We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range possess infinitely many periodic orbits. Such a range of energies is non-empty, for instance, in the physically relevant case where the Tonelli Lagrangian is a kinetic energy and the magnetic form is oscillating (in which case, is the minimal energy of the system).
This paper is the last chapter of a work started in  and further developed in [2, 5, 6, 8] devoted to studying the multiplicity of periodic orbits on generic low energy levels in magnetic Tonelli Lagrangian systems on surfaces. Such a study was based on a generalization of Bangert’s waist theorem [10, Theorem 4], classically formulated for geodesic flows on , to the magnetic Tonelli setting. Roughly speaking, a waist is a non-constant periodic geodesic (resp. a periodic orbit in the Tonelli case) which minimizes the length (resp. the action) among nearby curves. The original waist theorem says that a Riemannian 2-sphere possesses infinitely many closed geodesics provided it possesses a waist. Such a statement is a crucial ingredient for the proof that, indeed, every Riemannian 2-sphere possesses infinitely many closed geodesics [11, 18, 19].
Let us introduce the general setting in which we will work. If M is a closed smooth manifold, a Tonelli Lagrangian is a smooth function whose restriction to any fiber of is superlinear with positive definite Hessian, see e.g. [20, 17, 1]. A magnetic Tonelli system is a pair , where is a Tonelli Lagrangian and σ is a closed 2-form on M, which we refer to as the magnetic form. If denotes the projection of the cotangent bundle, the pair defines a flow on that is conjugated through the Legendre transformation to the Hamiltonian flow on of the dual Tonelli Hamiltonian , , see e.g. [4, 22, 8]. A particularly relevant special case of this setting is the electromagnetic one, when the Lagrangian L is of the form for some Riemannian metric g and some smooth potential . In this situation, the system models the motion of a particle on M with kinetic and potential energies described by L and under the further effect of a Lorentz force described by σ. When the potential U vanishes, the dynamics of the system is a so-called magnetic geodesic flow.
In this paper, we will focus on the case where . The energy function , , is preserved along the motion. Therefore it is natural to study the dynamics of a magnetic Tonelli flow on a prescribed energy hypersurface , and very different qualitative behaviors appear for different values of the energy e, see  and references therein. For our purposes two energy values will bear special significance: and . The former is the minimal energy e such that the corresponding energy hypersurface projects onto the whole . We postpone the precise definition of to Section 2. For now, we just mention that , and when σ is exact with primitive θ we have , where is the Mañé critical value of the universal cover of (see e.g. [15, 1] for the definition of Mañé critical values).
The periodic orbits problem for magnetic geodesics was first studied by Novikov [21, 22] in the early 1980s. The classical least action principle for the periodic orbits with prescribed energy is not directly available in this setting, due to the potential non-exactness of the magnetic 2-form. Novikov showed how to recover the variational principle in the universal cover of the space of periodic curves, and in his celebrated “throwing out cycles” method he proposed how to exploit the corresponding deck transformation in order to detect action values of periodic orbits (for the throwing out cycles method, see also ). For magnetic geodesics on closed surfaces, waists were first studied by Taimanov in a series of papers [24, 25, 26]. Taimanov’s result is that, given a kinetic Lagrangian and an oscillating magnetic 2-form σ on a closed two-dimensional configuration space, there exists a waist at the energy level e, for all (see also  for a different proof). When σ is exact, Abbondandolo, Macarini, Mazzucchelli and Paternain  employed Taimanov’s waist on any energy level e belonging to a full measure subset of in order to construct a sequence of minmax families giving an infinite number of (geometrically distinct) periodic orbits with energy e. Short afterwards, Asselle and Benedetti extended the result to non-exact σ on surfaces of genus at least one [5, 6]. The results in [24, 25, 26, 2, 5, 6] have been further extended by Asselle and Mazzucchelli  to the general magnetic Tonelli setting. In this note we complete the picture by treating the last case remained open for the multiplicity problem: the 2-sphere. Namely, we are going to prove the following result.
Let be a Tonelli Lagrangian, and σ a 2-form on . For almost every , the Lagrangian system of possesses infinitely many periodic orbits with energy e.
We wish to stress that the existence of infinitely many periodic orbits on all energy values in is still an open problem. In Theorem 1.1, as well as in [2, 5, 6], a negligible subset of energies must be excluded due to a lack of compactness in the variational setting that is employed. However, energy levels with only finitely many periodic orbits can be found above [27, 12] as well as below  the interval .
For closed surfaces M of genus at least one, any closed 2-form σ on M lifts to an exact 2-form on the universal cover of M. This allows to define the Mañé critical value of the universal cover for any Tonelli Lagrangian . We set if M has positive genus, and if . The combination of Theorem 1.1 together with the above mentioned results in [5, 6, 8] yields the following statement about the multiplicity of periodic orbits on general closed surfaces.
Let M be a closed surface, a Tonelli Lagrangian, and σ a 2-form on M. For almost every , the Lagrangian system of possesses infinitely many periodic orbits with energy e.
The open interval is not empty for instance if M is orientable, σ is oscillating, and the Lagrangian has the form of a kinetic energy for some Riemannian metric g (see ); in such case, is the minimal energy of the system. We recall that a 2-form σ on an orientable surface is oscillating when it satisfies and for some . On non-orientable surfaces, any non-zero 2-form lifts to an oscillating 2-form on the orientation double cover.
This paper is dedicated to the memory of Abbas Bahri. Bahri was interested in the problem of periodic orbits of magnetic geodesic flows. In a joint work with Taimanov , he established the existence of periodic magnetic geodesics with prescribed energy on closed configuration spaces of arbitrary dimension under the assumption that the analog of the Ricci curvature for the Lagrangian system is positive.
The paper is organized as follows. In Section 2 we recall the variational setting for our periodic orbits problem: we provide the definition of the action 1-form , and of its global primitive on the universal cover of the space of loops; at the end we will review the definition of the energy values and , and the notion of a waist for magnetic Tonelli systems. In Section 3 we provide the proof of Theorem 1.1.
2 The Primitive of the Free-Period Action Form
2.1 The Variational Principle
Let be a Tonelli Lagrangian with associated energy function , and σ a 2-form on . Since we will be interested in the Euler–Lagrange dynamics on a given energy hypersurface , for some fixed , we can modify the Tonelli Lagrangian far from and assume without loss of generality that each restriction coincides with a polynomial of degree 2 outside a compact set. Let , where is the 1-periodic circle. For each energy value , we consider the free-period action 1-form on given by
where denotes the free-period action functional
By the least action principle, vanishes at some if and only if the p-periodic curve is an orbit of the magnetic Tonelli system of , see e.g.  and references therein.
The 1-form is not exact if σ is not exact. In order to work with a primitive of , following Novikov [21, 22], we will lift it to the universal cover of . We see as the unit sphere in , oriented in the usual way, and we fix the point . We consider the universal cover
As usual, we realize as the space of homotopy classes relative to the endpoints of continuous paths starting at . Here, we see as the constant loop at . The projection map is given by . We have , where the functional
is defined as follows. Given , we write , where and for all . We see γ as a map of the form by setting . We then set
Assume that is a proper open subset, so that is exact with some primitive θ. Let be a connected component of the open set of those such that the periodic curve is contained in U. Up to an additive constant, the restriction is equal to , where is the free-period action functional associated with the Lagrangian , i.e.
It is well known that the fundamental group of the free loop space is isomorphic to , and therefore so is the fundamental group of . A generator of can be defined as follows. For each , consider the affine plane orthogonal to the vector and passing through . We denote by the closed curve with constant Euclidean speed whose support is precisely the intersection , its starting point is , and, for all , its orientation is such that the ordered pair agrees with the orientation of , see Figure 1.
We define . The group of deck transformations of the universal cover is generated by
where for all , and for all . The action varies under such a transformation as
2.2 Iterated Curves
For each , we denote by its m-fold iterate, where . The iteration map , , is smooth. We lift this map to a smooth map of the universal cover, so that the following diagram commutes:
For instance, we can set , where
A remarkable property of the iteration map is given by the non-mountain pass theorem for high iterates, which was first established for electromagnetic Lagrangians in [2, Theorem 2.6], and extended to general Tonelli Lagrangians in [8, Lemma 4.3 and proof of Theorem 1.2]. As we explained in Remark 2.1, coincides locally with the free-period action functional of a suitable Tonelli Lagrangian, and therefore the non-mountain pass theorem for high iterates holds for as well.
Theorem 2.2 (Non-mountain pass theorem for high iterates)
Let be a critical point of such that, for all , the critical circle of is isolated in the set of critical points of . There exists such that, for all integers , the following holds. There exists an (arbitrarily small) open neighborhood of the critical circle of such that, if we set , the inclusion induces an injective map between path-connected components
2.3 The Critical Values of the Energy
Let us single out two significant values of the energy. The first one is , that is, the minimal energy e such that the corresponding energy hypersurface projects onto the whole . The second value , which depends also on the magnetic form σ, is defined as the supremum of the energies verifying the following condition: there exists a finite collection such that the ’s are smooth pairwise disjoint loops, for all , the multicurve is the oriented boundary of a positively oriented compact embedded surface , and we have
We recall that reduces to the classical Mañé critical value of in case σ is exact with primitive θ, see .
The proof of Theorem 1.1 will build on the following existence result, which was originally proved by Taimanov [24, 26] in the case of electromagnetic Lagrangians (see also  for an alternative proof), and further extended by Asselle and Mazzucchelli [8, Theorem 6.1] to the general case of magnetic Tonelli systems.
For every energy value , the Lagrangian system of possesses a non-self-intersecting periodic orbit with energy e such that every element in is a local minimizer of the action functional .
3 Proof of the Main Theorem
3.1 Minmax Procedures
For each energy value , consider the local minimizer of given by Theorem 2.3, and choose an arbitrary . We fix an arbitrary energy value
such that, for all , the iterated critical point belongs to a critical circle that is isolated in (if there is no energy value with such a property, there are infinitely many periodic orbits on every energy level in the range ). The critical points are still local minimizers of , as they are iterates of a local minimizer, see [3, Lemma 3.1] and Remark 2.1.
Given any subset , for each we will write
The Palais–Smale condition holds locally for the free-period action functional of Tonelli Lagrangians, see [14, Proposition 3.12] or [1, Lemma 5.3]. This, together with Remark 2.1, implies that the functional satisfies the Palais–Smale condition locally as well. Therefore, a sufficiently small bounded open neighborhood of the critical circle of does not contain other critical circles of and satisfies
For any , we denote by the closure of the set of local minimizers of . For all and such that , we denote by
the family of continuous paths such that and . We define the corresponding minmax value
There is an open neighborhood of such that
is a non-empty compact set for all ,
for each , we have ,
for each and , the function is well defined and monotone increasing in I.
The proof is entirely analogous to the arguments in [2, Lemmas 3.1–3.3] and it will be omitted. ∎
3.2 The Valley of Short Curves with Low Period
We equip our sphere with an arbitrary Riemannian metric g, and with the Riemannian metric
where denotes the covariant derivative associated to g. The space is not complete with respect to the Riemannian metric (3.2), nor is its universal cover equipped with the pulled-back Riemannian metric. Indeed, there are Cauchy sequences such that . However, it turns out that this does not pose any problem while applying arguments from non-linear analysis to the functional . Indeed, the functional has a “valley” near the non-complete ends of , as we will review now (see [14, Section 3] and [7, Section 3] for analogous arguments in slightly different settings).
We write for the -norm of the derivative of any curve measured with respect to g, i.e.
We introduce the open subsets
If τ is small enough, is connected and evenly covered by . Namely, there exists a connected component such that can be written as a disjoint union
We choose such a connected component so that, for all with stationary curve at some point , we have
For all sufficiently small, we have
We cover the sphere with two open balls , and choose a primitive of σ on . Let be sufficiently small so that for any with length less than τ there exists such that γ is entirely contained in . The restriction of the functional to takes the following form: for each with , we have
Since we are assuming that the restriction of the Tonelli Lagrangian L to any fiber of is a polynomial of degree 2 outside a compact set, there exist constants such that, for all , we have
We denote by λ the 1-form on given by . The lower bound (3.4) implies that, for all with , we have
where the latter inequality follows from [1, Lemma 7.1]. This readily implies that on provided
Assume now that . If , we have
If , then , and therefore
Overall, this proves that .
Inequality (3.5) implies that, for all with , we have
where, as before, the second inequality follows from [1, Lemma 7.1]. This readily implies that as , which, together with the fact that on , also implies that . ∎
3.3 Essential Families
Let us fix an energy value . We say that a union of critical circles
is an essential family for when for every neighborhood of there exists a path whose image is contained in the union .
We denote by the subset of those such that the set of critical points is a union of isolated critical circles (that is, the periodic orbits with energy e are isolated). Notice that every energy level contains infinitely many periodic orbits. The existence of essential families can be guaranteed on generic energy levels in . The precise statement is the following.
There is a subset of full Lebesgue measure such that, for all , , and with , the space of paths admits an essential family.
The proof goes along the lines of the one of [2, Lemma 3.5], but the fact that we are working on the universal cover of with the functional requires some variations of the original argument, and therefore we provide full details for the reader’s convenience.
For all and such that , we denote by the subset of those such that the function
Being a countable intersection of full Lebesgue measure subsets of I, the subset has full Lebesgue measure as well.
Now, we fix and two distinct . In order to simplify the notation, we will just write and for and , respectively. We choose an arbitrary strictly decreasing sequence such that as , and we set . By definition of , there exists such that
For all such that and , the period of the curve can be bounded as
while the action can be bounded as
We introduce the subspaces
By the definition of the minmax value and by the estimates that we have just provided, for each there exists a path such that
We recall that, by the definition of the spaces of paths , we have that
Lemma 3.1 (ii) readily implies that we can attach two suitable tails to the path : we can find two continuous paths
such that , , , and ; see [2, Lemma 3.2] for a proof of this elementary fact. Since the open set is bounded, there exists large enough such that
We define the continuous path
Notice that , and as .
We claim that is an essential family for . Let be an arbitrary open set such that
Our goal for the remainder of the proof is to deform one of our paths , away from its endpoints, so that the modified path will have image inside . Notice that, since , if is small enough we have
and contains at most finitely many critical circles of . In particular, we can find a smaller open neighborhood of and some such that every smooth path with and has length at least . Here, the length is the one measured with respect to the pull-back of the Riemannian metric (3.2) to the universal cover .
Consider the open subsets introduced in (3.3), and the selected connected components of their preimage . Since , the set is the union of finitely many critical circles of . In particular, there exists small enough such that
If needed, we reduce so that the open subset is connected and evenly covered by . By Lemma 3.2, there exist and such that, for all ,
Finally, we fix an index large enough so that
In the following, we will denote by the Riemannian norm induced by the Riemannian metric (3.2). With a slight abuse of notation, we will denote by also the Riemannian norm that is pulled-back to the universal cover . Fix and introduce a vector field on of the form , for some suitable smooth function , such that
for all ,
for all such that .
We denote by the flow of V. This flow is complete. Indeed, since the vector field V is uniformly bounded, the flow lines that may not be defined for all positive times are those that enter all sets , for arbitrarily small. Since V is non-negatively proportional to , its flow lines are non-negative reparametrizations of those of . Finally, if a flow line of is not defined for all positive times, then it must enter the set (see [7, Proposition 3.1 (2)] for a proof of this fact), but this latter set is outside the support of V. Actually, since , we have
The free-period action form satisfies a generalized Palais–Smale condition on subsets of where the period is bounded from above and bounded away from zero, see [7, Theorem 2.1 (2)]. Moreover, for each sequence such that and as , we have as , see [7, Theorem 2.1 (1)]. In particular, belongs to for n large enough. This, together with (3.7), implies that there exists a constant such that
We fix an index large enough so that , which together with (3.9) implies
The composition belongs to . We claim that its image is contained in , which sets our goal for the proof. First of all, since does not increase along the flow lines of , we have
There are three possible cases to consider:
Overall, we showed that, for an arbitrary , if is not contained in , then it is contained in the sublevel set .∎
For each and , there exists a constant with the following property. Consider the critical circle of a critical point , where and . If is an essential family containing , then is an essential family for the same space of paths as well.
We set , where and . By Theorem 2.2, there exists such that, for all integers , the following statement holds. There exists an (arbitrarily small) open neighborhood of the critical circle of such that the inclusion induces an injective map between path-connected components
For every , we denote by the corresponding neighborhood of the critical circle of . Clearly, the inclusion induces an injective map
Now, assume that belongs to an essential family for . In particular, we have .
We require the neighborhood to be small enough so that for all neighborhoods of sufficiently small, we have . The existence of such a disjoint is guaranteed by the fact that the set of critical points of comes in isolated critical circles. Since, by Lemma 3.3, admits an essential family, there exists a continuous path whose image is contained in the union
Indeed, and belong to distinct critical circles that are isolated local minimizers of , and this latter functional satisfies the Palais–Smale condition locally.
Therefore, is also an essential family for . ∎
Now, let be the (possibly empty) subset of those energy values such that there are only finitely many (non-iterated) periodic orbits with energy e. In order to prove Theorem 1.1, all we need to do is to prove that the intersection is empty. We will show this in Theorem 3.7, after exploring what would happen on energy values in .
For each energy level and compact interval , there exists a finite union of critical circles such that, for all and with , contains an essential family for .
Let be the only non-iterated periodic orbits with energy e, where r is some natural number, and choose for all . Consider the constants given by Lemma 3.4, so that if we remove the critical circle of any with and from an essential family contained in , the result is still an essential family for the same space of paths. We set
Indeed, consider and such that . Let be an essential family for , whose existence is guaranteed by Lemma 3.3. By Lemma 3.4, if we remove from all the critical circles of periodic orbits of the form for , the resulting set is still an essential family for . Therefore, is an essential family for . ∎
Let and . For all , and we know that is bounded from below by . Hence, the following quantity is a well-defined real number:
The deck transformation induces a homeomorphism between the spaces of paths and , and we have
This readily implies
The infimum in the definition of is actually attained provided .
If , for all there exists such that and .
Let us fix , and set
Notice that . By Lemma 3.5, there exists a finite union of critical circles such that, whenever , contains an essential family for . We introduce the finite set of critical values
The value is the infimum of those belonging to the finite set F, and therefore it is a minimum. ∎
3.4 The Main Multiplicity Result
Theorem 1.1 is an immediate consequence of the following more precise statement.
The set is empty. Namely, for all energy values , there are infinitely many periodic orbits with energy e.
In the proof of Theorem 3.7, we will need the following abstract lemma established in [2, Lemma 2.5] for the free-period action functional. Being a local statement, such a lemma holds for the functional as well (see Remark 2.1).
Every isolated critical circle has an arbitrarily small open neighborhood such that the intersection has only finitely many connected components.
Proof of Theorem 3.7.
We assume by contradiction that there exists . We set
Lemma 3.5 provides a finite union of critical circles
In particular, contains an essential family for . For each , we consider an open neighborhood of the critical circle given by Lemma 3.8. We define
Notice that has finite cardinality according to Lemma 3.8. For each , there exists with the following property: there exist a path and such that the restriction is contained in the sublevel set , and . Since is finite, by the pigeonhole principle there exist distinct such that . In particular, .
Consider the path obtained by concatenation of three paths: the restricted path , some path connecting with within , and the restricted path traversed in the opposite direction. By construction, . However,
which contradicts the definition of . ∎
Dedicated to the memory of Abbas Bahri (1955–2016)
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: AB 360/2-1
Award Identifier / Grant number: SFB 878
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: WKBHJ (ANR-12-BS01-0020)
Award Identifier / Grant number: COSPIN (ANR-13-JS01-0008-01)
Funding statement: A.A. and L.A. are partially supported by the DFG grant AB 360/2-1 “Periodic orbits of conservative systems below the Mañé critical energy value”. G.B. is partially supported by the DFG grant SFB 878. M.M. is partially supported by the ANR grants WKBHJ (ANR-12-BS01-0020) and COSPIN (ANR-13-JS01-0008-01). Part of this project was carried out while M.M. was visiting the Sobolev Institute of Mathematics in Novosibirsk (Russia), under the Program “Short-Term Visits to Russia by Foreign Scientists” of the Dynasty Foundation; M.M. wishes to thank the Foundation and Alexey Glutsyuk for providing financial support, and Iskander A. Taimanov for the kind hospitality.
 Abbondandolo A., Macarini L., Mazzucchelli M. and Paternain G. P., Infinitely many periodic orbits of exact magnetic flows on surfaces for almost every subcritical energy level, preprint 2014, https://arxiv.org/abs/1404.7641; to appear in J. Eur. Math. Soc. (JEMS). 10.4171/JEMS/674Search in Google Scholar
 Abbondandolo A., Macarini L. and Paternain G. P., On the existence of three closed magnetic geodesics for subcritical energies, Comment. Math. Helv. 90 (2015), no. 1, 155–193. 10.4171/CMH/350Search in Google Scholar
 Asselle L. and Benedetti G., Infinitely many periodic orbits in non-exact oscillating magnetic fields on surfaces with genus at least two for almost every low energy level, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1525–1545. 10.1007/s00526-015-0834-1Search in Google Scholar
 Asselle L. and Benedetti G., The Lusternik–Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles, J. Topol. Anal. 8 (2016), no. 3, 545–570. 10.1142/S1793525316500205Search in Google Scholar
 Bahri A. and Taimanov I. A., Periodic orbits in magnetic fields and Ricci curvature of Lagrangian systems, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2697–2717. 10.1090/S0002-9947-98-02108-4Search in Google Scholar
 Contreras G., The Palais–Smale condition on contact type energy levels for convex Lagrangian systems, Calc. Var. Partial Differential Equations 27 (2006), no. 3, 321–395. 10.1007/s00526-005-0368-zSearch in Google Scholar
 Contreras G. and Iturriaga R., Global Minimizers of Autonomous Lagrangians, 22 Colóquio Brasileiro de Matemática, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1999. Search in Google Scholar
 Contreras G., Macarini L. and Paternain G. P., Periodic orbits for exact magnetic flows on surfaces, Int. Math. Res. Not. IMRN 2004 (2004), no. 8, 361–387. 10.1155/S1073792804205050Search in Google Scholar
 Fathi A., Weak KAM Theorem in Lagrangian Dynamics. Preliminary Version Number 10, Cambridge University Press, Cambridge, 2008. Search in Google Scholar
 Novikov S. P., Variational methods and periodic solutions of equations of Kirchhoff type. II, Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 37–52, 96. 10.1007/BF01106155Search in Google Scholar
 Novikov S. P., The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk 37 (1982), no. 5(227), 3–49, 248. Search in Google Scholar
 Taimanov I. A., The principle of throwing out cycles in Morse–Novikov theory, Dokl. Akad. Nauk SSSR 268 (1983), no. 1, 46–50. Search in Google Scholar
 Taimanov I. A., Non-self-itersecting closed extremals of multivalued or not everywhere positive functionals, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 2, 367–383. Search in Google Scholar
 Taimanov I. A., Closed non-self-intersecting extremals of multivalued functionals, Sibirsk. Mat. Zh. 33 (1992), no. 4, 155–162, 223. Search in Google Scholar
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