The Lusternik-Fet theorem
for autonomous Tonelli Hamiltonian systems
on twisted cotangent bundles
Let be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits for such a flow. Our main result asserts that if is not aspherical then contractible periodic orbits exist for almost all energies above the maximum critical value of .
Key words and phrases:Dynamical systems, Periodic orbits, Symplectic geometry, Magnetic flows
2010 Mathematics Subject Classification:37J45, 58E05
Let be a closed manifold and let be a closed two-form on . We refer to as the magnetic form. We consider the standard symplectic form on twisted by the pull-back of via the projection map :
Let be a Tonelli Hamiltonian; namely is uniformly convex and grows super-linearly in the fibres (see [Abb13]). We denote by the Hamiltonian flow of with respect to . It is generated by the vector field defined by
These flows are of special physical interest when the Hamiltonian is of mechanical type; namely when
where is the (dual) norm induced by a Riemannian metric on and is a smooth function. Indeed, these dynamical systems model the motion of a particle of unit mass and charge on a Riemannian manifold under the effect of the conservative force and of the stationary magnetic field . Among the mechanical Hamiltonians, we distinguish those which are purely kinetic (namely, those such that ). We denote one such Hamiltonian by . In this case we call the magnetic flow of the pair ; its trajectories are then called with slight abuse of terminology magnetic geodesics.
In this paper we are interested in showing the existence of contractible periodic orbits of the flow on a given energy level . Here and in the rest of the paper contractibility has to be understood in (or, equivalently, in when we refer to the projection of the orbit on the base manifold) and not in . Our main goal will be to prove a generalization to our setting of the classical theorem of Lusternik-Fet asserting that all closed not aspherical111A manifold is called aspherical if all its homotopy groups of degree bigger than are trivial (see [Lüc10] for a survey on this class of manifolds). Riemannian manifolds admit a non-trivial contractible closed geodesic [LF51] (see also [Bir17, Bir66] for the original idea of Birkhoff when is a sphere). To this purpose we shall define the energy value
If is not aspherical, then carries a contractible periodic orbit, for almost every .
When and the Hamiltonian is of mechanical type, Theorem 1.1 was proven for the first time by Novikov in [Nov82, Theorem, page 35] (see also [Koz85, Theorem 7]). There the author claims the existence for every , since he did not take into account some decisive compactness problems, which are today only partially solved (cf.[Con06, Mer10, Abb13]). These issues are also ultimately responsible for the fact that we can get existence only on almost every energy level.
Our strategy of proof is to put Novikov’s beautiful geometric idea on solid ground by using some techniques from the dynamics of autonomous Lagrangian systems pioneered by Contreras almost ten years ago [Con06]. First, we build a correspondence between the periodic orbits at level and the zeros of a closed one-form on the space of contractible loops in with arbitrary period. Second, using an argument à la Birkhoff and Lusternik-Fet, we employ a minimax method in order to construct zeros of as limits of sequences of loops such that .
One of the contributions of this paper is to show that such sequences do have limit points, provided the periods are uniformly bounded and bounded away from zero. Following [Con06] we obtain this bound for almost every value of by an adaptation of the so-called “Struwe monotonicity argument” [Str90]. It is an open problem to understand if this can be done for every .
In view of Theorem 1.1, it is natural to ask what happens to contractible periodic orbits when the manifold is aspherical. In this case and, therefore, is weakly exact; namely its lift to the universal cover of is exact. We define the Mañé critical value of the pair by
where is the lift of to the universal cover. Observe that is finite if and only if has a bounded primitive. This means that there exists with
where is the (dual) norm induced by the pull-back to of a Riemannian metric on . It is known that if is non-exact and is amenable, then does not admit bounded primitives [Pat06, Corollary 5.4], i.e. .
It is immediate to see that and for kinetic Hamiltonians we also have if and only if .
When the magnetic form is weakly exact and the Hamiltonian is of kinetic type, Will Merry proved the following result about the existence of contractible orbits (see [Mer10] and the forthcoming corrigendum [Mer]). To stick to his notation, we set , where is the Riemannian metric defining .
Theorem 1.2 (Merry, 2010).
Let be weakly exact and be a kinetic Hamiltonian. Then, for almost every , carries a contractible closed magnetic geodesic.
Using the techniques developed for Theorem 1.1 we can give an alternative proof of Merry’s result for an arbitrary Tonelli Hamiltonian. When is exact, respectively when , the theorem below was already shown by Contreras [Con06], respectively by Osuna [Osu05].
If is weakly exact and is a Tonelli Hamiltonian, then for almost every the level carries a contractible periodic orbit.
In general one cannot expect existence of contractible periodic orbits for energies above when is aspherical. Indeed, if is a metric on with non-positive sectional curvature, there are no non-constant contractible closed geodesics on . Hence, when , does not carry contractible periodic orbits, for any . However, to the authors’ knowledge it is an open question to determine whether a given aspherical manifold has a metric with no non-constant contractible closed geodesics. Observe, indeed, that there are examples of aspherical manifolds supporting no metric of non-positive sectional curvature [Dav83, Lee95].
As we have sketched above, the proof of Theorem 1.1 (and of Theorem 1.3) is based on variational methods for autonomous Lagrangian systems. In recent years, such methods have also been used to prove the existence of infinitely many (not necessarily contractible) periodic orbits on almost all low energy levels when is a closed surface, is exact, and (see [AMP15, AMMP14]). In a recent paper, the authors extended this latter result to the case in which is oscillating but not necessarily exact, under the further assumption that the surface is not (see [AB15, AB]). Settling the case of represents a challenging open problem.
Finally, we analyze the existence of contractible periodic orbits with energy below without assuming any additional condition on or on . To this purpose we recall that, if is a symplectic manifold, a set is said to be displaceable in if there exists a compactly supported Hamiltonian diffeomorphism , i.e. a time- map of a time-dependent Hamiltonian flow, such that . In [Con06] Contreras observed that, when , sublevels are displaceable for . This fact enables him to apply a result of Schlenk [Sch06, Corollary 3.2] which guarantees the existence of a contractible periodic orbit for almost every . Our simple remark leading to the next theorem is that Contreras’ observation carries over to the case of arbitrary and that Schlenk’s abstract result still applies.
The energy sublevel set is displaceable in , for every .
The energy level set carries a contractible periodic orbit for almost every .
The proof of Theorem 1.1 and 1.3 and the proof of Corollary 1.5 are substantially different. As we previously remarked, the former is tailored to the class of Tonelli Lagrangian systems while the latter pertains to the broader world of symplectic geometry. Building on work of Taĭmanov [Taĭ83], Abbondandolo showed how to use the first approach to give a proof of Corollary 1.5 when is exact [Abb13]. It is still not known whether his idea can be adapted for an arbitrary . Similarly, the second approach was used by Schlenk in [Sch06, Corollary 3.6] to reprove Theorem 1.1 and 1.3, when , and lies in the range , where
The fact that is positive follows from results of Laudenbach-Sikorav [LS94] and Polterovich [Pol95]. Moreover, in the weakly exact case Merry shows in [Mer11, Theorem 1.1] that . It is conjectured in [CFP10] that . Such equality holds for non-exact forms on tori since in this case and there exists a symplectomorphism , where is some symplectic manifold, as shown in [GK99, Theorem 3.1].
A common feature of both methods is that they yield existence results only for almost every level in a certain energy range. However, it is known that if, under the hypotheses of Theorem 1.1 or Theorem 1.3 or Corollary 1.5, some fixed energy level is stable in [HZ94, page 122], then there exists a closed contractible orbit with energy (see [HZ94, Ch. 4, Theorem 5] and [Abb13, Corollary 8.4]). This happens for instance whenever is not-aspherical, is weakly exact and , as follows at once from the analysis contained in [Mer10, Section 4].
In connection with such results, one would like to understand which energy levels are stable. For example, when is a closed orientable surface, is symplectic and , stability holds on every low energy level. Such statement can be found in [Ben14a] for (see also [FS07, Lemma 12.6] and [Pat09, Remark 2.2]). When the global angular form associated to any nowhere vanishing section of is a stabilizing form. Hence, in Corollary 5.1 we get a new proof of the existence of a contractible closed magnetic geodesic for every low energy, a result which was already proven by Ginzburg in [Gin87] and, when , by Schneider in [Sch11, Sch12a, Sch12b].
In this setting, one can also give better lower bounds for the number of contractible closed magnetic geodesics. If , this number turns to be infinite for every sufficiently low energy: for this was established independently in [LC06] and in [Hin09]; for surfaces of higher genus a proof was recently given in [GGM15]. On the other hand, for every low energy level carries either two or infinitely many closed magnetic geodesics [Ben14b, Ben15].
If has dimension higher than and is a symplectic form, proving that low energy levels are stable is an open problem (see [CFP10] where a proof of stability is given in the homogeneous case). However, existence of contractible magnetic geodesics for every low energy levels still holds as Usher proves in [Ush09] building upon previous work of Ginzburg and Gürel [GG09] (see also [Ker99] for multiplicity results, generalizing [Gin87], when is a Kähler form).
We end up the introduction by giving a summary of the content of this paper. In Section 2 we introduce the free-period action -form associated to the pair and we investigate the compactness properties of the vanishing sequences of as well as the completeness of the associated negative “gradient flow”. This leads naturally to studying the behaviour of on the set of short loops, which is the content of Section 3. The general analysis developed in Section 2 and the geometric information obtained in Section 3 allow us to prove Theorem 1.1 in Section 4 and Theorem 1.3 in Section 5 via minimax methods. Applications to stable energy levels on surfaces are given along the way. Finally, in Section 6 we turn to symplectic techniques and prove Theorem 1.4 and Corollary 1.5.
The authors are deeply grateful to Will Merry for making the draft of his corrigendum available to them prior to publication. They also warmly thank Alberto Abbondandolo, Viktor Ginzburg and Felix Schlenk for valuable discussions and the anonymous referee for precious suggestions, which helped to improve the article. Luca Asselle is partially supported by the DFG grant AB 360/2-1 ”Periodic orbits of conservative systems below the Mañé critical energy value”. Gabriele Benedetti is supported by the DFG grant SFB 878.
2. The free-period action 1-form
In this section we give a variational characterization of the periodic orbits on a given energy level in a suitable space of parametrized loops with arbitrary period. We allow also for non-contractible loops at this point of our discussion, since the results contained here naturally apply in this higher degree of generality.
In order for the variational problem to be well-defined, we require to be quadratic at infinity. This means that can be written as in (1.1) outside some compact set. This is by no means restrictive since in our arguments we will always take to lie inside a fixed bounded interval of energies and therefore one can easily find a Tonelli Hamiltonian quadratic at infinity which coincides with on the energy levels of interest. Moreover, this can be done preserving the value (and also the value , when is weakly exact). Thus, in the remainder of the paper we will assume that is quadratic at infinity and we will denote by the Riemannian metric determining the kinetic term at infinity.
2.1. General setting
Let be the Legendre transform induced by and the Fenchel dual of . We call the Lagrangian function of the system. Let denote the energy function and observe that
Since is quadratic at infinity, and are also quadratic at infinity. This means that outside a compact set of we have
where is the norm induced by on . In particular, satisfies
with , are suitable constants. This inequality will play a crucial role in the proof of Theorem 2.6. We push the flow to via and define
In particular, closed orbits of contained in correspond to closed orbits of contained in . In the next lemma, we recall that has an intrinsic definition in terms of a perturbed Euler-Lagrange equation.
Let be an open subset of such that admits a primitive on . The restriction of the flow on is the standard Euler-Lagrange flows of the Lagrangian . As a corollary, a curve satisfies in every coordinate chart the perturbed Euler-Lagrange equations
if and only if is a trajectory of .
Thanks to Equation (2.2) we will be able to describe periodic orbits of by a variational principle on some space of loops in that we now introduce.
Let and define . The set can be interpreted as the space of absolutely continuous loops with square integrable weak derivative and with arbitrary period. Indeed, every absolutely continuous loop with -weak derivative can be identified with the pair , where . Conversely, from we obtain a loop by . To ease the notation, we adopt the identification throughout the paper and, to avoid confusion, we denote with a dot the derivatives with respect to and with a prime the derivatives with respect to . We define
as the length, respectively the -energy of . We also consider the analogous quantities associated to . Clearly, we have and .
As it is shown in [AS09], has the structure of a complete Hilbert manifold. Its tangent space at is the set of all absolutely continuous vector fields along whose covariant derivative with respect to the Levi-Civita connection of is -integrable. We define the inner product on each tangent space by
Moreover, a local chart centered at any smooth loop can be constructed as follows. Let be the open ball of radius and let be a bi-bounded time-dependent chart for . This is a smooth map such that for all :
is an embedding;
and its inverse have bounded -norm.
To ease the notation, we write
Notice that, for each , is a smooth section of . Consider the maps
The required local chart around in is
The set has a natural Hilbert manifold structure given by the product between that of and the standard one on . This means that the tangent space of admits the splitting
and it carries the product metric
The distance function induced by is not complete as the second factor is not complete with the Euclidean distance. However, is complete on all subsets of the form , where is some positive number.
We denote the norm on induced by simply by . We also define local charts around by
and observe that the connected components of correspond to the free homotopy classes of loops in . Throughout the whole paper we will denote by the connected component of contractible loops.
We are now ready to introduce the action 1-form . First, we consider the free-period Lagrangian action functional associated with
The functional is well-defined, since is quadratic at infinity. We set
where is the standard projection and is the transgression of :
If we can compute how acts on using the splitting (2.3). If and , we find
In the direction of the period we find
A loop satisfies if and only if is a periodic orbit of contained in .
We write now in a local chart in order to investigate its regularity as a section of the bundle . Our computations follow Section 3 in [AS09]. We decided to include them here anyway since in the original paper, there is a little inaccuracy in the definition of the pull-back Lagrangian to a local chart.
Let be a bi-bounded time-dependent chart for and let be the associated local chart of . There exists a smooth function
such that , where is defined by
For any and for any the family of functions is Tonelli and quadratic at infinity uniformly in . Namely, there exist positive constants depending on such that for all
First, we express the velocity of in a local chart. Let be given by and compute
where and . If we define by
then we have
Using (2.13) we write in a local chart. If , then
Since , there is with and hence we get
For every , we can write
for suitable and . By definition of and we get
Thus, (2.9) holds if we set
The -form is locally uniformly Lipschitz. Moreover, its integral over any closed differentiable path depends only on the free homotopy class of . In this sense, we say that is .
In general is not globally exact on . To this purpose we observe that is weakly-exact if and only if vanishes over . As we discuss in Section 5, this means that is exact on if and only if is weakly-exact. However, even if is weakly exact it can still happen that is not globally exact on . For example, if and , then is not exact on any connected component of . On the other hand, if the lift of to the universal cover admits a bounded primitive, then is exact on (see [Mer10, Lemma 2.2]).
2.2. A compactness criterion for vanishing sequences
In view of Lemma 2.2, in order to find periodic orbits for we have to show that the set of zeros of is non-empty. The mechanism we use to construct such zeros is to look for limit points of vanishing sequences for . These sequences are the generalization of Palais-Smale sequences to our setting.
We call a for , if
Since is continuous, the set of limit points of vanishing sequences coincides with the set of zeros of . Therefore, it is crucial to know under which hypotheses a vanishing sequence has a limit point. Clearly, if or , the limit points set is empty. The following theorem shows that the converse is also true.
Let be a vanishing sequence for in a given connected component of . Then there exists such that
As a consequence, the following two statements hold:
If tends to zero, then .
If is uniformly bounded and bounded away from zero, then has a converging subsequence.
Since the vector field has norm , we have
Hence, by (2.1) and the fact that is a vanishing sequence, we get
On the other hand, since is quadratic at infinity, (2.1) yields
hence proving (2.15). Statement (1) follows at once from what we have just proved.
We now show (2). Since the periods are by assumption uniformly bounded from above, we know by (2.15) that is also uniformly bounded. Hence, the curves are uniformly -Hölder continuous. By the Ascoli-Arzelà theorem, up to a subsequence they converge uniformly to a continuous curve . Thus, there exists a smooth curve such that and, up to a subsequence, all the belong to the image of . Let us define and as and . Taking in (2.13) and using the fact that is equivalent to the Euclidean metric on , we see that is a bounded sequence in . Hence and, up to a subsequence, converges -weakly to . Exploiting once again that is a vanishing sequence, we have
Now one argues as in the part of the proof of Lemma 5.3 in [Abb13] after Equation and the thesis follows. ∎
2.3. The action variation along a path and the gradient of the action
We know that when the -form is non-exact, a global primitive of on does not exist. However, if is of class , the variation of along the path is always well defined. It is given by the formula
Then, since is closed, we extend the definition of to any continuous path by uniform approximation with paths of class . Observe that if is a smooth map such that admits a primitive , then for every path , there holds
The next lemma describes how changes under deformation of paths with the first endpoint fixed. The proof follows from the fact that is a closed form.
Let and suppose that is a homotopy of paths. Denote by and the paths in obtained keeping one of the variables fixed. If is constant, then
We now proceed to define the desired gradient. First we consider the vector field , where is the duality between and given by the metric . By Corollary 2.4 is locally uniformly Lipschitz and hence we have local existence and uniqueness for solutions of the associated Cauchy problem for positive times. However, the maximal solutions of the Cauchy problem might not be defined for all positive times for two reasons. First, is not metrically complete so that the period could go to zero in finite time. Second, is not uniformly bounded on so that the trajectories of the flow could escape to infinity in finite time. To avoid the latter problem we consider the normalized vector field
We define to be the positive semi-flow of on . Its flow lines are then the maximal solutions
of the Cauchy problem associated to with initial condition . Here is any element in and is some number in . By definition, we have
We say that is complete, if for all . It is shown in [Con06, Example 6.8] that the semi-flow is not complete on , if . However, the next result poses some restrictions on the flow lines with finite maximal interval of definition. Our proof follows [Con06, Lemma 6.9].
Let be a maximal flow line of . If , then there exists a sequence such that
First, arguing by contradiction we prove that
Thus, we assume that for every and observe that
Since is complete and the derivative of is bounded by the above inequality, there exists the limit
As is locally uniformly Lipschitz, there exists a neighbourhood of , such that the solutions to the Cauchy problem with initial data in all exist in a small fixed interval . This yields a contradiction as soon as and . Suppose now that (2.23) holds. In this case there is a sequence such that
Using (2.1), we find
which gives the required bound for the energy. ∎
The previous proposition shows that the only source of non-completeness of the semi-flow are trajectories that go closer and closer to the subset of constant loops. We have encountered a similar phenomenon in Theorem 2.6(1), where we saw that a possible class of vanishing sequences without limit points are those with infinitesimal length. It is therefore necessary to take a better look to the behaviour of on short loops; we will do this in the next section.
Before moving on to this task, we end this subsection by generalizing to our setting one of the standard estimates for gradient flows (see for example [Con06, Lemma 6.9]), namely the bound of the -Hölder norm of a flow-line in terms of the action variation and the length of the interval. This result will be used in the proof of Proposition 4.5 in order to show that the period bounds obtained by the Struwe’s monotonicity argument are preserved by the semi-flow.
If is a flow line of , then