JJo Nelson From Dynamics to Contact and Symplectic Topology and Back
Symplectic and contact topology is an active area of mathematics that combines ideas from dynam-ical systems, analysis, topology, several complex variables, as well as differential and algebraicgeometry. Symplectic and contact structures first arose in the study of classical mechanical systems ,allowing one to describe the time evolution of both simple and complex systems such as springs,planetary motion and wave propagation [3]. Understanding the evolution and distinguishingtransformations of these systems led to the development of global invariants of symplectic andcontact manifolds.The equations of motion in classical mechanics are determined by the notion of a conservedquantity, energy . A related quantity is action , which is minimized by solutions to the equations ofmotion. For a closed system, such as the Kepler problem whose solutions describe paths of planetsorbiting the sun, the energy is the sum of the kinetic and potential energy in the system, and theaction is given by the (minimized) mean value of kinetic minus potential energy. Symplectic andcontact structures emerge as we investigate these systems by unpacking the information hiddenin the notions of energy and action.The position of a particle in a mechanical system is a point x = ( x , .., x n ) in Euclidean space,and the vector space R n defined by these coordinates is called the configuration space . The positionand momentum of a particle allows us to predict the particle’s motion at all future times withina system. The phase space of a system is precisely this space that represents all possible states ofthe system, consisting of both the position and momentum of a particle. In the case that there are n degrees of freedom, the phase space is R n . The assumption that the trajectories of a particle x ( t ) minimize an action functional gives rise to a system of n second-order differential equationscalled the Euler-Lagrange equations , discovered in 1808 by Joseph-Louis Lagrange [32].These equations grew out of Lagrange’s observation that the possible elliptic motions of asingle planet under the sun’s gravitational pull can be described by six real parameters. How-ever, the influence of other planets perturbs this ellipticity. In order to describe the variation, onemust study the derivatives of these real parameters. These three equations are extremely compli-cated, but they can be simplified by introducing
Lagrange brackets , which are combinations of thederivatives with respect to position and velocity at fixed time. Lagrange then showed that theseequations can be transformed into what is now known as a
Hamiltonian system of six first-order dif-ferential equations that conserve energy [33]. At the time, his notion of energy was a “disturbingfunction,” which described the variance from elliptic motion. Moreover, these Lagrange bracketsturn out to be the coefficients of humanity’s oldest symplectic structure [48].Figure 1:
The symplectic gradient X H , [34] In the mid 1800s, William Rowan Hamiltonand Carl Jacobi realized the theoretical conse-quences of Lagrange’s work, in particular that the n Euler-Lagrange equations can be transformedinto a Hamiltonian system of 2 n equations [49].The Hamiltonian system is governed by the con-servation of an energy function, called the Hamil-tonian function H ( x , y ) , which defines the Hamil-tonian vector field X H . The flow lines of this vectorfield are solutions to Hamilton’s equations of mo- a r X i v : . [ m a t h . S G ] N ov o Nelson From Dynamics to Contact and Symplectic Topology and Back tion,˙ x = ∂ H ∂ y , ˙ y = − ∂ H ∂ x .In the coordinates z = ( x , ... x n , y , ... y n ) ∈ R n the Hamiltonian system can be written in theform of a system of 2 n differential equations, J ˙ z = ∇ H ( z ) .where ∇ H denotes the gradient of H and J is the 2 n × n matrix J = − .The Hamiltonian vector field or symplectic gradient of H , seen in Figure 1, is defined by X H = − J ∇ H : R n → R n .Systems whose Hamiltonian function explicitly depends on time, such as those describingthe motion of a charged particle in a time-dependent electric field, use extended phase space , whichincludes the 2 n -phase space plus the time variable. Extended phase space results in the notion of a contact structure . In this setting, solutions to equations of motion yield flows of a Hamiltonian-likevector field, called the Reeb vector field.Contact structures appear naturally in other areas of mathematics and physics, including ther-modynamics [2]. In particular, contact geometry allows one to understand geodesic flow on thetangent bundle of a Riemannian manifold. Geodesics are locally the shortest distance betweenpoints, where distance is defined in terms of a metric intrinsic to a manifold. An n -dimensional manifold is a smooth object that locally looks like R n . One can interpret a Riemannian manifoldas a model for an optical medium, in which case geodesics with respect to the metric correspondto light rays. This in turn yields
Huygens’s principle , which states that every point on a wavefrontis a source of wavelets, which spread forward at the same speed.
Figure 2:
At the infinitesi-mal level, ω measures orientedarea spanned by vectors X and Y at a point p . To study more general even dimensional Hamiltonian systems weneed to allow symplectic manifolds to serve as the phase space. In clas-sical mechanics, replacing the standard 2 n -dimensional phase spacewith a 2 n -manifold results in a canonical symplectic structure on themanifold, reflecting the conservation of energy. Formally, a symplec-tic form ω is a closed nondegenerate 2-form. It allows one to measuretwo dimensional area in a well-defined way, as seen in Figure 2, andas a result forces symplectic manifolds to be even dimensional. Usingthe symplectic form one can define the Hamiltonian vector field , X H , ona symplectic manifold by ω ( X H , · ) = dH ( · ) . For example, the surface of a donut or beach ball is a 2-manifold. If we cut out a small piece of either surface and“zoomed in” it would look like a flat sheet of paper, e.g. R . o Nelson From Dynamics to Contact and Symplectic Topology and Back The name symplectic arose in 1939 due to Hermann Weyl, who studied the symplectic lineargroup. This group manifests itself when one studies the canonical transformations of a Hamiltoniansystem, which are changes of coordinates that preserve Hamilton’s equations. Weyl recalls in afootnote on page 165 [50], “The name complex group formerly advocated by me in allusion to linecomplexes, as these are defined by the vanishing of antisymmetric bilinear forms, has becomemore and more embarrassing through collision with the word complex , [a Latin adjective], in theconnotation of complex number. I therefore propose to replace it by the corresponding Greekadjective symplectic .” Figure 3: An integrable (right) & contact (left) structure on R . Many contact manifolds arise as hy-persurfaces or boundaries of symplecticmanifolds, and the geometry of contactand symplectic manifolds is closely inter-twined. A contact structure ξ is a maxi-mally nonintegrable hyperplane distribu-tion. In three dimensions, this means thatthe planes of ξ twist so much that even lo-cally there is never a surface whose tan-gent planes are all contained in ξ , which is in contrast to the notion of an integrable hyperplanedistribution, seen in Figure 3. An integrable hyperplane distribution is one in which all the planesare given by tangent planes of a submanifold. Any 1-form α whose kernel defines a contact struc-ture is called a contact form .Figure 4: The flows of two Reeb vector fields; theright is on S and is parametrized by S . Credit:Patrick Massot (left), Niles Johnson (right) The
Reeb vector field R α depends on the choice ofcontact form α and is defined by α ( R α ) = d α ( R α , · ) = R α preserves the form α and hence thecontact structure ξ . It can also follow very complexpatterns, as in Figure 4.Moreover, the flows Reeb vector fields of differ-ent contact forms defining the same contact struc-ture may have wildly different properties.An interesting result about symplectic and con-tact manifolds is Darboux’s theorem, which statesthat locally all contact structures look like the kernel of the standard contact form on R n + , ξ = ker α = ker (cid:32) dz + n ∑ i = x i dy i (cid:33) ,and that locally all symplectic forms look like the standard symplectic form on R n , ω = n ∑ i = dx i dy i .Hence, there can be no local invariants of symplectic and contact manifolds, a stark contrast toRiemannian geometry where the notion of curvature provides local invariants. In the symplectic Now such isomorphisms are called symplectomorphisms, due to Souriau’s contributions [42]. o Nelson From Dynamics to Contact and Symplectic Topology and Back realm, the absence of local invariants means that there is an infinite dimensional group of dif-feomorphisms that preserve the symplectic structure and a discrete set of nonequivalent global symplectic structures in each cohomology class. Analogously in the contact realm, there is an in-finite dimensional group of diffeomorphisms that preserve the contact structure and a discrete setof nonequivalent global contact structures in each planar homotopy class.The ability to distinguish contact structures in a planar homotopy class is not obvious. Oneof the first results along these lines is the celebrated theorem of Yakov Eliashberg from 1989 [11],which states that the 3-sphere admits two homotopy classes of contact structures which are ho-motopic as plane fields but which are not homotopic via contact structures.Figure 5: The overtwisted (left, Patrick Massot) and stan-dard (right, Otto van Koert) contact structure.
One of these structures is the standard struc-ture , given in cylindrical coordinates ( r , θ , z ) ∈ R by ξ std = ker α = ker ( dz + r d θ ) ,and the other is the overtwisted contact struc-ture , ξ OT = ker ( cos rdz + r sin rd θ ) .These are visualized in the z = ξ std and ξ OT are horizontal along the z -axis and along any ray they both turn counterclockwise as one moves outward from the z -axis.However, the rotation angle of ξ std approaches (but never reaches) π /2, while the contact planesof ξ OT make infinitely many complete turns. A closed orbit of a vector field X on a manifold M is a map, γ : R / T Z → M ,for some T >
0, which satisfies the ordinary differential equation˙ γ ( t ) = X ( γ ( t )) .One is then led to wonder when a (smooth) vector field X on a closed manifold M admits a closedorbit. For some special three manifolds like the 3-torus, it is easy to construct vector fields with noclosed orbit. On the other hand, when M is the 3-sphere, this question turns out to be incrediblydifficult and not always possible; see [29] for a brief history.The Weinstein conjecture is one of the most famous questions in regard to the existence of pe-riodic orbits [47]. It originated from work in the 1970s by Alan Weinstein, who demonstrated theexistence of periodic orbits on convex compact hypersurfaces in R n [46], and Paul Rabinowitz,who demonstrated the existence of periodic orbits on star shaped hypersurfaces in R n [39]-[41].In reading Rabinowitz’s papers, Weinstein realized that there was a simple geometric feature com-mon in the different results, namely that we he called contact type , which is a special contact hy-persurface in a symplectic manifold. Weinstein’s realization connected the existence of periodicorbits of Hamiltonian systems to contact geometry, spurring further interest in the study of contactmanifolds. o Nelson From Dynamics to Contact and Symplectic Topology and Back The Weinstein conjecture:
Let ( M , ξ ) be a closed co-oriented contact manifold. Then for anycontact form α for ξ , the Reeb vector field R α admits a closed periodic orbit.At the same time, Rabinowitz’s paper [40] had a profound effect on a young graduate student,Helmut Hofer. Helmut reminisced at his sixtieth birthday conference:Why did I come into symplectic geometry? I had the flu, and the only thing to readwas a copy of Rabinowitz’s paper where he proves the existence of periodic orbits onstar-shaped energy surfaces [40]. It turned out to contain a fundamental new idea,which was to study a different action functional for loops in the phase space ratherthan for Lagrangians in the configuration space. Which actually if we look back, ledto the variational approach in symplectic and contact topology, which is reincarnatedin infinite dimensions in Floer theory and has appeared in every other subsequentapproach. The flu turned out to be really good.This variational approach led to further progress by Claude Viterbo in 1987 for hypersurfacesof contact type in R n [45], which was extended further by Hofer-Viterbo ([23] in 1987), Hofer-Zehnder ([28] in 1988), and Struwe ([43] in 1990).Meanwhile, the Arnold conjecture haunted the dreams of geometers. The Arnold conjecture.
A symplectomorphism on a closed symplectic manifold that is gener-ated by a time-dependent Hamiltonian vector field should have at least as many fixed points as afunction on the manifold must have critical points.The minimal number of critical points is a topological invariant, which means that it is un-changed under homeomorphisms. Thus, the very flexible topology of the manifold determinesqualitative aspects of Hamiltonian flows. In 1983, Charles Conley and Eduard Zehnder provedthis conjecture for tori of arbitrary dimension via a finite dimensional approximation of the sym-plectic action functional on the loop space [9]. The other affirmative result was due to Eliash-berg in 1979, who proved it for closed two-dimensional symplectic manifolds, Riemann surfaces.At this point, the variational methods involving finite-dimensional approximations of the actionfunctional on the loop space stalled.Fortunately, in 1985, Mikhail Gromov pioneered the study of moduli spaces of pseudoholo-morphic curves [21] to prove his celebrated nonsqueezing theorem, demonstrating that symplecticmappings are very different from volume-preserving ones.
The Gromov nonsqueezing theorem . A standard symplectic ball cannot be symplecticallyembedded into a thin cylinder.Andreas Floer’s subsequent breakthrough was to marry the variational methods of Conleyand Zehnder with Gromov’s theory of pseudoholomorphic curves, by adapting ideas from Ed-ward Witten’s interpretation of Morse theory [51]. Floer realized that the gradient trajectories In 1976, Moser wrote that this action functional was “certainly not suitable for an existence proof, [36, (1.5)].”Rabinowitz, on the other hand, showed more optimism than his former advisor in 1977, [39, Remark 4.44]. This brings to mind the anecdote of how Edward Witten, a physicist, came to develop his unique perspective ofMorse theory. Raoul Bott recalls first exposing Witten to Morse theory in [6]. “In 1979 I gave some lectures at Carg`ese onequivariant Morse theory...to a group of very bright physicists, young and old, most of whom took a rather detachedview of the lectures. ‘Beautiful and oh so far from Physics’ was Wilson’s reaction, I remember. On the other hand,Witten followed the lectures like a hawk, asked questions, and was clearly very interested. I therefore thought I haddone a good job indoctrinating him, so that I was rather nonplussed to receive a letter from him some eight monthslater, starting with the comment, ‘Now I finally understand Morse theory!’” o Nelson From Dynamics to Contact and Symplectic Topology and Back counted in Morse theory didn’t need to come from a flow, but instead just needed to satisfy a suf-ficiently nice partial differential equation with appropriate asymptotics, see [13] - [16]. Gromov’spseudoholomorphic curves are maps between closed Riemann surfaces and symplectic manifoldsthat satisfy the Cauchy-Riemann equation, a nonlinear elliptic partial differential equation. Floermodified them, studying moduli spaces of noncompact pseudoholomorphic curves perturbed bya Hamiltonian term. These Floer trajectories are maps from the cylinder to a symplectic manifoldthat converge at the ends to 1-periodic solutions of the associated Hamiltonian vector field.At first, Gromov was skeptical of Floer’s ideas. Floer however, successfully formulated thenonlinear Fredholm theory describing his Floer trajectories as the zero set of an infinite dimen-sional bundle, thereby realizing the gradient trajectories of the highly degenerate action functionalon the loop space. This led to the creation of what is now called
Floer theory , an infinite dimen-sional extension of Witten’s reformulation of Morse theory. Floer used his new theory and itsvariants to define symplectic invariants [18] and prove the Arnold conjecture in many cases [17].
In 1993, Hofer realized he could study moduli spaces of pseudoholomorphic maps from the com-plex plane to the symplectization of a contact 3-manifold to prove the Weinstein conjecture for S [22]. However, the study of the moduli spaces of pseudoholomorphic planes is not straightfor-ward due to additional difficulties in establishing compactness and transversality. Clifford Taubeswent on to prove the Weinstein conjecture in dimension three in 2007, relying on deep results inSeiberg-Witten theory [44].During the later ’90s, Helmut Hofer, Kris Wysocki, and Eduard Zehnder continued their studyof pseudoholomorphic curves in contact geometry, leading to a wealth of new dynamical results.This work led Hofer, together with Eliashberg, to the concept of contact homology. In 2000, con-structions of these moduli based theories looked promising with the advent of a comprehensive symplectic field theory announced in [12], a generalization of Floer theory and Gromov Witten the-ory [35]. This field theory involves the study of pseudoholomorphic curves from punctured Rie-mann surfaces to noncompact symplectic manifolds with cylindrical ends.These curves are still the zero set of an infinite dimensional bundle, but there is typically afailure of transversality. As a result, one must perturb the zero set describing these curves, usingeither the ambient geometry or an abstract functional analytic framework. Otherwise the resultingmoduli spaces will not yield well-defined invariants. Hofer, Wysocki, and Zehnder have devel-oped the abstract analytic framework, collectively known as polyfolds , to systematically resolvethese issues, see [24]-[27], and provide foundations for symplectic field theory. My research,
These lectures led to Witten’s 1982 paper [51], which used ideas from quantum physics to streamline Morse theory.He recalled the evolution of these ideas in his Commemorative Lecture for the 2014 Kyoto Prize:“...Trying to get to thebottom of things, I considered simpler and simpler models, each of which turned out to contain the same puzzle. Afterpondering this for a long time, I eventually remembered – I think while in a swimming pool in 1981 – a lecture that Ihad heard by Raoul Bott about two years earlier...I am sure that just like me, most of the physicists at that school hadnever heard of it, and had no idea what it might be good for in physics. And I had probably not heard of Morse theoryagain until that day in 1981 when – dimly managing to remember part of what Bott had told us – I realized that Morsetheory was behind what I had been puzzling over.” In 1997, when Gromov was awarded the Steele Prize for a Seminal Contribution to Research for his pseudoholo-morphic curves, he recalled: “Floer has morsified them [pseudoholomorphic curves] by breaking the symmetry, and Istill cannot forgive him for this. (Alas, prejudice does not pay in science.)” [7]. The symplectization of ( M , ker α ) is ( R × M , d ( e t α )) . This development indicates some clairvoyance on the part of George David Birkhoff, who in 1938 indicated his o Nelson From Dynamics to Contact and Symplectic Topology and Back in part joint with Michael Hutchings, makes use of geometric perturbation methods to providecomplete foundations for a subset of symplectic field theory known as cylindrical contact homol-ogy. These geometric methods require additional assumptions on the underlying space, but arepreferable for computations and applications [30, 31, 37, 38].Recent work has shown that the three body problem can be studied via contact geometry[1, 8, 19]. As a result, the modern methods of pseudoholomorphic curves are expected to giveinsight into the movement of satellites, allowing one to make predictions about the existence andnumber of energy efficient orbits that cannot be found by classical methods [4]. It would then befitting to conclude with the words of an anonymous, albeit optimistic, symplectic geometer: “Thefuture of contact and symplectic geometry looks so bright that we all have to wear shades.”
References [1] P. Albers, J.W. Fish, U. Frauenfelder, H, Hofer, O. van Koert,
Global surfaces of section in theplanar restricted 3-body problem . Arch. Ration. Mech. Anal. 204 (2012), no. 1, 273–284.[2] V. I. Arnold,
Contact geometry: the geometrical method of Gibbs’ thermodynamics , Proceedingsof the Gibbs Symposium (New Haven, 1989), American Mathematical Society, Providence(1990), 163–179.[3] V.I. Arnold,
Mathematical Methods in Classical Mechanics . Springer-Verlag, Berlin (1978).[4] E. Belbruno,
Capture dynamics and chaotic motions in celestial mechanics. With applications to theconstruction of low energy transfers.
Princeton University Press, 2004.[5] G. D. Birkhoff,
Fifty years of American mathematics, Semicentennial Addresses of Amer. Math. Soc. ,1938, p. 307.[6] R. Bott,
Morse theory indomitable . Inst. Hautes ´Etudes Sci. Publ. Math. No. 68 (1988), 99-114.[7] M. Berger,
Encounter with a Geometer, Part II.
Notices of the AMS 47 (3): 326-340.[8] K. Cieliebak, U. Frauenfelder, and O. van Koert,
The Finsler geometry of the rotating Keplerproblem.
Publ. Math. Debrecen 84 (2014), no. 3-4, 333–350.[9] C. Conley and E. Zehnder,
The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol’d.
Invent. Math. 73 (1983), no. 1, 33-49.[10] Y. Eliashberg,
Estimates on the number of fixed points of area preserving transformations.
SyktyvkarUniversity Preprint. 1979.[11] Y. Eliashberg,
Classification of overtwisted contact structures on 3-manifolds . Invent. Math. 98(1989), no. 3, 623-637.[12] Y. Eliashberg, A. Givental, and H. Hofer,
Introduction to symplectic field theory . GAFA (2000),560-673. “disturbing secret fear that geometry may ultimately turn out to be no more than the glittering intuitional trappingsof analysis” [5]. On the other hand in 1980, Weinstein noted, “The recent success of symplectic geometric methods inlinear partial differential equations suggests that one might need the glitter to find the gold” [49]. o Nelson From Dynamics to Contact and Symplectic Topology and Back [13] A. Floer, A relative Morse index for the symplectic action.
Comm. Pure Appl. Math. 41 (1988), no.4, 393-407.[14] A. Floer,
The unregularized gradient flow of the symplectic action . Comm. Pure Appl. Math. 41(1988), no. 6, 775-813.[15] A. Floer,
Morse theory for Lagrangian intersections . J. Differential Geom. 28 (1988), no. 3, 513-547.[16] A. Floer,
Witten’s complex and infinite-dimensional Morse theory . J. Differential Geom. 30 (1989),no. 1, 207-221.[17] A. Floer,
Symplectic fixed points and holomorphic spheres. Comm. Math. Phys. , 120 (1989), 37-88[18] A. Floer,
Cuplength estimates on Lagrangian intersections.
Comm. Pure Appl. Math. 42 (1989),no. 4, 335-356.[19] U. Frauenfelder,
Dihedral homology and the moon . J. Fixed Point Theory Appl. 14 (2013), no. 1,55-69.[20] H. Geiges,
An Introduction to Contact Topology . Cambridge Studies in Advanced Mathematics109, Cambridge University Press, 2008.[21] M. Gromov,
Pseudoholomorphic curves in symplectic manifolds . Invent. Math. (1985), 307-347.[22] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjec-ture in dimension three . Invent. Math., (1993), 515-563.[23] H. Hofer and C. Viterbo,
The Weinstein conjecture in cotangent bundles and related results . Ann.Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 3, 411-445 (1989).[24] H. Hofer, K. Wysocki, and E. Zehnder,
A General Fredholm Theory I: A Splicing-Based DifferentialGeometry , JEMS 9:841-876, 2007.[25] H. Hofer, K. Wysocki, and E. Zehnder,
A general Fredholm theory II: Implicit function theorems ,GAFA 19:206-293, 2009.[26] H. Hofer, K. Wysocki, and E. Zehnder,
A general Fredholm theory III: Fredholm Functors andPolyfolds , Geom. & Topol. 13:2279-2387, 2009.[27] H. Hofer, K. Wysocki, and E. Zehnder,
Applications of Polyfold Theory I: Gromov-Witten Theory ,arXiv: 1107.2097[28] H. Hofer and E. Zehnder,
Periodic solutions on hypersurfaces and a result by C. Viterbo.
Invent.Math. 90 (1987), no. 1, 1-9.[29] M. Hutchings,
Taubes’ proof of the Weinstein conjecture.
Bulletin of the AMS 47 (2010), 73-125.[30] M. Hutchings and J. Nelson,
Cylindrical contact homology for dynamically convex contact forms inthree dimensions , to appear in Jour. Symp. Geom. (30 pg), arXiv:1407.2898[31] M. Hutchings and J. Nelson,
An integral lift of cylindrical contact homology without contractibleReeb orbits , in preparation. o Nelson From Dynamics to Contact and Symplectic Topology and Back [32] J. L. Lagrange, M´emoire sur la th´eorie des variations des ´e´ements des plan`etes , M´em. Cl. Sci. Math.Phys. Inst. France (1808), 1-72.[33] J. L. Lagrange,
Second m´emoire sur la th´eorie de la variation des constantes arbitraires dans lesprobl`emes de m´ecanique , M´em. Cl. Sci. Math. Phys. Inst. France (1809), 343-352.[34] D. McDuff and D. Salamon,
Introduction to Symplectic Topology . Oxford University Press,(1995).[35] D. McDuff and D. Salamon,
J-holomorphic Curves and Symplectic Topology , AMS ColloquiumPublications, 2004.[36] J. Moser,
Periodic orbits near an equilibrium and a theorem by Alan Weinstein.
Comm. Pure Appl.Math. 29 (1976), no. 6, 724-747.[37] J. Nelson,
Automatic transversality in contact homology I: Regularity . Abh. Math. Semin. Univ.Hambg. 85 (2015), no. 2, 125-179.[38] J. Nelson,
Automatic transversality in contact homology II: Invariance and computations . In prepa-ration.[39] P. Rabinowitz,
A variational method for finding periodic solutions of differential equations . Nonlin-ear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977), 225-251.[40] P. Rabinowitz,
Periodic solutions of Hamiltonian systems , Comm. Pure Appl. Math 31 (1978),157-184.[41] P. Rabinowitz,
Periodic solutions of a Hamiltonian system on a prescribed energy surface.
J. Differ-ential Equations 33 (1979), no. 3, 336-352.[42] J.-M. Souriau,
Structure des syst´emes dynamiques , Dunod, Paris 1970.[43] M. Struwe,
Existence of periodic solutions of Hamiltonian systems on almost every energy surface.
Bol. Soc. Brasil. Mat. (N.S.) 20 (1990), no. 2, 49-58.[44] C. Taubes,
The Seiberg-Witten equations and the Weinstein conjecture.
Geom. Topol. 11 (2007),2117-2202.[45] C. Viterbo,
A proof of Weinstein’s conjecture in R n , Annales de l’Institut Henri Poincar´e Anal.Non Lin´eaire, 4 (1987), 337–356.[46] A. Weinstein, Periodic orbits for convex Hamiltonian systems , Ann. of Math. (2) 108 (1978), 507–518.[47] A. Weinstein,
On the hypothesis of Rabinowitz periodic orbit theorems , J. Differential Equations 33(1979), 353–358.[48] A. Weinstein,
Lectures on symplectic manifolds.
CBMS Regional Conference Series in Mathe-matics, 29. American Mathematical Society, 1979.[49] A. Weinstein,
Symplectic geometry . Bull. Amer. Math. Soc. 5 (1981), no. 1, 113. o Nelson From Dynamics to Contact and Symplectic Topology and Back [50] H. Weyl, The Classical Groups. Their Invariants and Representations , Princeton University Press,1939.[51] E. Witten
Supersymmetry and Morse theory . J. Diff. Geom. 17 (1982), no. 4, 661-692.. J. Diff. Geom. 17 (1982), no. 4, 661-692.