Introduction to symplectic and hamiltonian geometry. This theorem shows that the symplectic invariants called symplectic capac. While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Symplectic invariants and hamiltonian dynamics core. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and hamiltonian dynamics. These symplectic invariants include spectral invariants, boundary depth, and partial symplectic quasistates. Dan cristofarogardiner institute for advanced study university of colorado at boulder january 23, 2014 dan cristofarogardiner what can symplectic geometry tell us about hamiltonian dynamics. Symplectic invariants and hamiltonian dynamics springerlink. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. The discoveries of the last decades have opened new perspectives for the old field of hamiltonian systems and led to the creation of a new field. Symplectic maps to projective spaces and symplectic invariants.
Floer homology in symplectic geometry and in mirror symmetry. We know that elliptic and hyperbolic orbits have no symplectic. Download symplectic invariants and hamiltonian dynamics. Surprising rigidity phenomena demonstrate that the nature of symplectic mappings is very different from that of volume preserving mappings. In this paper we present an attempt to better understand the space of all symplectic capacities, and discuss some further general. Differential invariants for symplectic lie algebras realized. Symplectic maps to projective spaces and symplectic invariants denisauroux abstract. Symplectic invariants and hamiltonian dynamics helmut. Symplectic invariants and hamiltonian dynamics helmut hofer. The nonlocal symplectic vortex equations and gauged gromov.
Periodic orbits for symplectic twist maps of t n x ir n. The main purpose of this paper is to give a topological and symplectic classification of completely integrable hamiltonian systems in terms of characteristic classes and other local and global invariants. Hamiltonian system 1 isnt necessary to be symplectic, and not all symplectic integrator can preserve the quadratic invariants of hamiltonian system 1 6, 16. Differential invariants for symplectic lie algebras. For some handson experience of the standard map, download meiss simulation code 4. Symplectic invariants and hamiltonian dynamics helmut hofer, eduard zehnder auth. As it turns out, these seemingly differ ent phenomena are mysteriously related. This text covers foundations of symplectic geometry in a modern language. What can symplectic geometry tell us about hamiltonian. The discoveries of the past decade have opened new perspectives for the old field of hamiltonian systems and led to the creation of a new field. We present a very general and brief account of the prehistory of the. Dec 18, 2007 we construct symplectic invariants for hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolichyperbolic type. We present applications to approximation theory on symplectic manifolds and to hamiltonian dynamics. Symplectic topology and floer homology is a comprehensive resource suitable for experts and newcomers alike.
On the other hand, due to the analysis of an old variational principle in classical mechanics, global periodic phenomena in hamiltonian systems have been established. Prominent among them are the socalled symplectic capacities. Happily, it is very well written and sports a lot of very useful commentary by the authors. Would it for instance provide any advantage to studying hamiltonian dynamic. Symplectic twist maps advanced series in nonlinear dynamics.
In hamiltonian dynamical system, any time evolution is defined by hamiltonian equations and expressed by canonical transformations or symplectic diffeomorphisms on phase spaces. C0limits of hamiltonian paths and the ohschwarz spectral. B2r zeclz symplectic vectorspaces v, 09, and w, cow the product is defined by. We show how these series are related to the singular. Bayesian inference from symplectic geometric viewpoint. One of the links is a class of symplectic invariants, called symplectic capacities. The nonlocal symplectic vortex equations and gauged. The proof of the nontriviality of these invariants involves various flavors of floer theory. Hamiltonian dynamics on the symplectic extended phase. Symplectic and contact geometry and hamiltonian dynamics.
The flows are symplectomorphisms and hence obey liouvilles theorem. This was soon generalized to flows generated by a hamiltonian over a poisson manifold. Symplectic invariants and hamiltonian dynamics modern. Symplectic invariants and hamiltonian dynamics is obviously a work of central importance in the field and is required reading for all wouldbe players in this game. Jun 08, 2007 hamiltonian dynamics on convex symplectic manifolds frauenfelder, urs. We start by describing symplectic manifolds and their transformations, and by explaining connections to topology and other geometries. Symplectic invariants near hyperbolichyperbolic points. Gromovwitten invariants of symplectic quotients and adiabatic limits gaio, ana rita pires and salamon, dietmar a.
I ceremade, universit6de parisdauphine, place du m. These invariants are the main theme of this book, which includes such topics as basic symplectic geometry, symplectic capacities and rigidity, periodic orbits for hamiltonian systems and the action principle, a biinvariant metric on the symplectic diffeomorphism group and its geometry, symplectic fixed point theory, the arnold conjectures and. Symplectic invariants and hamiltonian dynamics reprint of the 1994 edition helmut hofer institute for advanced study ias school of mathematics einstein drive princeton, new jersey 08540 usa email protected eduard zehnder departement mathematik eth zurich leonhardstrasse 27 8092 zurich switzerland email protected. Recall that hamiltonian mechanics is based upon the flows generated by a smooth hamiltonian over a symplectic manifold. On the other hand, analysis of an old variational principle in classical mechanics has established global periodic phenomena in hamiltonian systems. Spectral invariants in rabinowitzfloer homology and global hamiltonian perturbations. Introduction to symplectic and hamiltonian geometry by ana cannas da silva. As the initial research of contact hamiltonian dynamics in this direction, we investigate the dynamics of contact hamiltonian systems in some special cases including invariants, completeness of phase flows and periodic behavior. The nonlocal symplectic vortex equations and gauged gromovwitten invariants a dissertation submitted to eth zurich for the degree of doctor of sciences presented by andreas michael johannes o t t dipl. It is partially based on a twosemester course, held by the author for thirdyear students in physics and mathematics at the university of salerno, on analytical mechanics, differential geometry, symplectic manifolds and integrable systems.
Jun 10, 2005 different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and hamiltonian dynamics. We show that if two hamiltonians g,h vanish on a small ball and if their flows are sufficiently c 0close, then using the above result, we prove that if. Symplectic topology and floer homology by yonggeun oh. There is a mysterious relation between rigidity phenomena of symplectic geometry and global periodic solutions of hamiltonian dynamics. We consider an explicitly timedependent hamiltonian h that is defined on a finitedimensional contact manifold with its closed, generally degenerate contact 2form. Symplectic invariants for parabolic orbits and cusp. This paper studies how symplectic invariants created from hamiltonian floer theory change under the perturbations of symplectic structures, not necessarily in the same cohomology class. Symplectic invariants and hamiltonian dynamics springer. They are defined through an elementarylooking variational problem involving poisson brackets. Symplectic invariants and hamiltonian dynamics reprint of the 1994 edition helmut.
It is partially based on a twosemester course, held by the author for thirdyear students in physics and mathematics at the university of salerno, on analytical mechanics, differential geometry, symplectic manifolds and. Hamiltonian dynamics on convex symplectic manifolds urs frauenfelder1 and felix schlenk2 abstract. This is not only a matter of was to free classical mechanics from the constraints of specific coordinate systems and to. Symplectic structure perturbations and continuity of.
Salamon, propagation in hamiltonian dynamics and relative symplectic homology, duke math. In mathematics, nambu mechanics is a generalization of hamiltonian mechanics involving multiple hamiltonians. Indeed, since both the rungekutta and the olms are equivariant under linear symmetry groups, being symplectic implies the preservation of quadratic invariants of hamiltonian systems by a result of feng and ge 6. Symplectic and contact structure, lagrangian submanifold. Symplectic topology of integrable hamiltonian systems, ii. Section 3 expresses the hamiltonian dynamics in its historical 2. One of the links is provided by a special class of symplectic invariants discovered by i. Symplectic invariants and hamiltonian dynamics mathematical. The first volume covered the basic materials of hamiltonian dynamics and symplectic geometry and the analytic foundations of gromovs pseudoholomorphic curve theory.
It is now understood to arise naturally in algebraic geometry, in lowdimensional topology, in representation theory and in string theory. I v ijr potential function i qi, pi positions and momenta of atoms i m i atomic mass of ith atom in molecular dynamics. Hamiltonian dynamics and the canonical symplectic form. Denote by 2 v the power set of v, and by bzr the euclidean ball of radius r in c, i. This is an introduction to the contributions by the lecturers at the minisymposium on symplectic and contact geometry.
We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable hamiltonian systems with two degrees of freedom. To this end we first establish an explicit isomorphism between the floer homology and the morse homology of such a manifold, and then use this isomorphism. Symplectic invariants and hamiltonian dynamics symplectic invariants and hamiltonian dynamics modern birkh. In this paper we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of.
Symplectic and contact geometry and hamiltonian dynamics mikhail b. On an exact symplectic manifold, there exists a 1form. The origins of symplectic topology lie in classical dynamics, and the search for periodic orbits of hamiltonian systems. However, in general, we cannot assume that these coordinates x, y. Zehnder, symplectic invariants and hamiltonian dynamics birkhauser, 1995. Download fulltext pdf download fulltext pdf download fulltext pdf on symplectic dynamics article pdf available in differential geometry and its applications 202. These invariants consist in some signs which determine the topology of the critical lagrangian fibre, together with several taylor series which can be computed from the dynamics of the system. Phase spaces and equations of motion are abstract symplectic manifolds and hamiltonian vector fields respectively.
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