closed characteristics of second-order lagrangians

CLOSED CHARACTERISTICS OF SECOND-ORDER LAGRANGIANS

W.D. KALIES AND R.C.A.M. VANDERVORST

ABSTRACT. We study the existence of closed characteristics on three-dimensional en-ergy manifolds of second-order Lagrangian systems. These manifolds are always non-compact, connected, and not necessarily of contact type. Using the specific geometry ofthese manifolds, we prove that the number of closed characteristics on a prescribed en-ergy manifold is bounded below by its second Betti number, which is easily computablefrom the Lagrangian.

1. PREFACE

Second-order Lagrangian systems are defined variationally by extremizing actionfunctionals of the form � ��

�� . The Lagrangian � depends not only on

the state variable � and its first derivative, but also on its second derivative, which is notthe usual situation for variational problems in classical mechanics. The Euler-Lagrangeequations of such systems are given by

(1.1)��

��

��

��

��

��

��

and are in essence fourth-order differential equations. These systems have recently beenused in many models in physics and engineering, and the literature pertaining to them isextensive. We refer the reader to [6, 7, 8] and the references therein for more informa-tion.

Under the natural hypothesis that� is convex in ��, a second-order Lagrangian systemis equivalent to a two degrees of freedom Hamiltonian system in �� endowed with itsstandard symplectic form �. The Hamiltonian is given by

(1.2) ��

��

��

��

��

��

��

��

��

Introducing the symplectic coordinates � �� , the Hamiltonian becomes�� , where �� is the Legendre transform of � with respect to��. Hamilton’s equations of motion are equivent to (1.1) and yield a dynamical system � on �

� . The Hamiltonian � foliates �� with three-dimensional energy manifolds��

� � �� . These manifolds are invariant under the flow �, andthe dynamical behavior of the system can be studied on an individual energy manifold.An energy manifold is regular if � is a regular value of � . In the context of second-order Lagrangians this is equivalent to the condition that �� whenever�� . See Section 4 for a discussion of the singular case. For more detailson second-order Lagrangians see [11].

Recently the analysis of periodic orbits, or closed characteristics, on given energymanifolds has become an important issue in the study of general Hamiltonian systems

Date: September 24, 2003.1

2 W.D. KALIES AND R.C.A.M. VANDERVORST

[5, 9, 13]. In this paper, we study the geometric structure specific to second-order La-grangian systems and place it in this broader context.

Given an arbitrary �� -dimensional manifold � embedded in �� , with �the standard symplectic form, one can construct a Hamiltonian system for which � isthe energy manifold for � � �. The particular choice of the Hamiltonian is not intrinsicto the problem of finding periodic orbits, which can be phrased in more geometric terms.The specific geometry of� and the 2-form � define a characteristic linebundle,

��

The vector field of any Hamiltonian � satisfying � � �� is a section of thisbundle. The trajectory of a periodic orbit can thus be regarded as a closed characteristicof the linebundle, i.e. an embedding � �� of the circle into� for which

��

This formulation of the problem relates the existence of periodic orbits of the differentialequation (1.1) to the geometric and topological properties of its energy manifolds.

Motivated by a novel result by Rabinowitz [9], Weinstein [13] conjectured in the1970’s, that any compact hypersurface � � �� , with the additional requirementthat

��

for some 1-form � with �� , i.e. � is of contact type relative to �, has at least oneclosed characteristic. This conjecture was later proved by Viterbo [12].

Energy manifolds determined by second-order Lagrangians do not fit within this the-ory for two reasons, they are always non-compact and they are not necessarily of contacttype in �� , as was proved in [2]. Even with a more general formulation via Reebvector fields, the latter issue cannot necessarily be resolved, see [2]. However, in thispaper we show that these manifolds possess certain geometric and topological propertieswhich guarantee the existence of closed characteristics.

In order to reduce the amount of technical detail, we restrict ourselves, for now, toLagrangians that satisfy the following hypotheses:

(H1) �� .

(H2) �� ,where �� is locally bounded. Note that (H2) is a lower bound on�; an upper boundis not necessary. These hypotheses can be weakened as discussed in Section 4. We nowformulate the main result of this paper.

Theorem 1.1. Let� be a regular energy manifold of a second-order Lagrangian systemwith �� Lagrangian � satisfying hypotheses (H1) and (H2). Then the number of closedcharacteristics on� is bounded below by the second Betti number �� .

The proof of Theorem 1.1 also provides a method for computing the second Bettinumber of � . Indeed, for any Lagrangian system with ��

� � � � �, the homotopytype of � can be determined directly from the sign of its “potential”, �� ,see Section 4 and [2].

Theorem 1.1 is a generalization of the situation for first-order Lagrangian systems��

�� where � � �

�� . An energy manifold � is one-

dimensional, and each compact component of (regular) � consists of a single periodicsolution. Thus the number of closed characteristics is exactly �� . For second-order Lagrangians �� is only a lower bound. One can easily give examples

CLOSED CHARACTERISTICS OF SECOND-ORDER LAGRANGIANS 3

of systems with infinitely many different closed characteristics, see [11]. Note that forLagrangians of the special form � � �� Hypothesis (H2) becomes void and thesimilarity between first and second-order systems becomes even stronger.

In [11], a version of Theorem 1.1 (Theorem 12, p. 1408) was proved under an ad-ditional hypothesis that the Lagrangian satisfies a twist property, defined in the nextsection. For some systems this property can be verified ([11] Lemma 9, p. 1405), butin many cases it cannot. Theorem 1.1 is an improvement of this previous result remov-ing the twist hypthesis. However, we do draw on the results for twist systems to proveTheorem 1.1.

2. GEOMETRY OF SECOND-ORDER LAGRANGIANS

To establish the existence of closed characteristics on energy manifolds of second-order Lagrangian systems, we use their variational structure. A closed characteristic isequivalent to a periodic solution � which are found as critical points of the action, i.e

Æ��

��

��

where � � � is the period of �. Note that variations are taken in � as well as �.We consider functions which have a simple profile consisting of two monotone laps,

�� which increases from some minimal value �� to a maximal value �� and �� whichdecreases from �� back to ��, with �� at �� and ��. If �� and �� are solutions, thentheir concatenation �� is called a ‘broken geodesic’, and the extrema �� and �� willbe called concatenation points. Note that a broken geodesic need not be a solution to(1.1) at its concatenation points, since the third derivatives need not match, see [11].

We obtain a periodic solution from the method of broken geodesics in two steps. Firstwe must determine when monotone laps exist between given values of �, and this isaccomplished in Section 3 via minimization. Then it must be shown that there existsa broken geodesic which is a solution to (1.1), which follows from the geometric andtopological properties of� as we now explain.

From the Hamiltonian (1.2), solutions must satisfy

��

��

at points where �� . We denote this level set in the �� -plane by � . Note that� is the section of � defined by � � �� . Indeed � � �� is the cylinder� �, where the �-variable is determined by ��-coordinate. Due to the convexity of� with respect to ��, the manifold � consists of two graphs in the �� -plane. Inparticular, the projection � of � onto the �-axis can be characterized by �� , and the sets � � �� and � � �� aregraphs over �� . A particular connected component of �� will be denoted by � , andwill be referred to as an interval component.

We will consider broken geodesics whose values lie in a single interval component � .Given such a component � of �� , define � �� . For givenlaps �� and �� let �� , �

�� , and �� , �

�� be the ��-values at the concatenation points

respectively. As shown in [11], if the condition

(2.3) ��

�� and ��

�

��


is satisfied at the concatenation points, then �� is a periodic solution, and thus aclosed characteristic.

Let �� and ��

� �. Consider the trajectory ��

�� of the Hamiltonian flow. Here �� is a function of �� since the

initial point has � � �� , and hence is in � . Thus there are two choices for ��,and we will choose �� .

��

��

��

�

��

��

��

��

FIGURE 2.1. The increasing and decreasing laps �� and �� respectively.

Define !�� and "�� to be the values of � and �� at the first maximum of�� respectively, see Figure 2.1. As �� , then !�� . The maps !�and "� are well-defined for fixed �� with decreasing �� as long as !�� .In addition the !� and "� are smooth in ��

�� on the domain of definition #�. We

can define analogous maps !�� and "�� as the values of � and �� at thefirst minimum of a decreasing lap, see Figure 2.1, with domain of definition #�. Let

�� !�� "��

�� !�� "��

be subsets of the cotangent bundle � � . Then �� are two-dimensional submanifoldsof � � given as graphs over the �� -plane and the �� -plane respectively, seeFigure 2.2.

��

��

�

� � ��

��

FIGURE 2.2. The contangent bundle � � and the intersecting La-grangian submanifolds �� and ��.


The submanifolds �� are in fact Lagrangian submanifolds of � � . Condition (2.3)implies that intersection points of the manifolds �� correspond to broken geodesicswhich are periodic solutions, i.e. �� !�� , !�� , �� "�� ,and "�� .

In the special case that there exist unique laps �� and �� for all �� , thenthe system is a twist system, as mentioned in the introduction. In this case, �� areexact Lagrangian submanifolds of � � , i.e. �� are the graphs of exact 1-forms on provided by generating functions for the laps [11]. In this case a direct variationalprinciple exists in terms of just the extrema �� and ��. This case is analyzed in detail in[11], and will be used here to prove the main result. If the Lagrangian system is not atwist system, a generating function can still be found by considering the full action ��as in [1]. However, in this paper we will use a continuation principle to study �� via continuation to a twist system.

We denote by � � � the canonical projection onto the base and by �� theprojection onto the �� coordinates of a point in � � . The following lemma isproved by Theorem 3.12 in Section 3 and shows that, for any Lagrangian satisfying(H1) and (H2), the projections �� cover the base, which plays a crucial role in ourintersection theory.

Lemma 2.1. Let � satisfy Hypotheses (H1) and (H2), then for each pair �� there exist increasing and decreasing laps �� and �� (in ��) respectively. In particular�� .

The next lemma establishes that the intersection set �� is always strictlycontained in for Lagrangians satisfying (H1) and (H2).

Lemma 2.2. Let � satisfy Hypotheses (H1) and (H2), then

��

where � � � �� Moreover, if �� is compact then there exists acompact set � �� such that �� .

Proof:Let �� $ be a fiber in � � . For each point �� Lemma 2.1

implies that $�� . Take a point �� , and consider the points ��

$ � �� and �� $ � ��. Lemma 7 in [11] implies that for each pair ��

��

and �� either ��

��

or ��

has a definite sign (strictly negative). Thus,for any boundary point �� we have

(2.4) ��$ � �� $ � ��

which implies that �� .Now assume that �� is compact, hence � is a compact interval component. Define

Æ � �� Æ� �� Æ�. From Lemma 8 in [11],there exists Æ� � � such that for all Æ � Æ� the boundaries �� Æ� and�� Æ� satisfy (2.4). This proves that �� Æ � � where � Æ

is defined in same way as � . Define the diagonal � � �� .Suppose now that there exists a sequence of points ��

�� accumulating at �� Æ��,

then it follows from Lemma 5 in [11] that

��

��

��

��


for any pair ��

��

��

�� in �$� � �� $� � ��, where $� � ��

��.

The latter combined with the behavior of �� on � Æ now implies that there existsa compact set � int �� Æ int �� such that �� .

To study�� we use the intersection number %��. Our approach is to define%�� via the Brouwer degree by constructing proper equations on � � whose zerosets are ��. This can be done in many ways and the intersection number %�� doesnot depend on the particular choice of the defining equations.

Let

�� !�� "�� and

�� !�� "��

where the domain of definition of �� is �� #�, �� , and thedomain of definition of �� is �� #�, �� . Then �� are the levelsets ��

� �� in � � . Define � �� on #� #�. Then the zeroset of � is �� , which is bounded and contained in int �#� #��. Thelatter follows from Lemma 2.2. Indeed, for any intersection point, �� , we have�� int � . If �� #�, then �� , a contradiction. Thus �� int#�.Similarly it follows that �� int#�. Since �� the boundedness of�� follows from continuity. These facts combined justify the definition

%�� #� #��

c.f [3]. Since �� , we have %�� %��. We are now ready toprove the main result.

Proof of Theorem 1.1: Let� be a regular energy manifold corresponding to�� .We compare the Lagrangian system determined by ��

�� with the

system determined by �� . The latter system is of Swift-Hohenberg

type which is shown to be a twist system in [11]. The two systems are related by contin-uation. Specifically, define �� &�� &��. Then the energy manifolds�� areregular for all & � �� Hence each�� is homotopy equivalent to� ��. Moreover,from the definition of �� it is clear that the sections �� and the base manifolds � � for all & � �� .

Since �� and �� are homotopy equivalent, the Betti numbers �� and�� , ' � �, are equal. In Section 7 of [2] it was shown that �� is equal to the number of compact components of the section� , which can be computeddirectly from the graph of the potential�� , i.e. the number of compact intervals onwhich�� .

Since �� defines a twist system, the results in [11], specifically Lemma 8 illustrated inFigure 2, imply via a Conley index argument that for each compact component of� theoverall degree �� #� #�� is��, hence %��

�� . The sign of %��

��

depends on the orientations of �� induced by their definition as level sets of ��. Since�� satisfies hypotheses (H1) and (H2) and �� for all & � �� , the results ofLemma 2.1 apply for all & � �� . Moreover Lemma 2.2 implies that ��

��

for some compact set � uniformly for all & � �� .Now, the continuation property of the degree can be used to show that the intersection

number can be continued for all & � �� . This fact requires a little argument. Foreach &� � �� there exists a (�&�� and compact )�� #� #� such that


��

�� )�� for all & � �&� � (�&�� &� � (�&�� . Therefore �� # �

� # �� )�� , i.e. the function %��

�� is locally constant on �� . Since

�� is connected, %��

�� is globally constant on �� . In particular %��

��

%��

�� .

Hence for each compact component of � , the energy manifold � contains a closedcharacteristic. Therefore the number of closed characteristics is at least �� .

3. EXISTENCE OF LAPS IN THE REGULAR CASE

3.1. Properties of the Lagrangian Action. Fix � � � and a compact componentinterval component � . Given u � �� and b � � � ��*�� *��

� *�*� �� and ��*�� *�� +��, define ,� �u� b� � �� - �� -� ��

�� *�� -� � *�� and �� for � � �� -�� and

��

��

��

��

��

which is well-defined on ,�u� b� � ��,� �u� b�. To prove that Lemma 2.1, weconsider the following minimization problem,

��u� b� � ��

��

��

and establish the existence of minimizers.Minimization �� requires a growth condition on the Lagrangian. Hypothesis (H2)

implies the following property.

Lemma 3.1. If hypothesis (H2) holds, then for every ( � � there exists �� such that�� (�� for all � � � and � � �.

Proof: Since � is bounded, we have�� . Thus�� (�� (�� for all � � � and � � �. Since �� and�� is bounded for � � � , there exists �� such that �� (�� for all � � � and � � �.

Lemma 3.2. If � � ,�u� b�, then��

�

�� b��

��

��

�� b��

��

Proof: Since � is monotone, we can reparametrize by �� and let .�� . Transforming to �� .�-variables yields

�� .��

��

��

��.��

�� .��

.��

��

with . � / � �� where / is a smooth function satisfying .�� *

�� and

.�� *�� . Hence . is absolutely continuous with .�� .��

.��0� �0 for all


� � �� , which implies �.�� *��

�.�� . Note that under this

transformation

��

��

�

��

�.�� and

��

��

��

�.��

Therefore,

��

��

�

��

�.��

�

��

��

�. � *��

�

��

��

.� �� *��

��

. �� *��

�

��

��

��

.� �� *��

��

� ��

.� ��

��

��

� *��

�

�� *��

��

��

.� ��*��

��

�� b��

��

��

�� b��

��

Now we use this inequality to prove that �� is bounded below on,�u� b�, so that theminimization problem is well-posed, i.e. �� .

Lemma 3.3. There exists a constant �� b�� such that �� forall � � ,�u� b�.


Proof: Applying Lemma 3.1 and �� we obtain

��

��

��

��

��

�� b��

��

��

�� b��

��

��

��

�

��

��

�

��

��

��b��

� �

��

�� b��

��

� �� b��

��

which implies that �� is bounded below on ,�u� b�.

Define the sublevel set ��u� b� � �� ,�u� b� �� 1�.

Lemma 3.4. There exists positive constants �� and �� depending on 1� �� and �b�� such that for any � � � ��u� b� we have - � �� and�� .

Proof: We have

1 � ��

�

��

��

��

��

�

�� b��

��

�

��

� ��

�� b��

��

Therefore,��

�� 1� �b�� , which also implies��

��

��1� �b�� . As for a lower bound on - we argue as follows. Integrating�� over �� - �, we find that �� - �� - �� - ��.

Lemma 3.5. There exists ��-� 1� u� �b�� such that �� for all � ��u� b�.

Proof: By Cauchy-Schwarz,��

�� 1� �b�� - �� which implies that

�� 1� �b�� - �� and��

�� 1� �b�� - ��

��- . Therefore, �� 1� �b�� -�.

To find a minimizer we need to establish that �� is coercive and weakly lower semi-continuous along a minimizing sequence. Lemma 3.5 implies coercivity provided that -is uniformly bounded, which is proved in Subsection 3.2 for the regular case. We now


show that �� is sequentially weakly lower semicontinuous along sequences for which -is bounded.

Lemma 3.6. Suppose u� � ,�u� b� with both �� and -� uniformly bounded.Then �� for some � � �� - ��.

Proof: We can rescale � to separate the dependence on - from variations in �. Let�,� �u� b� � �2 � �� 2�� 2�� 2�� *�+-� 2

�� *�+-� and2��3� �� for 3 � �� . Let �,�u� b� � �� ,� �u� b� �� . Then

��2� �

��

��

�- ��2��3�� -�

�2�3��

2��3�

-

�� -�

��3

for 2 � �,�u� b�.The functions 2��3� � ��-3� are uniformly bounded in �� , hence we

can extract a weakly convergent subsequence 2� 4 2 with -� - . Observe that

the functional��

- ��2� 2�+-� � �� 3 is continuous in - � � and weakly continu-

ous in 2 � �� . The functional ��

��

�2�� 3 separates the variables - and 2

and is continuous in - and sequentially weakly lower semicontinuous in 2. Hence,

��

��

�2�� 3 � ��

��

��

�2�� 3. Therefore ��2� � �� 2��.

Lemma 3.7. If ��u� b� � �� for some � � ,�u� b�, then � � �� - �� satisfiesthe Euler-Lagrange equation (1.1) and�� .

Proof: This follows from standard regularity theory, c.f [6].

Lemmas 3.5 and 3.6 imply that a minimizer exists in � �� - �� provided that - isbounded along some minimizing sequence. Lemma 3.7 states that a minimizer belong-ing to ,�u� b� is a solution to the Euler-Lagrange equations. Therefore, we must showthat minimizing sequences exist for which - is bounded and the weak limit belongs to,�u� b�. This issue will be addressed in Subsection 3.2 for the regular case. We con-clude this subsection with a technical lemma concerning the continuity of the infima��u� b� with respect to the parameter b.

Lemma 3.8. Suppose b� b � � and u� � ,�u� b�� with -� -� �� u� b�� and �� in �� - ��. Then �� u� b�.

Proof: Again we can rescale � to separate the dependence on - from variations in �.Let /�-� b�� be a smooth strictly monotone function satisfying /�� /�� /

�� *�+-� and /�� *�+- , and define

��2� - � b� �

��

�

�2 � /�

2� � /�

-�2�� /��

- �

�- �3

for 2 � �2 � �� 2��3� � /��3� �� for 3 � �� . Then �� 2� - � b� ��u� b�. The family /�-� b� can be chosen to vary continuously in b, the family offunctionals ��2� - � b� is continuous in b for each fixed 2 and - . Therefore, the infimum��u� b� is upper semicontinuous with respect to b, cf. [10].


Let 2��3� � ��-�3� � /�-�� b��3�. Since �� b� is continuous, we have��2� - � b� � �� 2�� -�� b�� u� b�� u� b� � ��2� - � b�.Therefore �� 2� - � b� � ��u� b�.

3.2. The Existence of Minimizers. In this section we prove the existence of minimiz-ers when u � �� int for a single interval component � , and hence we willassume that �� is regular. In this case the following property is due to continuity.

(P3) There exist 5 � � and Æ� � � such that �� 5 � � for all �� Æ�� Æ��.

Lemma 3.9. Under hypotheses (H1) and (H2), there exists a constant �� , dependingon 1� �b�� +Æ�, and �+5, such that for any � � � �� we have - � ��.

Proof: Let �Æ� � �� - � �� Æ��, where Æ� is chosen in (P3). Since

��Æ��Æ��

��

�� , we have ��Æ�� 1� Æ�� +Æ��. Let ( � �. Then,

1 � ��

��Æ�

��

��Æ�

��

� 5�- � ��Æ� �� (��

��Æ�

��

which implies that - � ��1� Æ�� +Æ�� +5�, by Lemma 3.4.

Lemmas �� and �� imply that action is bounded below on ,�u� b�, and the time -is bounded on sublevel sets of ��. Therefore, Lemma �� implies that �� is coerciveand sequentially weakly lower semicontinuous along any sequence in,�u� b� on which�� is bounded. Let ��,� �u� b� � �� - �� 4 � for some sequence�� ,�u� b��. Functions � � ��,�u� b� � ��,� �u� b� are monotone, possiblywith critical inflection points. We have shown that the minimization problem is well-posed in the sense that a minimizer exists in ��,�u� b�. However we must still show thatthis minimizer lies in ,�u� b� to apply Lemma 3.7.

Without loss of generality, we can assume that the following condition holds.

(P4) The constant Æ� � � in (P3) can be chosen such that�� is nonincreasingin � for all �� Æ�� Æ��.

Property (P4) is not a restriction on�. Consider the family of Lagrangians ��+�� . Then �� for all � � �. Hence, the minimizersof �� are the same for all � � �. Since �� is compact, we can choose � � � suchthat �� is strictly negative for all � � �� . Then, replacing �� by�� will satisfy (P4) without changing the minimization problem, and since� � �, the growth condition (H2) is still satisifed. Also, property (P3) still holds withpossibly smaller values of 5 and Æ�. Property (P4) is used in the following lemma whichimplies that a minimizer must lie in ,�u� b�.

Lemma 3.10. Suppose �� . Let � � �� ,� �� b�� for some b� � �*� *�with � � �*� � Æ�. Define �- � �� +* � � and � � ,

�� b�� by �� *� � ��.Then �� and �� . If �� , then �� .


Proof: As in the proof of Lemma 3.2, transforming �� and�� into �� .�-variables,we have

��

�

��

�.��

��

�� .��

.��

��

�

��

�� *��

�*�

�� (3.5)

Here we have used properties (P3) and (P4).

Corollary 3.11. If � � ��,�u� b� is a minimizer of ��, then �� on �� - �, hence� � ,�u� b�.

Proof: Suppose � has a critical point at ��. Since � is monotone, �� is contained insome maximal compact interval of critical points � . By continuity, for any *� � �*� *�with �*� sufficiently small, there is an interval �� containing � such that �� *. Let �� and �� . Then using Lemma ��, we can constructa function � � ,�� b�� such that �� . Replacing �� by � yieldsa function �� - �� such that �� , which contradicts the fact that � is aminimizer.

We have proved the following theorem which implies Lemma 2.1.

Theorem 3.12. Suppose � satisfies (H1) and (H2) on an interval component ��. If u � and b � �, then there exists a strictly monotone minimizer � � ,�u� b� � � �� - ��of �� which satisfies the Euler-Lagrange equation ��.

In fact the above results prove Theorem 3.12 for u � int . In order to include allof one can choose a sequence of minimizers �� ,�u�� b� with u� u � � .To obtain a limit in ,�u� b� we need to argue that -� is uniformly bounded. Supposenot, i.e. -� �. Since �� we would obtain, after appropriate shifts, asolution asymptotic to either �� or ��, or both, which is a contradiction.

4. EXTENSIONS AND CONCLUDING REMARKS

4.1. More General Lagrangians. Essentially, the hypotheses (H1) and (H2) in Sec-tion 1 are stated in the manner most convenient for implementing the minimization inSection 3 without too many technical details. The analysis in that section is needed toestablish the surjectivity of the projection �� onto the base , which ensures that thecontinuation to a twist system is well-defined. The geometric and topological consider-ations in Section 2, other than surjectivity, require merely the convexity of � in � ��.

Thus, the hypotheses (H1) and (H2) can be weakened. For example the conditions

(H1’) � � � � �� for all ��

(H2’) �� ,

where �� is locally bounded, would also be sufficient. The hypothesis (H1’) impliesthat the action �� is well-defined on the Sobolev space �� and (H2’) implies thatthe the action is bounded below. Moreover, the use of other function spaces would allowsuper-quadratic growth of � in ��.


4.2. Sharp Lower Bounds. Consider the Lagrangian �� +�� +and � � �. The corresponding action is not bounded below. Indeed if �� 6 �� +��+� � then

� ��

��

��

��

�� 6�

� �� 6�

� ��

Thus for 6 large enough � �� as � �. This example shows that the mini-mization procedure can fail when � � in hypothesis (H2).

This problem is not just a failure of a particular method. For � � � in the previousexample, it is not difficult to show that there are values �� and �� for which no monotonelaps exists. The growth condition (H2) is a geometric restriction. Since � is non-compact, it is inevitable that some such restriction is necessary.

4.3. The Topology of Energy Manifolds. For Lagrangians that satisfy the convexityhypothesis �� , the topology of the energy manifolds�� can becompletely determined from the sign changes in the potential �� . Considerthe homotopy

�� &�� &��

�of Lagrangians with corresponding Hamiltonians �� . If �is regular, then it is immediately clear that �� is regular for all & � �� . So ��defines a cobordism between � � �� and �� +� � �� .A straightforward calculation shows that the height function �&� � & has no criticalpoints, and hence standard Morse theory implies that � is homotopy equivalent to ��

for all & � �� . In fact they are diffeomorphic.In [2] the homotopy type of �� was computed for the regular case, which implies

�� has simple zeros. There is a deformation retraction of�� onto a bouquetof circles and 2-spheres. Consequently, the homology of �� is determined by its Bettinumbers with 7� � � and 7� � � for � � �. The second Betti number 7� is thenumber of compact components of � , i.e. the number of compact intervals in � onwhich �� . The first Betti number is the number of compact intervals onwhich �� , which depends on the behavior of �� as �� . Inany case, 7� � �7� � �� 7�� 7� � ��.

A simple example shows that the lower bound �� in Theorem 1.1 is sharp.Let �� +��+� and � � �. Then�� with �� , and solving the (linear) Euler-Lagrange equation explicitly shows that there are noclosed characteristics.

4.4. Singular Manifolds. Singularities in� occur at critical points of �� .Depending on the eigenvalues of these points as equilibrium points of the flow 8 �, thereare three types of singularities: saddle (four real eigenvalues), saddle-focus (four com-plex eigenvlaues), and center (four imaginary eigenvalues).

Consider an energy manifold � with (isolated) singular points in the interior of acompact component � of �� . The techniques in this paper imply that for each compo-nent of � � �singular points� there is a closed characteristic independent of the type ofthe singularities. Note that this is already different from the first-order Lagrangian casewhere singular manifolds cannot contain closed characteristics.


However, in the second-order case depending on the type of the singularities, evenmore closed characteristics must exist. It is shown in [11] that if the twist property holdson each component of � � �singular points� and the singular points are either of saddle-focus or center type, then the twist property holds on all of � , and additional closedcharacteristics exist with nonzero intersection number.

The arguments of this paper should be applicable in this case by continuation of asingular manifold with saddle-focus or center type singularities to a twist system. Themain issue is whether the surjectivity criterion in Lemma 2.1 holds over all of � . Weleave the details for future work, but we do not forsee any major problems in applyingthe techniques of [7, 6], which provide exactly the tools required to minimize in thepresence of a saddle-focus or center equilibrium, to show, again by minimization as inSection 3, that the surjectivity condition holds.

4.5. Forcing of Additional Closed Characteristics. For twist systems, it is shown in[4] that the existence of certain closed characteristics can force the existence of a mul-titude of closed characteristics due to their braiding and knotting. The above continua-tion method does not always immediately apply because the intersection numbers cor-responding to these additional closed characteristics can be trivial, but in certain casesthe topological information obtained from the braid type will imply nontrivial intersec-tion number. In those cases, the arguments of this paper will imply the existence ofmore closed characteristics. One might also attempt to prove the existence of multi-ple solutions by more carefully studying the intersections using the fact that they areintersections of Lagrangian manifolds, which we leave for future work.

REFERENCES

[1] B. Aebischer, M. Borer, M. Kalin, C. Leuenberger, and H. M. Reimann. Symplectic Geometry.Birkhauser Verlag, Basel, Boston, Berlin, 1994.

[2] S. Angenent, J.B. VandenBerg, and R.C.A.M VanderVorst. Contact and non-contact type Hamilton-ian systems generated by second-order Lagrangian systems. Preprint, 2002.

[3] W. Fulton. Intersection Theory. Springer-Verlag, New York, Heidelberg, Berlin, 1984.[4] R.W. Ghrist, J.B. VandenBerg, and R.C.A.M VanderVorst. Morse theory on spaces of braids and

Lagrangian dynamics. Preprint, 2001.[5] H. Hofer and E. Zehnder. Symplectic Invariants and Hamiltonian Dynamics. Birkhauser Verlag,

Basel, Boston, Berlin, 1994.[6] W.D. Kalies, J. Kwapisz, J.B. VandenBerg, and R.C.A.M VanderVorst. Homotopy classes for stable

periodic and chaotic patterns in fourth-order Hamiltonian systems. Comm. Math. Phys., 214:573–592, 2000.

[7] W.D. Kalies, J. Kwapisz, and R.C.A.M VanderVorst. Homotopy classes for stable connections be-tween Hamiltonian saddle-focus equilibria. Comm. Math. Phys., 193:337–371, 1998.

[8] R. Peletier and W. Troy. Spatial Patterns: Higher-Order Models in Physics and Mechanics, vol-ume 45 of Progress in Nonlinear Differential Equations and their Applications. Birkhauser Verlag,Basel, Boston, Berlin, 2001.

[9] P. Rabinowitz. Periodic solutions of a Hamiltonian system on a prescribed energy surface. J. Diff.Eq., 33:336–352, 1979.

[10] K.R. Stromberg. An Introduction to Classical Real Analysis. Wadsworth Inc., Belmont, CA, 1981.[11] J.B. VandenBerg and R.C.A.M. VanderVorst. Second-order Lagrangian twist systems: simple closed

characteristics. Trans. Amer. Math. Soc., 354:1393–1420, 2002.[12] C. Viterbo. A proof of Weinstein’s conjecture in �� . Ann. Inst. H. Poincare – Anal. Nonl., 4:337–

356, 1987.[13] A. Weinstein. On the hypothesis of Rabinowitz’ periodic orbit theorems. J. Diff. Eq., 33:353–358,

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closed characteristics of second-order lagrangians

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