On an integrable magnetic geodesic flow on the two-torus
aa r X i v : . [ m a t h . D S ] F e b On an integrable magnetic geodesic flow onthe two-torus ∗ I.A. Taimanov † The motion of a charge in a magnetic field on a configuration space M n isdescribed by the Euler–Lagrange equations for the Lagrangian function L ( q, ˙ q ) = 12 g ik ˙ q i ˙ q k + A i ˙ q i where the first term is the kynetic energy calculated by using the Riemannianmetric g ik on M n and the second term describes the interaction of a chargewith the magnetic field F which is a closed two-form F on M n such that F = dA . If F is non-exact, then the one-form A is defined only locally. If F = 0, then we get the Lagrangian function for geodesics on M n and forthis reason solutions q ( t ) for the Euler–Lagrange equations with a generalmagnetic field are called magnetic geodesics.Trajectories lying on the energy level E = 12 g ik ˙ q i ˙ q k = constsatisfy to the Euler–Lagrange equations for the reduced Lagrangian function L E ( q, ˙ q ) = √ E p g ik ˙ q i ˙ q k + A i ˙ q i . ∗ The work was supported by RSF (grant 14-11-00441). † Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia, and Department ofMechanics and Mathematics, Novosibirsk State University, 630090 Novosibirsk, Russiae-mail: [email protected].
1n this article we consider the magnetic geodesic flow on the two-torus T = R / (2 π Z ) , where the coordinates x and y are defined modulo 2 π , with the flat Rieman-nian metric ds = dx + dy and with the magnetic field given by the 2-form F = cos x dx ∧ dy. This form is exact: F = dA with A = sin x dy. Therewith we1. completely integrate the flow and, in particular, describe all trajectoriesin terms of elliptic functions (Theorem 1);2. show that for all contractible periodic magnetic geodesics the reducedaction S E ( γ ) = Z γ L E dt is positive: S E > for E < find the minimizers of the action S E extended to submani-folds Σ of T as follows: S E (Σ , f ) = √ E length ( ∂f (Σ)) + Z Σ f ∗ ( F ) , where f : Σ → T is the embedding (Theorem 3 ); This extension of S E for the space of films S E was introduced in [25, 26]. Theorem 3describes the minimal films for the case E < , i.e. when S E attains negative values. explicitly describe all contractible periodic magnetic geodesics and, inparticular, show that they exist only for E < and simple (not iteratesof others) contractible periodic magnetic geodesics form two S -families (Theorem 4).The initial intention for writing this article was to supply the study ofperiodic problem for magnetic geodesics with a non-trivial explicit exampleof a magnetic geodesic flow with interesting properties.The study of the periodic problem for magnetic geodesics was initiatedby Novikov in early 1980s [18, 19, 20] in the framework of qualitative studyof certain Hamiltonian systems from classical mechanics. It appears that anapplication of the classical Morse theory to proving the existence of periodicmagnetic geodesics meets many obstacles which can not be overgone by clas-sical methods [20]. We gave a survey of that as well as of the first results inthis direction in our survey [24].For dealing with these difficulties new ideas methods arising from sym-plectic geometry, dynamical systems and fixed points theory were proposed(see, for instance, [1, 5, 9, 10, 11, 12, 13, 14, 15, 16, 17, 21, 22, 23, 25, 26]).One of them consists in a proposal to extend the reduced action functional S E for closed curves in an oriented two-manifold to the space of films, andthen to establish the existence of a minimal film whose boundary componentswould be locally minimal periodic magnetic geodesics. That was done in[25, 26] where the existence of nontrivial periodic magnetic geodesics wasestablished for strong magnetic fields.Let us assume that a magnetic field F is exact. Then it is easy to satisfythe strength condition from [25, 26] by multiplying F by a sufficiently largeconstant λ or by considering only low energy levels E . In fact, the ratio λ √ E has to be sufficiently large, or, in other words, E < E where E is someconstant. However if E > E with E a constant, then S E is the lengthfunctional for a Finsler metric and the periodic problem can be studied bythe classical Morse theory. It was established in [11] that E = E = C ,the constant C is the Mane critical level and on this level the existence of aperiodic magnetic geodesic is derived from the Aubry–Mather theory. Hencethe existence of a periodic magnetic geodesic was established for all energylevels for exact magnetic fields on oriented closed two-manifolds.Recently the existence of infinitely many periodic magnetic geodesics wasestablished for exact magnetic fields on two-manifolds for almost all subcrit-ical energy levels, i.e. for almost all E < C [2, 3]. For other recent results3hich clarify the periodic problem on two-manifolds we refer to [4, 6, 8].To finish the introduction we are left to make two
Remarks.
1) Given an exact magnetic field, if the functional S E attainsnegative values on certain closed contractible curves, then the whole manifoldof one-point curves M n can be overthrown into domain { S E < } and henceevery k -cycle from M n generates a nontrivial cycle from H k +1 ( P ( M n ) , { S E ≤ } ) where P is the space of contractible closed curves in M n ( “the principleof throwing out cycles” ) [23, 27]. It would be interesting to understand howthe contractible periodic magnetic geodesics given by Theorem 4 fit into thispicture and how to compute H ∗ ( P ( M n ) , { S E ≤ } ) by using them.2) The existence of a minimal film proved in [25, 26] does not say anythingabout the existence of contractible periodic magnetic geodesics. A priori ifwe have a nonselfintersecting closed curve γ with S E ( γ ) <
0, then it is notnecessary that there exists a minimizer of S E in the class of nonselfintersectingclosed curves with S E <
0. Theorems 2 and 3 say that the flow in studysupplies an explicit example of such a situation. Actually this article arisesas an answer to the question by A. Abbondandolo on such an example askedby him after my talk in Bochum in May 2015.
The motion of a charge in T in the magnetic field F is defined by theLagrangian function L ( q, ˙ q ) = 12 g ik ˙ q i ˙ q k − A i ˙ q i = ˙ x + ˙ y x ˙ y, where q = ( x, y ) ∈ T .The corresponding Lagrangian system takes the form¨ x = cos x ˙ y, ¨ y = − cos x ˙ x. (1)Since the Lagrangian function is independent on y , we have Proposition 1
The system (1) has two functionally independent first inte-grals:1) the energy E = ˙ x + ˙ y , ) the momentum corresponding to y : p := p y = ˙ y + sin x. By this Proposition, we have˙ y = p − sin x, ˙ x = p E − ( p − sin x ) , and therefore these equations are integrable in quadratures: t = Z xx dv p E − ( p − sin v ) ,y ( t ) = y + pt − Z t sin x ( τ ) dτ, (2)where the constants p and E are defined by the initial data: x = x (0) , y = y (0) , ˙ x (0) , ˙ y (0) . To find x ( t ) we have to invert the the integral in the first equation from (2.To do that we make the substitution z = sin x and derive t = Z sin x sin x dzw where w = (1 − z )(2 E − ( p − z ) ) . (3)This elliptic curve is reduced to the Legendre normal form by the standardprocedure (see §
5) and x ( t ) is expressed in terms of the Jacobi elliptic func-tions. Theorem 1 sin x ( t ) = sn (cid:18) t + DC , k (cid:19) ,y ( t ) = y + pt − Z t sn (cid:18) τ + DC , k (cid:19) dτ, where D = R sin x dzw and the constants k and D are determined by the re-duction of the elliptic curve (3) to the Legendre normal form (see (14) and(15)). emark. This flow is a simplest example of a magnetic analog of thegeodesic flow on a surface of revolution. The general family is given by pairsconsistsing of a metric g ( x )( dx + dy ) and a two-form h ′ ( x ) dx ∧ dy wherethe functions g and h are both periodic. In this case the additional conser-vation law needed for complete integrability takes the form of the “Clairautintegral”: ∂L∂ ˙ y = ˙ yg ( x ) + h ( x ) . We think that in this class one can find other interesting examples of magneticgeodesic flows.
The Lagrangian is not homogeneous in velocities and therefore the restric-tions of the system on different energy levels are not trajectorically isomor-phic as in the case of the geodesic flow. In fact, the restriction of the systemon the energy level is described by the Lagrangian L E = √ E p g ik ˙ q i ˙ q k + A i ˙ q i = √ E p ˙ x + ˙ y + sin x ˙ y. Indeed, the Lagrangian L E is homogeneous of first order in velocities and thecorresponding trajectories are defined up to parameterizations. The Euler–Lagrange equations takes the form ddt √ Eg ik ˙ q k p g ik ˙ q i ˙ q k ! = ∂L E ∂q i and for E = g ik ˙ q i ˙ q k both left- and -right-hand sides reduces to the left- andright-hand sides for the Euler–Lagrange equation for L .The following statement is easily checked by straightforward computa-tions Proposition 2
The Lagrangian function L E ( q, ˙ q ) is positive for all ˙ q = 0 ifand only if E > .If E = , then L E vanishes at { ˙ x = 0 , ˙ y > , sin x = − } and { ˙ x = 0 , ˙ y < , sin x = 1 } . L E takes negative values for every E < . emark. The particular level E = may be treated as the Mane criticallevel of the system. For magnetic geodesic flows the Mane critical level C is defined for magnetic fields F on Riemannian manifolds M such that thepullback π ∗ F of F onto the universal covering f M of M is exact. In this caseit is defined as C = inf θ sup M H ( q, θ )where H ( q, p ) = | p | is the Hamiltonian function of the system and dθ = π ∗ F . In our case we have C = inf θ sup R
12 ( θ + θ )where θ = θ dx + θ dy . Since dθ = cos x dx ∧ dy , we have θ = f x , θ = f y + sin x , where df = f x dx + f y dy and f : R → R is a smooth function.The restriction of f y onto the line sin x = 1 is 2 π -periodic and R π f y dy =0. Therefore on this line f y achieves its minimum at which f y = 0 and( f y + sin x ) = 1 . That implies that sup R H ( q, θ ) ≥ . This lower boundbecomes exact at f ≡ C = .Closed trajectories of the flow lying on the energy level E are extremalsof the functional S E ( γ ) = Z γ L E dt defined on the space of closed curves. However this functional is not alwaysbounded from below in contrast to the length functional studied in the clas-sical Morse theory and moreover for non-exact magnetic fields, i.e. whenthe closed 2-form is not globally represented as a differential dA of a 1-form,this functional is multi-valued. These differences with the Morse they werediscussed in detail by Novikov [20] (see also [24]).In our case the magnetic field F is exact however, by Proposition 2, S E is not bounded from below for small values of E .The variational study of the periodic problem is hindered also by the factthat fit is not known does S E satisfies the Palais–Smale type conditions. Apriori the deformation decreasing S E may diverge even when S E approachesa critical level.In [25, 26] for studying periodic magnetic geodesics in two-dimensionalmanifolds we introduced the spaces of films which are embeddings f : Σ → M of oriented two-manifolds Σ with boundaries into a two-dimensional7losed manifold M endowed with a Riemannian metric and with a magneticfield F which is not necessarily exact. For such films f : Σ → M there is defined an action functional S E (Σ , f ) = √ E length ( f ( ∂ Σ)) + Z Σ f ∗ ( F ) , which for exact magnetic fields reduces to S E = R f ( ∂ Σ ) L E dt . It was provedin [25, 26] that if1. F is exact, or F is non-exact and R M F ‘ > (as we assume withoutloss of generality),2. there is a film with S E < , then there exists a film Σ on which S E attains its minimal value among filmsand the boundary of Σ consists of closed magnetic trajectories which arelocally minimal for S E . By (2), y ( t ) = y (0) + Z ( p − z ) dz p (1 − z )(2 E − ( p − z ) ) , z = sin x ( t ) . Therefore for the closed trajectory γ ( t ) of the flow we have∆ y = Z γ ˙ y dt = Z γ ( p − z ) dz p (1 − z )(2 E − (( p − z ) )where ∆ y is the increment of y along the pullback of the trajectory onto theuniversal covering R → T . If a closed trajectory is contractible, then∆ y = 0 . Such a field is called strong if F is exact or oscillating if F is non-exact. S E ( γ ) = Z γ L E dt = Z γ ( √ E p ˙ x + ˙ y + sin x ˙ y ) dt == Z γ ( √ E p ˙ x + ˙ y + sin x ˙ y ) dt = Z γ ( √ E p ˙ x + ˙ y + ( p − ˙ y ) ˙ y ) dt == Z γ ( √ E p ˙ x + ˙ y − ˙ y )) dt + p Z γ ˙ y dt. Since √ E p ˙ x + ˙ y = ˙ x + ˙ y , p is constant, and R γ ˙ y dt = ∆ y , we have S E ( γ ) = Z γ ˙ x dt + p ∆ y ≥ . (4) Proposition 3
Given the energy level E , if γ is a nontrivial (different froma one-point contour) closed magnetic geodesic with ˙ x ≡ , then it is one ofthe following orbits: γ ±± = ( x ( t ) = ± π , y ( t ) = ±√ E t, ≤ t ≤ π r E ) , and no one of these orbits is contractible. The proof of this Proposition immediately follows from (1). Togetherwith (4) this Proposition implies
Theorem 2
For a contractible periodic magnetic geodesic γ which is differ-ent from a one-point contour and lies on the energy level E , we have S E ( γ ) = Z γ ˙ x dt > . Let us consider a film Π formed by the embedding of the cylinderΠ = (cid:26) π ≤ x ≤ π , ≤ y ≤ π (cid:27) into T . Its image is the closure of the domain on which F <
0. Theboundary of Π is formed by a pair of closed trajectories: ∂ Π = γ − + ∪ γ + − S E (Π) = √ E length ( ∂ Π) + Z π dy Z π/ π/ cos xdx = 4 π ( √ E − . Theorem 3
For every
E < the functional S E on the space of films attainsits minimal value on Π . Proof.
By the results of [25] mentioned in § S E attains its minimumon a film Σ whose boundary components are local minima of S E .Let all the boundary components are contractible. This is possible if Σconsists in components diffeomorphic to discs with holes. The boundary ofevery hole is a contractible magnetic geodesic γ and, by Theorem 2, S E > e Σ with S E ( e Σ) = S E (Σ) − S E ( γ ) , i.e. decrease the value of S E which contradicts to the definition of Σ as aglobal minimum.Hence ∂ Σ contains a non-contractible component γ and, since ∂ Σ realizesa trivial class in 1-homologies, it has to contain another non-contractiblecomponent γ . Therefore S E (Σ) ≥ length ( γ ) + length ( γ ) + Z Ψ F. However the lengths of noncontractible contours are at least √ E π , i.e.,the length of γ + − and of γ − + which are minimal non-contractible geodesicson T . Therefore S E (Σ) ≥ π + Z Σ F and the right-hand side achieves its minimum on the film Π. Q.E.D.Now let us describe all nontrivial (different from a point) contractibleclosed trajectories.By (2), every periodic in x trajectory is obtained by the inversion of theintegral Z dz p (1 − z )(2 E − ( p − z ) ) , z = sin x, (5)10here z goes along the bounded real oval, i.e. the contour, on the Riemanniansurface w = P ( z ) = (1 − z )(2 E − ( p − z ) ) , which covers the interval I such that P ( z ) ≥ P ( z )vanishes at its ends. It is clear that I ⊂ [ − , . Moreover it is clear that all roots of P ( z ) are different otherwise the integral(5) diverges and does not correspond to an x -periodic solution.On every periodic trajectory there exist a pair of points q , q such that˙ x ( q ) = ˙ x ( q ) = 0 , ˙ y ( q ) > , ˙ y ( q ) < . (6)The condition ˙ x = 0 is equivalent to2 E − ( p − z ) = 0 . (7)If I contains only one root of this equation then at the corresponding pointson an x -periodic trajectory ˙ y = p − z has the same values and therefore thereare no points q and q meeting (6). In this case the x -periodic trajectorydoes not close up and there is a nontrivial translation period in y . Hence fora periodic trajectory the roots z < z of (7) lie inside I : − < z < z < . We are left to check the last condition that the translation period in y vanishes. By (2), it is equal to∆ y = 2 Z z z ( p − z ) dz p (1 − z )(2 E − ( p − z ) ) == 2 Z z z ( p − z ) dz p (1 − z )( z − z )( z − z ) , z + z = 2 p. We have z = p − a, z = p + a, a > , and by substitution u = p − z we derive that∆ y = 2 Z a − a u du p (1 − ( p − u ) )( a − u ) .
11y comparing the values of the integrand at ± u , we infer that∆ y > p < < p >
0= 0 for p = 0 . Let us summarize these facts in the following
Theorem 4
1. For E ≥ there are no contractible closed orbits.2. For every E such that < E < , there exist two S -families of simpleperiodic magnetic geodesics. These families are invariant with respectto translations by x : x → x + const , and obtained by the inversion ofthe integral t = Z dz p (1 − z )(2 E − z ) , −√ E ≤ z = sin x ( t ) ≤ √ E, and by solving the equation for dynamics in y : ˙ y = − sin x. All other nontrivial contractible periodic magnetic geodesics are iteratesof these simple closed magnetic geodesics.3. These families lie in the domains separated by the contours on which sin x = ± . In particular, no one contractible closed orbit intersectsthese contours.4. These families degenerate to the pair of contours { sin x = 0 } formed byone-point closed curves as E → .5. For these simple periodic magnetic geodesics S E = 2 Z a − a p E − sin x dx, a = arcsin √ E. Statement 4 is quite evident from the physical point of view: for verysmall energies closed orbits are trapped near critical points of the magneticfield, i.e. of the function f such that F = f dx ∧ dy .Statement 5 is derived by straightforward computations from Theorem 2.12 Appendix: The Legendre normal form ofan elliptic curve and elliptic integrals
In this section we recall some facts on the reduction of an elliptic curve tothe Legendre normal form (for more details see, for instance, [7]). This isnecessary for deriving Theorem 1.Let P ( z ) = ( z − a )( z − a )( z − a )( z − a ) , be a polynomial with four different real zeroes a , . . . , a . We recall how totransform the Riemann surface (elliptic curve) w = P ( z ) (8)to the Legendre form η = (1 − ξ )(1 − k ξ ) . (9)We enumerate the zeroes as follows: a < a < a < a and decompose P ( x ) into a product P ( z ) = Q ( z ) Q ( z ) of two quadraticpolynomials of the form Q ( z ) = ( z − a )( z − a ) , Q ( z ) = ( z − a )( z − a ) . Let us consider two cases:1) Q ( z ) = z − a , Q ( z ) = z − a . Then the transformation ξ = za , η = wa a reduces the equation (8) to the form (9) with k = a a .2) If the case 1) does not hold, then there exist λ and λ such that Q ( z ) − λ Q ( z ) = (1 − λ )( z − µ ) , Q ( z ) − λ Q ( z ) = (1 − λ )( z − ν ) . These constants λ , are determined as the eigenvalues of the pair of quadraticforms defined by Q and Q , i.e., as the zeroes of the equationdet (cid:18) − λ − a + a + λ a + a − a + a + λ a + a a a − λa a (cid:19) = 0 . Q ( z ) = B ( z − µ ) + C ( z − ν ) , Q ( z ) = B ( z − µ ) + C ( z − ν ) . (10)The constants B , B , C , C are trivially computed as B j = Q j ( ν )( µ − ν ) , C j = Q j ( µ )( µ − ν ) , j = 1 , , (11)and a substitution of them into the equations B + C = B + C = 1 leadsto the following relations2( µν + a a ) = ( µ + ν )( a + a ) , µν + a a ) = ( µ + ν )( a + a ) . (12)Since µ and ν are different from the zeroes a j , j = 1 , . . . ,
4, the latter relationsare rewritten as ν − a ν − a = − µ − a µ − a , ν − a ν − a = − µ − a µ − a . (13)The solutions µ and ν to (13) are obtained as the zeroes of the quadraticequation: λ − Aλ + B = 0where, by (12), we have A = 2 a a − a a a + a − a − a , B = a a ( a + a ) − a a ( a + a ) a + a − a − a . It is easy to check that µ and ν have to correspond to different real ovals of(8). This means, without loss of generality, that a < ν < a and µ < a or µ > a . By (10), we have w = ( B ( z − µ ) + C ( z − ν ) )( B ( z − µ ) + C ( z − ν ) ) == B ( z − µ ) (cid:18) C B ( z − ν ) ( z − µ ) (cid:19) B ( z − µ ) (cid:18) C B ( z − ν ) ( z − µ ) (cid:19) =14 B B ( z − µ ) (1 − ξ )(1 − k ξ )for ξ = r − C B z − νz − µ , k = B C B C . We are left to put η = w √ B B ( z − µ ) to reduce the curve (8) to the Legendre normal form (9). By (11) and (13),we have p B B = p P ( ν )( µ − ν ) > ,k = (cid:18) ν − a µ − a (cid:19) (cid:18) µ − a ν − a (cid:19) , (14) λ = r − B C = s ( ν − a )( ν − a )( µ − a )( a − µ ) , λ a − νa − µ = ξ ( a ) = ± . Since λ is defined up to a sign, let us put λ = a − νµ − a to achieve ξ ( a ) = − . Therewith we have ξ = 1 λ z − νz − µ = µ − a a − ν z − νz − µ ,ξ ( a ) = 1 , ξ ( a ) = − k , ξ ( a ) = 1 k where k > { ( z, w ) , P ( z ) ≥ } are mapped into real ovals by thetransformation ( z, w ) → ( ξ, η ), we conclude that k < . By dξdz = ν − µλ ( z − µ ) ,
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