamiltonian systems.

1. Introduction and statement of the main results.The existence of homoclinic and heteroclinic orbits for Hamiltonian systems by using the variational methods and critical point theory has been studied by many authors (see for instance, [4]- [6], [10,11], [14]- [18] and [20]).We must say that Rabinowitz has given fundamental contributions to this field.Our present work is motivated by [14] and [11].
In [14] Rabinowitz studied the autonomous second order Hamiltonian system q + V q (q) = 0, q = (q 1 , . . ., q n ) ∈ R n , with the function V : R n → R satisfying the assumptions (R 1 ) V ∈ C 1 (R n , R), and V (q) ≤ 0 for all q ∈ R n .(R 2 ) V is periodic in q i with a period T i , 1 ≤ i ≤ n.
(R 3 ) The set U = {y ∈ R n ; V (y) = 0} consists only of isolated points.Then for every x ∈ U, there exist at least two heteroclinic orbits of (1) joining x to U \ {x}.Moreover, at least one of these orbits emanates from x and at least one terminates at x.As mentioned in [11] Rabinowitz's proof strongly depends on the fact that the system is autonomous.
They proved that for every x ∈ W there exists at least one heteroclinic solution of (2) such that q(−∞) = x and q(∞) ∈ W \ {x}, i.e. q emanates from x and terminates at a point in W \ {x}.
We note that for n = 1 the function V (t, q) =: V 1 (t, q) = (1 + e −ct 2 )(cos q − 1) satisfies the assumptions (R 1 ) − (R 3 ) of [14] if c = 0, and satisfies the five conditions (A 1 ) − (A 5 ) if c < 0. We also note that Izydorek and Janczewska's result cannot be applied to the function V 1 (t, q) with c > 0, because it does not satisfy the assumption (A 4 ).In this paper we want to extend the results of [14,11] to Hamiltonian systems on a manifold with the potential function V being in a broad class.
Let M be a C 1 smooth connected Riemannian manifold of dimension n with a complete metric g = (g ij (q)), where q ∈ M. Then on its cotangent bundle T * M there exists a natural symplectic structure.Assume that H is a Hamiltonian on M given by where (g ij (q)) is the inverse matrix of (g ij (q)) and V is a C 1 smooth potential function.We note that the Hamiltonian function (3) on M has been studied by several authors, see for instance [2] page 647.The aim of this paper is to study the existence of heteroclinic orbits of the following Hamiltonian systems where we have used the Einstein summation.Suppose that 0 for all t ∈ R} and assume that (b 1 ) #V ≥ 2 and σ = 1 3 min{ρ(x, y) : x, y ∈ V, x = y} > 0, where ρ(•, •) denotes the Riemannian distance of two points on M; We remark that our condition (c) is weaker than (A 3 ), because V (t, q) := V 2 (t, q) = 1 1 + t 2/3 (sin q − 1), satisfies (c) but not (A 3 ).We note that for n = 1 the functions V 1 (t, q) and V 2 (t, q) both satisfy the conditions (a), (b) and (c).But V 1 with c < 0 satisfies (d 1 ) but not (d 2 ).The functions V 1 with c > 0 and V 2 both satisfy (d 2 ) but not (d 1 ).
Our main result is the following.
Theorem 1.1.Suppose that the conditions (a), (b), (c) and either (d 1 ) or (d 2 ) hold.Then for any x ∈ V there exists y ∈ V \ {x} for which the Hamiltonian system (4) has a heteroclinic orbit connecting x and y, i.e. there exists a complete solution q(t) of (4) satisfying In the next section we will prove this theorem.Now we give an application of Theorem 1.1 to a classical non-autonomous Hamiltonian on the cotangent bundle of a two-dimensional torus.
2. Proof of Theorem 1.1.From [1] the Hamiltonian system (4) is equivalent to the Lagrangian system via the Legendre transformation p T = (g ij (q)) qT , where p = (p 1 , • • • , p n ), q = ( q1 , • • • , qn ) and T denotes the transpose of a matrix.The solution of the Lagrangian system ( 6) is a critical point of the functional in the Hilbert space with the norm where | q| 2 1 = g ij (q) qi qj and | • | is one of the equivalent norms in the n-dimensional Euclidean space.Recall that W 1,2 loc (R, M) consists of functions belonging to the Hilbert space W 1,2 (J, M) where J ⊂ R has compact closure in R (see e.g.[9] page 154).We note that if q ∈ H, then q ∈ C(R, M), the space of continuous functions having images in M. In what follows we use the notation → w to denote the convergence of sequences in H under the weak topology induced by the norm of H.But the notation → denotes the convergence of sequences in H under the strong topology induced by the L ∞ norm of L ∞ (R, M).Recall that L ∞ (R, M) denotes the Banach space of bounded functions (see e.g.[9]

page 145).
In what follows we try to find some critical points of the functional I(q) by minimizing it in H.The proof of Theorem 1.1 follows from a series of lemmas.We must say that we extend the ideas of the proof of [11] and [14] to our Hamiltonian systems.
Lemma 2.1.For 0 < ε ≤ σ.Assume that q ∈ H and q(t) where l(q[a i , b i ]) denotes the length of the curve q(t) when t arranges from a i to b i .
Proof.From the definition of the length of a curve on a Riemannian manifold (see e.g.[8] page 43 or [3] page 117), by direct calculations we get that , ), and we have used the Cauchy-Schwartz inequality.So we have Under the assumptions of Lemma 2.1, it follows that Lemma 2.2.For any q ∈ H we have Proof.It follows from the fact that In order to simplify notations we will use q(a, b) to denote the set {q(t); t ∈ (a, b)} for any a, b ∈ R and a < b.
Proof.If the manifold M is bounded, the conclusion follows easily from the definition of L ∞ (R, M).We assume that M is unbounded.
Firstly we claim that the orbit q(R) intersects at most finitely many elements of {∂B σ (x); x ∈ V}.Otherwise there exist sequences This means I(q) = ∞, a contradiction.The claim follows.
On the contrary if q ∈ L ∞ (R, M) we can choose a base point q 0 ∈ M and a real number R > 0 such that q(R) where t 1 := sup{q(t) ∈ ∂B R (q 0 )}.For any n ∈ N there exists a t n ∈ R such that ρ(q(t n ), q(t 1 )) > n and t 1 < t n < t n+1 → ∞ as n → ∞.By Lemma 2.1 and (9) we have where the limit is taken in the topology induced by the Riemannian metric on M. We call α(q(t)) and β(q(t)) the α and ω limit sets of q(t), respectively.By Lemma 2.3, α(q(t)) = ∅ and β(q(t)) = ∅ because q(t) is bounded on M. We now prove that β(q) ⊂ V and that it contains a unique point.The same conclusion holds for α(q).
On the contrary we assume that β(q) V, then there is a Hence by (c) it follows from the boundedness of q(t) that We get from Lemma 2.1 that for any k ∈ N This means that I(q) = ∞, a contradiction.Hence we should have Then an analogous argument as in the last paragraph implies that I(q) = ∞.This contradiction forces that β(q) has a unique element.
First we claim that H 1,ε (y) is bounded in H. Indeed, for any q ∈ H 1,ε (y) In addition there exists a c > 0 such that ρ(q(0), x) ≤ c for all q ∈ H 1,ε (y).
Otherwise for any k ∈ N there is a q k ∈ H 1,ε (y) such that ρ(q k (0), x) ≥ k.Set t k := max{t < 0; q k (t) ∈ ∂B ε (x)}.By Lemma 2.1 and (9) we have a contradiction.This implies that there exists a c * such that |q(0)| 2 ≤ c * for all q ∈ H 1,ε .Hence from (8) we have Recall that | • | and • are the norm defined in (8) and in the paragraph following it.This proves the claim.

The claim follows.
Let {q m } be a minimizing sequence for (10).Then from the definition of H 1,ε (y) and the fact that lim m→∞ I(q m ) = µ ε (y), we can assume without loss of generality that {q m } ⊂ H 1,ε (y).Then {q m } is bounded in H, and consequently there is a subsequence which is convergent to some element, saying q ε,y , of the Hilbert space H under the weak topology.In order to simplify the notation we suppose without loss of generality that q m → w q ε,y , otherwise instead of {q m } we must have a subsequence.From the last two claims, the sequence {q m } is uniformly bounded and equicontinuous.It follows from the Arzela-Ascoli Theorem (see e.g.[19] page 245) that q m → q ε,y in L ∞ loc (R, M).Next we prove that I(q ε,y ) ≤ µ ε (y).For any fixed a, b ∈ R and a < b, define Then it follows from Lemma 3.1 of the Appendix that I a,b (q) is lower semi-continuous in the weak topology of H. Since we have Since q ε,y ∈ H, and so is continuous in R. By the arbitrary of a, b ∈ R, we should have I(q ε,y ) ≤ µ ε (y) by letting a → −∞ and b → ∞.
Finally we prove that q ε,y ∈ H ε (y).We claim first that {q ε,y (t); t ∈ R} ∩ B ε (V \ {x, y}) = ∅.Otherwise there exist ξ ∈ V \ {x, y} and t 0 ∈ R such that q ε,y (t 0 ) ∈ B ε (ξ), then there exists M > 0 so that for m > M we have q m (t 0 ) ∈ B ε (ξ) because q m → q ε,y uniformly on any given neighborhood of t 0 under the strong topology.This is in contradiction with q m ∈ H ε (y).Hence the claim follows.
Next we only need to prove that q ε,y (−∞) = x and q ε,y (∞) = y.Lemma 2.4 shows that q ε,y (−∞) and q ε,y (∞) both exist and belong to V. Moreover we get from the last claim that q ε,y (−∞), q ε,y (∞) ∈ {x, y}.On the contrary we assume that q ε,y (∞) = x.Case 1.The condition (d 1 ) holds.Since q ε,y (∞) = x, there is a t * ∈ R such that if t ≥ t * we have q ε,y (t) ∈ B ε/2 (x).For any s > t * , since q m → q ε,y uniformly on [t * , s] under the strong topology, there exists M (s) > 0 such that q m (t) ∈ B ε (x) for m > M (s) and t ∈ [t * , s].We choose one of such m's, and denote it by m(s).By the definition of H ε (y) we have q m(s) (∞) = y, so we can set t 1 (s) = max{t ∈ R; q m(s) (t) ∈ ∂B ε (x)} and t 2 (s) = min{t ∈ R; q m(s) (t) ∈ ∂B ε (y)}.Then we have s < t 1 (s) < t 2 (s).Now we obtain the first inequality follows from Lemma 2.2 and I(q m ) ≤ µ ε,y + 1, and the second from (b 1 ).So we get from the Mean Value Theorem for integration that where and t s > s, by the boundedness of {q m } ⊂ H 1,ε in the strong topology we obtain from the condition (d 1 ) that I(q m(s) ) → ∞ as s → ∞.This is in contradiction with the fact that I(q m ) ≤ µ ε,y +1 for all m ∈ N. Hence we have q ε,y (∞) = y.Working in a similar way we can prove that q ε,y (−∞) = x.Case 2. The condition (d 2 ) holds.First it is easy to check that the case q ε,y (t) ≡ x does not happen, because each q m connects x and y, and q m → w q ε,y in H. Thus we have a t 0 such that q ε,y (t 0 ) = x.Then there exist k ∈ N, k ≥ 2 and t * 1 ∈ R such that q ε,y (t * 1 ) ∈ ∂B ε/(k−1) (x).By the assumption (d 2 ) we can choose a δ > 0 satisfying δ < ε/(4k) and Since q ε,y (∞) = x there exists a t * 2 > t * 1 such that q ε,y (t) ∈ B δ (x) for all t ≥ t * 2 .The sequence q m → q ε,y uniformly on [t * 1 , t * 2 ] under the strong topology implies that there is a M > 0 such that for m > M we have q m (t * 1 ) ∈ M \ B ε/k (x) and q m (t * 2 ) ∈ B 2δ (x).Consequently, for m > M , where g m (t) is the minimal geodesic connecting x to q m (t * 2 ), whose existence follows from Corollary 10.8 of [12] via the δ being chosen sufficiently small.Clearly Q m ∈ H ε (y).Since | ġm (t)| ≡ constant and ρ(q m (t * 2 ), x) < 2δ, we have This implies inf q∈Hε(y) a contradiction.So we have q ε,y (∞) = y.
Summarizing the above proof we get that q ε,y ∈ H 1,ε ⊂ H ε .Consequently I(q ε,y ) = µ ε (y).This proves that q ε,y is a minimal of I in H ε (y).
We now consider the regularity of the minimal.Lemma 2.6.For every y ∈ V \{x}, the minimal q ε,y of Proof.: We only need to prove that q ε,y is C 2 in any interval (r, s) ⊂ R \ R(ε, y).For φ ∈ H with supp φ ⊂ (r, s), and |δ| sufficiently small, we have q ε,y + δφ ∈ H ε .Hence I(q ε,y ) ≤ I(q ε,y + δφ), i.e. q ε (y) is a local minimum of I(q).This implies that if the first variation exists, it must vanish.
Recall that L(t, q, q) is the Lagrangian function defined at the beginning of this section, and L q and L q denote the partial derivatives of L with respect to q and q, respectively.For the second equality we have used the fact that φ has the compact support belonging to (r, s).
From the form of L we get for |δ| ≪ 1 and s ∈ [0, 1] that on supp φ where c 1 is a constant depending only on q ε,y , φ and the given interval (r, s) This implies that f (t, δ) φ, g(t, δ)φ ∈ L 1 [r, s], because the majorants c 1 (1+| ẋ|+| φ|) φ and Moreover by the Lebesgue's Dominated Convergence Theorem (see e.g.[19] page 26) the limit lim δ→0 (I(q ε,y + δφ) − I(q ε,y )) /δ exists.Consequently the first variation vanishes.The above proof yields lim δ→0 I(q ε,y + δφ) − I(q ε,y ) δ = s r L p (t, q ε,y , qε,y ) φ + L q (t, q ε,y , qε,y )φ dt where c is a constant of integration.By the arbitrariness of φ we have which is called the integrated Euler equation.We can check easily that the function on the right hand side of ( 13) is absolutely continuous.Since L q q is positively definite we get from the Implicit Function Theorem that qε,y (t) is C 0 on [r, s], and hence q ε,y ∈ C 1 [r, s].It follows that the function on the right hand side of ( 13) is C 1 .Again by the Implicit Function Theorem we have qε,y ∈ C 1 [r, s], and consequently From the relation between the Euler equation and the Hamiltonian equation we get that q ε,y is a C 2 solution of system (4) on the interval [r, s], and hence on R \ R(ε, y).This proves the lemma.Lemma 2.7.For any M > 0 and y ∈ V \ {x}, if q ∈ {q ∈ H; q(−∞) = x, q(∞) = y and I(q) ≤ M }, there exists constant K depending only on M , σ and α σ such that ρ(q(t), x)) ≤ K := 3M √ 2α σ + 3σ.
Proof.For any given t ∈ R we assume without loss of generality that q(t) ∈ B σ (V).
We note that the bound K in Lemma 2.7 is independent of the choice of y.
In addition a contradiction, where we have used the fact that lim inf m→∞ t * jm = −∞.This proves the boundedness of {t jm }.
Since {q jm (t)} and {t jm } are both bounded, we get from the continuity of V (t, q) that {V (t, q jm (t)); t ∈ [t jm , t jm + ε jm ], m ∈ N} is uniformly bounded.This implies that the third term goes to zero as m → ∞.
The above proofs show that for all m sufficiently large In case (ii) we can define a sequence of functions {Q jm (t)} as those given in case (i), but instead of t jm by τ jm , and of η by y.Then analogous arguments as those in the proof of case (i) show that I(Q jm ) < I(q jm ) = µ εj m for m large enough, a contradiction.We note that in this case {Q jm (t)} ⊂ H εj m (y).
Summarizing the above proofs, we have obtained that q j (R)∩∂B εj (V \{x, y}) = ∅ for j large enough.This proves the lemma.Following Lemmas 2.8 and 2.9 we have finished the proof of the theorem.

Appendix.
Lemma 3.1.For any given a, b ∈ R with a < b, the functional I a,b (q) defined in (11) is lower semi-continuous in the weak topology restricted to H 1,ε .
Proof.The main idea of the proof follows from [13].We claim that for {q m } ⊂ H 1,ε , if q m → w q in H as m → ∞ then where we have used the fact that q m , q ∈ H 1,ε and the Cauchy-Schwartz inequality.Taking the supremum limit on the lase inequality as m → ∞, then the claim follows from the arbitrariness of ε.
m − q)dt → 0 for φ ∈ L[a, b] and b a φ( qm − q)dt → 0 for φ ∈ L 2 [a, b].First q m → q in C[a, b].Indeed from the proof of Lemma 2.5 the sequence {q m } is an equicontinuous family of uniformly bounded functions in C[a, b].So combining the Arzela-Ascoli Theorem and the assumption q m → w q, we obtain thatq m → q uniformly in C[a, b].The first claim follows from b a φ(q m − q)dt ≤ q m − q L ∞ b a |φ|dt ,and q m − q L ∞ → 0 uniformly as m → ∞.Here and after, | • | denotes the absolute value of real numbers or of real functions.For the second claim if φ ∈ C 1 [a, b] the claim follows easily from the integration by parts, the Schwartz inequality and the fact that q m → q uniformly in C[a, b].Since C 1 [a, b] is dense in L 2 [a, b], for any ε > 0 there exists ψ ∈ C 1 [a, b] such that φ − ψ L 2 < ε.Then we have b a φ( qm − q)dt ≤ b a ψ( qm − q)dt + 4ε(µ ε (y) + 1),

2
and |L q | ≤ c (1 + | q|) on the interval [a, b].This implies that L q ∈ L 1 [a, b] and L q ∈ L 2 [a, b].Hence from the claim at the beginning of the proof of this lemma we get that lim inf m→∞ (I a,b (q m ) − I a,b (q)) ≥ 0.
jm , q jm (t * jm )), where t * jm ∈ [s jm , r jm ] comes from the Mean Value Theorem for integration.By Lemma 2.7 {q jm } is bounded.So it follows from the fact q jm (t * jm ) ∈ B εj m (V) and the condition (d 1 ) that