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# Honors Calculus MATH 20

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A CHARGE IN A MAGNETIC FIELD ARNOLD S PROBLEMS 19819 198224 19844 199414 199435 199617 AND 199618 VIKTOR L GINZBURG 1i PROBLEMS The following problems are from Zadachi Arnol da Arnolldls problems by Vili Arnol d preface by MiBi Sevryuk and ViBi Filippov lzdatel stvo FAZIS Moscow 2000 in Russian The comments below are prepared for the English translation of this book 19819 Consider closed contractible bounding a disk on the universal covering curves of constant geodesic curvature K 0 on a surface M2i There are at least as many such curves as critical points of a function on M Counterexample horo cycles on a surface of constant negative curvature However for T2 and 52 this conjecture has not been disproved 198224 Can the center of mass of a convex domain in a homogeneous sphere coincide with the center of the sphere Since it cannot it makes sense to try to prove the existence of two closed curves magnetic trajectories of constant positive geodesic curvature on the sphere as follows Fiber the space of convex discs over 52 nd constrained critical points along bers using variational methods and then apply Morse theory techniques to look for critical points along the base 19844 Prove that on T2 there are generically at least four closed on the universal covering curves of constant geodesic curvature K gt 0 199414 Consider a particle in a magnetic eld on a Riemannian manifold of an arbitrary dimension The magnetic eld is given by a closed twoform twisting the symplectic form of the phase space In the case of a strong magnetic eld large curvature trajectories apply the averaging method and at least formulate conjectures on topological lower bounds for the number of periodic orbits These conjectures should generalize the theorem on the existence of 29 2 curves of large geodesic curvature on a surface of genus gr 199435 Find lower bounds for the number of periodic orbits of a charge in a magnetic eld where the motion of the charge is con ned to a surface and the magnetic eld is orthogonal to the surface Conjecturally on a surface of genus g a charge should generically have at least 29 periodic orbits From a mathematical perspective this is a problem about closed curves with given positive geodesic cur vature on the surface When the magnetic eld is suf ciently strong the conjecture is proved cfi Problem 1994 14 Date December 26 2001 2 VIKTOR L GINZBURG 2 COMMENTS As I see it now the work will consist of an introduction of about sixty pages the trans lation proper some two hundred printed pages and over three hundred pages of var ious notes and commentaries 7 Vladimir Nabokov Letter to Henry Allen Moe March 1953 see Na p 135 1 The magnetic problem First recall the Hamiltonian description of the motion of a charge in a magnetic eld on a manifold Let be a Riemannian manifold and let a be a closed twoform on M the magnetic eld Consider the twisted symplectic form to we 7r a on T M Here we is the standard symplectic form on T M and Tr T M A M is the natural projection The motion of a unit charge on M in the magnetic eld a is given by the Hamiltonian ow of the standard kinetic energy H T M A R In what follows we will refer to this ow as a twisted geodesic flow If M is a surface the integral curves of the twisted geodesic ow project to the curves on M whose geodesic curvature is k ad A where dA is the area form Hence the problems in question all concern the existence of periodic orbits of twisted geodesic ows Note that k need not be constant We emphasize that in contrast with the geodesic ow the case when a 0 the dynamics of the twisted geodesic ow on a level H 5 depends on c When 5 is large the twisted geodesic ow is close to the geodesic ow although not necessarily equivalent to it On the other hand when 5 gt 0 is small the ow is in some sense governed y a To illustrate this point consider the twisted geodesic ow on a closed surface of constant negative curvature 71 and a dA Arl For c E 012 the ow is periodic and the period goes to in nity as c A 127 In this case all orbits are periodic and contractible in M For c 12 the ow is the horocycle ow which is known to have no periodic orbits He This is a counterexample to the conjecture from Problem 199435 cf Problem 19819 and GiSD When 5 gt 12 the ow is smoothly equivalent to the geodesic ow In particular every free nontrivial homotopy class of a map 5391 A M contains the projection to M of a periodic orbit and there are no periodic orbits whose projections are contractible Hence when studying the existence of periodic orbits it makes sense to treat separately the cases of high and low energy levels For an arbitrary magnetic eld a the existence question for periodic orbits on low energy levels is still poorly understood see however 130 and we will focus mainly on the case where a is symplectic Note also that periodic orbits contractible in M may persist for all values of H This is true for example for a at torus and positive is Ar2 K01 Below we focus exclusively on the symplectic geometry approach to the existence problem for periodic orbits originating from Ar2 A different approach based on the MorsePNovikov theory is not discussed here see eg GN No NT Ta1 Ta2 2 Twisted geodesic ows on surfaces In this section we brie y list results on the existence of periodic orbits of twisted geodesic ows on surfaces with non vanishing magnetic eld Thus throughout this section we assume that k f 0 As A CHARGE IN A MAGNETIC FIELD 3 have already been pointed out for a at torus every level H c carries at least three periodic orbits four if the orbits are nondegenerate whose projection to the torus is contractible Ar2 Kolli This result essentially solving Problem 19844 was obtained by Arnold in the mideighties as a consequence of the Conley7Zehnder theorem CZ Since then it has served as the main motivation for applications of symplectic techniques to the study of periodic orbits of twisted geodesic ows In a similar vein consider an arbitrary closed orientable surface M of genus g with any metrici Then there are at least three periodic orbits on every low energy level two if M S2 and at least 29 2 when the orbits are nondegenerate See Ar2 Gil Gi3 K02 and the survey GiQ for further details and references This result is a partial solution of Problem 19819 cfi Problems 198224 and 199618 The dynamics on low energy levels can also be studied by using the averaging method ch Problem 199617 We refer the reader to Ar3 BS Ca Li Tr and the references therein for a detailed discussion of this method applications of KAM and adiabatic invariantsi 3 Twisted geodesic ows in higher dimensions Before formulating a conjecture concerning the lower bounds for the number of periodic orbits of twisted geodesic ows in higher dimensions let us discuss some of the relevant results Throughout this section M 0 denotes a compact symplectic manifold of dimen sion 2721 Note that Ma is then a symplectic submanifold of T Mwi Denote by N the vector bundle over M formed by symplectic orthogonals to M in T Mi For every I E M we have the Hamiltonian ow of d2H on N75 and as a result a berwise linear ow on N called the limiting owi The Hamiltonian ow on H e for a small 6 gt 0 is close after suitable rescaling to the limiting owi When the eigenvalues of d2HlNz do not bifurcate iiei periodic orbits of the limit ing ow form smooth manifolds in d2H e the results of the previous section extend to higher dimensions GKL Keli Assume for instance that the metric is conformal to an almostKahler metric on Ma iiei HX X f aXJX where f is a positive function and J is an almost complex structure compatible with 0 This assumption implies that all orbits of the limiting ow are closed and holds automatically when M is a surface Then on a low energy level there are at least CLM m periodic orbits Ke and at least SBM if the orbits are nondegenerate GKlli Here CLM stands for the cuplength of M over R and SBM denotes the sum of Betti numbersi Under different nonbifurcation conditions the lower bounds can be improved for example when at every point of M all eigenvalues are distinct there are at least mCLM periodic orbits The lower bounds from Ke are obtained using Moser s method Mo which can be viewed as a higheridimensional version of the averaging over the limiting ow ch Problem 19944 These results lead to the conjecture that in general for a symplectic magnetic eld and low energy levels the number of periodic orbits is no less than CLM1 or even CLMm and SB when the orbits are nondegeneratei Note that it is still unknown whether or not periodic orbits exist on a dense set of low energy levels However one can show that there are contractible actually small periodic orbits for a sequence of energy values converging to zero GKZ The main difficulty which arises in showing by symplectic topology methods that every low energy level carries a periodic orbit lies in the fact that it is hard to 4 VIKTOR L GINZBURG nd a tractable variational principle which would pick up periodic orbits on a xed energy level of noncontact type Clearly the berwise convexity of H should be used here in an essential way see the comments to Problem 199413 However the existence of periodic orbits for a dense set of levels appears to be accessible by using symplectic or Floer homologyi Finally to obtain a lower bound on the number of periodic orbits one needs a way to show that the action functional in question has sufficiently many critical points corresponding to geometrically distinct periodic orbits A different perspective on the magnetic problem in higher dimensions comes from the WeinsteiniMoser theoremi Consider the following question let W be a 2m dimensional symplectic manifold and let H W A R be a proper smooth function which has a MorsePBott nondegenerate minimum H 0 along a compact symplectic submanifold M of W Does the Hamiltonian flow of H have a periodic orbit on every energy level H e where e gt 0 is small We will refer to the affirmative answer to this question as the generalized WeinsteiniMoser conjecture According to Moser and Weinstein when W R2 and hence M is a point every low energy level carries at least n periodic orbits M0 We On the other hand taking W T M and H and w as above we see that the magnetic problem is just a particular case of the generalized WeinsteiniMoser conjecture In fact the results of GKL GKZ Ke are all proved in the context of this conjecture and apply to a broader class of Hamiltonian systems than the motion of a charge in a magnetic eld 4 High energy levels and degenerate magnetic elds In this section we only mention some of the relevant results For any metric on M T and any magnetic eld 0 almost all in the sense of measure theory levels of H carry at least one periodic orbit See Jil Ji2 GKl Lu Mai This fact is established by showing that T M has bounded Hoferi Zehnder capacity For any weakly exact a on a compact manifold M there exists a sequence of positive numbers ck A 0 such that every level H ck carries a contractible periodic orbit Mac2 Po As has been shown by J Mather for M T2 with any a and a nonflat metric there is a noncontractible periodic orbit on every high energy level see Gi2i It is a simple consequence of the Viterbo theorem Vi that for any metric and a on 52 there is a periodic orbit on every high energy level cf Problem 199618 Furthermore the results of Bialy Bi on Hopf rigidity also serve as indirect evidence of existence of contractible periodic orbits when M is a torus and a f 0 The dynamics of twisted geodesic flows is much better understood for exact magnetic elds For example in this case there is a periodic orbit on every high energy level as is easy to see eg from A sharper result can be obtained by directly applying variational methods ClPPl Note in this connection that for low energy levels the action functional need not satisfy the Palais7Smale condition see eg ClPPZDi The energy lower bound arising here is closely related to Ma e s critical value see ClPPl ClPPQ Man PP3i For exact magnetic elds on surfaces the sharpest results for low energy levels come from the MorsePNovikov theory GN Tal Ta2i Finally we refer the reader to the forthcoming paper CMP for a theorem closing the gap between the dynamics on low and high energy levels for exact twisted geodesic flows on surfaces A CHARGE IN A MAGNETIC FIELD 5 Topological entropy of twisted geodesic ows was studied in eigi7 NiL Ni2 Macl PPl PP2li Acknowledgments The author is deeply grateful to Vladimir Arnold Basak Giirel and Debra Lewis for numerous remarks and suggestions Arl ArZ Ar3 REFERENCES Vl Arnold Some remarks on ows of line elements and frames Soviet Math Dokl 2 1961 5627564 Vl Arnold On some problems in symplectic topology in Topology and Ceomet 7 Rochlin Seminar OYa Viro Editor Lect Notes in Math vol 1346 Springer 1988 VI Arnold Remarks on the Morse theory of a divergenceefree vector eld the averaging method and the motion of a charged particle in a magnetic eld Proc Stehloy Inst Math 216 1997 3713 G Benettin and P Sempio Adiabatic invariants and trapping of a point charge in a strong nonuniform magnetic eld Nonlinearity 7 1994 2817303 Bialy Rigidity for periodic magnetic elds Ergodic Theory Dynam Systems 20 2000 161971626 C Castilho The motion of a charged particle on a Riemannian magnetic eld J Di erential Equations 171 2001 1107131 Conley and E e er Birkho iLewis xed point theorem and a conjecture of VI Arnold Inventiones Math 73 1983 33419 G Contreras R lturriaga GP Paternain M Paternain Lagrangian graphs minimizing measures an Mane7s critical value Ceom Funct Anal 8 1998 78809 G Contreras R lturriaga GP Paternain M Paternain The Palaisesmale condition and Ma7s critical values Ann Henri Poincare 1 2000 65584 G Contreras L lturriaga GP Paternain Periodic orbits for exact magnetic ows on surfaces in preparation VL Ginzburg New eneralizations of Poincare7s geometric theorem Functional Anal Appl 21 2 1987 1007106 VL Ginzburg On closed trajectories of a charge in a magnetic eld An application of symplectic geometry in Contact and Symplectic Geometry Cambridge 1994 CB Thomas Editor Publ Newton Inst 8 Cambridge University Press Cambridge 1996 p 1317148 VL Ginzburg On the existence and nonexistence of closed trajectories for some Harnile tonian ows Math Z 223 1996 3977409 VL Ginzburg and E Kerman Periodic orbits in magnetic elds in dimensions greater than two in Geometry and topology in dynamics Eds M Barge and K Kuperberg Contemp Math 246 Amer Math Soc Providence RI 1999 111121 VL Ginzburg and E Kerman Periodic orbits of Hamiltonian ows near symplectic extrema Preprint 2000 mathDG0011011 to appear in Paci c Journal of Mathematics PG Grinevich and SP Novikov Nonsel ntersecting magnetic orbits on the plane Proof of the overthrowing of cycles principle in Topics in topology and mathematical physics 592 Amer Math Soc Transl Ser 2 170 Amer Math Soc Providence RI 1995 GA Hedlund Geodesics on a twoedimensional Riemannian manifold with periodic co e icients Annals of Math 33 1932 7197739 H Hofer and C Viterbo The Weinstein conjecture for cotangent bundles and related results Ann Scuola Norm Sup Pisa Cl Sci 4 15 1988 no 3 4114145 1989 H Hofer and E Zehnder Symplectic invariants and Hamiltonian dynamics Birkhauser Advanced Texts BaseleBostoneBerlin 1994 MrY Jiang Hofer7Zehnder symplectic capacity for Zedimensional manifolds Proc Roy Soc Edinbuiyh 123A 1993 9457950 7Y iang Periodic solutions of Hamiltonian systems on hypersurfaces in a torus Manuscripta Math 85 1994 3077321 E Kerman Periodic orbits of Hamiltonian ows near symplectic critical submanifolds Internat Math Res Notices 1999 17 9537969 surface under a nonzero A CHARGE IN A MAGNETIC FIELD ARNOLD S PROBLEMS 19819 198224 19844 199414 199435 199617 AND 199618 VIKTOR L GINZBURG 1i PROBLEMS The following problems are from Zadachi Arnol da Arnolldls problems by Vili Arnol d preface by MiBi Sevryuk and ViBi Filippov lzdatel stvo FAZIS Moscow 2000 in Russian The comments below are prepared for the English translation of this book 19819 Consider closed contractible bounding a disk on the universal covering curves of constant geodesic curvature K 0 on a surface M2i There are at least as many such curves as critical points of a function on M Counterexample horo cycles on a surface of constant negative curvature However for T2 and 52 this conjecture has not been disproved 198224 Can the center of mass of a convex domain in a homogeneous sphere coincide with the center of the sphere Since it cannot it makes sense to try to prove the existence of two closed curves magnetic trajectories of constant positive geodesic curvature on the sphere as follows Fiber the space of convex discs over 52 nd constrained critical points along bers using variational methods and then apply Morse theory techniques to look for critical points along the base 19844 Prove that on T2 there are generically at least four closed on the universal covering curves of constant geodesic curvature K gt 0 199414 Consider a particle in a magnetic eld on a Riemannian manifold of an arbitrary dimension The magnetic eld is given by a closed twoform twisting the symplectic form of the phase space In the case of a strong magnetic eld large curvature trajectories apply the averaging method and at least formulate conjectures on topological lower bounds for the number of periodic orbits These conjectures should generalize the theorem on the existence of 29 2 curves of large geodesic curvature on a surface of genus gr 199435 Find lower bounds for the number of periodic orbits of a charge in a magnetic eld where the motion of the charge is con ned to a surface and the magnetic eld is orthogonal to the surface Conjecturally on a surface of genus g a charge should generically have at least 29 periodic orbits From a mathematical perspective this is a problem about closed curves with given positive geodesic cur vature on the surface When the magnetic eld is suf ciently strong the conjecture is proved cfi Problem 1994 14 Date December 26 2001 2 VIKTOR L GINZBURG 2 COMMENTS As I see it now the work will consist of an introduction of about sixty pages the trans lation proper some two hundred printed pages and over three hundred pages of var ious notes and commentaries 7 Vladimir Nabokov Letter to Henry Allen Moe March 1953 see Na p 135 1 The magnetic problem First recall the Hamiltonian description of the motion of a charge in a magnetic eld on a manifold Let be a Riemannian manifold and let a be a closed twoform on M the magnetic eld Consider the twisted symplectic form to we 7r a on T M Here we is the standard symplectic form on T M and Tr T M A M is the natural projection The motion of a unit charge on M in the magnetic eld a is given by the Hamiltonian ow of the standard kinetic energy H T M A R In what follows we will refer to this ow as a twisted geodesic flow If M is a surface the integral curves of the twisted geodesic ow project to the curves on M whose geodesic curvature is k ad A where dA is the area form Hence the problems in question all concern the existence of periodic orbits of twisted geodesic ows Note that k need not be constant We emphasize that in contrast with the geodesic ow the case when a 0 the dynamics of the twisted geodesic ow on a level H 5 depends on c When 5 is large the twisted geodesic ow is close to the geodesic ow although not necessarily equivalent to it On the other hand when 5 gt 0 is small the ow is in some sense governed y a To illustrate this point consider the twisted geodesic ow on a closed surface of constant negative curvature 71 and a dA Arl For c E 012 the ow is periodic and the period goes to in nity as c A 127 In this case all orbits are periodic and contractible in M For c 12 the ow is the horocycle ow which is known to have no periodic orbits He This is a counterexample to the conjecture from Problem 199435 cf Problem 19819 and GiSD When 5 gt 12 the ow is smoothly equivalent to the geodesic ow In particular every free nontrivial homotopy class of a map 5391 A M contains the projection to M of a periodic orbit and there are no periodic orbits whose projections are contractible Hence when studying the existence of periodic orbits it makes sense to treat separately the cases of high and low energy levels For an arbitrary magnetic eld a the existence question for periodic orbits on low energy levels is still poorly understood see however 130 and we will focus mainly on the case where a is symplectic Note also that periodic orbits contractible in M may persist for all values of H This is true for example for a at torus and positive is Ar2 K01 Below we focus exclusively on the symplectic geometry approach to the existence problem for periodic orbits originating from Ar2 A different approach based on the MorsePNovikov theory is not discussed here see eg GN No NT Ta1 Ta2 2 Twisted geodesic ows on surfaces In this section we brie y list results on the existence of periodic orbits of twisted geodesic ows on surfaces with non vanishing magnetic eld Thus throughout this section we assume that k f 0 As A CHARGE IN A MAGNETIC FIELD 3 have already been pointed out for a at torus every level H c carries at least three periodic orbits four if the orbits are nondegenerate whose projection to the torus is contractible Ar2 Kolli This result essentially solving Problem 19844 was obtained by Arnold in the mideighties as a consequence of the Conley7Zehnder theorem CZ Since then it has served as the main motivation for applications of symplectic techniques to the study of periodic orbits of twisted geodesic ows In a similar vein consider an arbitrary closed orientable surface M of genus g with any metrici Then there are at least three periodic orbits on every low energy level two if M S2 and at least 29 2 when the orbits are nondegenerate See Ar2 Gil Gi3 K02 and the survey GiQ for further details and references This result is a partial solution of Problem 19819 cfi Problems 198224 and 199618 The dynamics on low energy levels can also be studied by using the averaging method ch Problem 199617 We refer the reader to Ar3 BS Ca Li Tr and the references therein for a detailed discussion of this method applications of KAM and adiabatic invariantsi 3 Twisted geodesic ows in higher dimensions Before formulating a conjecture concerning the lower bounds for the number of periodic orbits of twisted geodesic ows in higher dimensions let us discuss some of the relevant results Throughout this section M 0 denotes a compact symplectic manifold of dimen sion 2721 Note that Ma is then a symplectic submanifold of T Mwi Denote by N the vector bundle over M formed by symplectic orthogonals to M in T Mi For every I E M we have the Hamiltonian ow of d2H on N75 and as a result a berwise linear ow on N called the limiting owi The Hamiltonian ow on H e for a small 6 gt 0 is close after suitable rescaling to the limiting owi When the eigenvalues of d2HlNz do not bifurcate iiei periodic orbits of the limit ing ow form smooth manifolds in d2H e the results of the previous section extend to higher dimensions GKL Keli Assume for instance that the metric is conformal to an almostKahler metric on Ma iiei HX X f aXJX where f is a positive function and J is an almost complex structure compatible with 0 This assumption implies that all orbits of the limiting ow are closed and holds automatically when M is a surface Then on a low energy level there are at least CLM m periodic orbits Ke and at least SBM if the orbits are nondegenerate GKlli Here CLM stands for the cuplength of M over R and SBM denotes the sum of Betti numbersi Under different nonbifurcation conditions the lower bounds can be improved for example when at every point of M all eigenvalues are distinct there are at least mCLM periodic orbits The lower bounds from Ke are obtained using Moser s method Mo which can be viewed as a higheridimensional version of the averaging over the limiting ow ch Problem 19944 These results lead to the conjecture that in general for a symplectic magnetic eld and low energy levels the number of periodic orbits is no less than CLM1 or even CLMm and SB when the orbits are nondegeneratei Note that it is still unknown whether or not periodic orbits exist on a dense set of low energy levels However one can show that there are contractible actually small periodic orbits for a sequence of energy values converging to zero GKZ The main difficulty which arises in showing by symplectic topology methods that every low energy level carries a periodic orbit lies in the fact that it is hard to 4 VIKTOR L GINZBURG nd a tractable variational principle which would pick up periodic orbits on a xed energy level of noncontact type Clearly the berwise convexity of H should be used here in an essential way see the comments to Problem 199413 However the existence of periodic orbits for a dense set of levels appears to be accessible by using symplectic or Floer homologyi Finally to obtain a lower bound on the number of periodic orbits one needs a way to show that the action functional in question has sufficiently many critical points corresponding to geometrically distinct periodic orbits A different perspective on the magnetic problem in higher dimensions comes from the WeinsteiniMoser theoremi Consider the following question let W be a 2m dimensional symplectic manifold and let H W A R be a proper smooth function which has a MorsePBott nondegenerate minimum H 0 along a compact symplectic submanifold M of W Does the Hamiltonian flow of H have a periodic orbit on every energy level H e where e gt 0 is small We will refer to the affirmative answer to this question as the generalized WeinsteiniMoser conjecture According to Moser and Weinstein when W R2 and hence M is a point every low energy level carries at least n periodic orbits M0 We On the other hand taking W T M and H and w as above we see that the magnetic problem is just a particular case of the generalized WeinsteiniMoser conjecture In fact the results of GKL GKZ Ke are all proved in the context of this conjecture and apply to a broader class of Hamiltonian systems than the motion of a charge in a magnetic eld 4 High energy levels and degenerate magnetic elds In this section we only mention some of the relevant results For any metric on M T and any magnetic eld 0 almost all in the sense of measure theory levels of H carry at least one periodic orbit See Jil Ji2 GKl Lu Mai This fact is established by showing that T M has bounded Hoferi Zehnder capacity For any weakly exact a on a compact manifold M there exists a sequence of positive numbers ck A 0 such that every level H ck carries a contractible periodic orbit Mac2 Po As has been shown by J Mather for M T2 with any a and a nonflat metric there is a noncontractible periodic orbit on every high energy level see Gi2i It is a simple consequence of the Viterbo theorem Vi that for any metric and a on 52 there is a periodic orbit on every high energy level cf Problem 199618 Furthermore the results of Bialy Bi on Hopf rigidity also serve as indirect evidence of existence of contractible periodic orbits when M is a torus and a f 0 The dynamics of twisted geodesic flows is much better understood for exact magnetic elds For example in this case there is a periodic orbit on every high energy level as is easy to see eg from A sharper result can be obtained by directly applying variational methods ClPPl Note in this connection that for low energy levels the action functional need not satisfy the Palais7Smale condition see eg ClPPZDi The energy lower bound arising here is closely related to Ma e s critical value see ClPPl ClPPQ Man PP3i For exact magnetic elds on surfaces the sharpest results for low energy levels come from the MorsePNovikov theory GN Tal Ta2i Finally we refer the reader to the forthcoming paper CMP for a theorem closing the gap between the dynamics on low and high energy levels for exact twisted geodesic flows on surfaces A CHARGE IN A MAGNETIC FIELD 5 Topological entropy of twisted geodesic ows was studied in eigi7 NiL Ni2 Macl PPl PP2li Acknowledgments The author is deeply grateful to Vladimir Arnold Basak Giirel and Debra Lewis for numerous remarks and suggestions Arl ArZ Ar3 REFERENCES Vl Arnold Some remarks on ows of line elements and frames Soviet Math Dokl 2 1961 5627564 Vl Arnold On some problems in symplectic topology in Topology and Ceomet 7 Rochlin Seminar OYa Viro Editor Lect Notes in Math vol 1346 Springer 1988 VI Arnold Remarks on the Morse theory of a divergenceefree vector eld the averaging method and the motion of a charged particle in a magnetic eld Proc Stehloy Inst Math 216 1997 3713 G Benettin and P Sempio Adiabatic invariants and trapping of a point charge in a strong nonuniform magnetic eld Nonlinearity 7 1994 2817303 Bialy Rigidity for periodic magnetic elds Ergodic Theory Dynam Systems 20 2000 161971626 C Castilho The motion of a charged particle on a Riemannian magnetic eld J Di erential Equations 171 2001 1107131 Conley and E e er Birkho iLewis xed point theorem and a conjecture of VI Arnold Inventiones Math 73 1983 33419 G Contreras R lturriaga GP Paternain M Paternain Lagrangian graphs minimizing measures an Mane7s critical value Ceom Funct Anal 8 1998 78809 G Contreras R lturriaga GP Paternain M Paternain The Palaisesmale condition and Ma7s critical values Ann Henri Poincare 1 2000 65584 G Contreras L lturriaga GP Paternain Periodic orbits for exact magnetic ows on surfaces in preparation VL Ginzburg New eneralizations of Poincare7s geometric theorem Functional Anal Appl 21 2 1987 1007106 VL Ginzburg On closed trajectories of a charge in a magnetic eld An application of symplectic geometry in Contact and Symplectic Geometry Cambridge 1994 CB Thomas Editor Publ Newton Inst 8 Cambridge University Press Cambridge 1996 p 1317148 VL Ginzburg On the existence and nonexistence of closed trajectories for some Harnile tonian ows Math Z 223 1996 3977409 VL Ginzburg and E Kerman Periodic orbits in magnetic elds in dimensions greater than two in Geometry and topology in dynamics Eds M Barge and K Kuperberg Contemp Math 246 Amer Math Soc Providence RI 1999 111121 VL Ginzburg and E Kerman Periodic orbits of Hamiltonian ows near symplectic extrema Preprint 2000 mathDG0011011 to appear in Paci c Journal of Mathematics PG Grinevich and SP Novikov Nonsel ntersecting magnetic orbits on the plane Proof of the overthrowing of cycles principle in Topics in topology and mathematical physics 592 Amer Math Soc Transl Ser 2 170 Amer Math Soc Providence RI 1995 GA Hedlund Geodesics on a twoedimensional Riemannian manifold with periodic co e icients Annals of Math 33 1932 7197739 H Hofer and C Viterbo The Weinstein conjecture for cotangent bundles and related results Ann Scuola Norm Sup Pisa Cl Sci 4 15 1988 no 3 4114145 1989 H Hofer and E Zehnder Symplectic invariants and Hamiltonian dynamics Birkhauser Advanced Texts BaseleBostoneBerlin 1994 MrY Jiang Hofer7Zehnder symplectic capacity for Zedimensional manifolds Proc Roy Soc Edinbuiyh 123A 1993 9457950 7Y iang Periodic solutions of Hamiltonian systems on hypersurfaces in a torus Manuscripta Math 85 1994 3077321 E Kerman Periodic orbits of Hamiltonian ows near symplectic critical submanifolds Internat Math Res Notices 1999 17 9537969 surface under a nonzero

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