Hamiltonian dynamics on the symplectic extended phase. Symplectic maps to projective spaces and symplectic invariants denisauroux abstract. Prominent among them are the socalled symplectic capacities. This theorem shows that the symplectic invariants called symplectic capac. Symplectic invariants and hamiltonian dynamics core. I ceremade, universit6de parisdauphine, place du m. For some handson experience of the standard map, download meiss simulation code 4. I v ijr potential function i qi, pi positions and momenta of atoms i m i atomic mass of ith atom in molecular dynamics. One of the links is provided by a special class of symplectic invariants discovered by i. In hamiltonian dynamical system, any time evolution is defined by hamiltonian equations and expressed by canonical transformations or symplectic diffeomorphisms on phase spaces.
One of the links is a class of symplectic invariants, called symplectic capacities. 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. Zehnder, symplectic invariants and hamiltonian dynamics birkhauser, 1995. We know that elliptic and hyperbolic orbits have no symplectic. We show how these series are related to the singular. 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. Bayesian inference from symplectic geometric viewpoint. On an exact symplectic manifold, there exists a 1form. This is not only a matter of was to free classical mechanics from the constraints of specific coordinate systems and to. Periodic orbits for symplectic twist maps of t n x ir n. Happily, it is very well written and sports a lot of very useful commentary by the authors. Section 3 expresses the hamiltonian dynamics in its historical 2.
What can symplectic geometry tell us about hamiltonian. Symplectic topology and floer homology is a comprehensive resource suitable for experts and newcomers alike. 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. C0limits of hamiltonian paths and the ohschwarz spectral. 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. Spectral invariants in rabinowitzfloer homology and global hamiltonian perturbations. On the other hand, analysis of an old variational principle in classical mechanics has established global periodic phenomena in hamiltonian systems.
Symplectic invariants and hamiltonian dynamics helmut hofer. While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Dec 18, 2007 we construct symplectic invariants for hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolichyperbolic type. This was soon generalized to flows generated by a hamiltonian over a poisson manifold. We present a very general and brief account of the prehistory of the.
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. The nonlocal symplectic vortex equations and gauged. 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. 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. 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. 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.
Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. Download symplectic invariants and hamiltonian dynamics. Hofer born february 28, 1956 is a germanamerican mathematician, one of the founders of the area of symplectic topology he is a member of the national academy of sciences, and the recipient of the 1999 ostrowski prize and the 20 heinz hopf prize. In mathematics, nambu mechanics is a generalization of hamiltonian mechanics involving multiple hamiltonians. Symplectic and contact geometry and hamiltonian dynamics. They are defined through an elementarylooking variational problem involving poisson brackets. We consider an explicitly timedependent hamiltonian h that is defined on a finitedimensional contact manifold with its closed, generally degenerate contact 2form. Recall that hamiltonian mechanics is based upon the flows generated by a smooth hamiltonian over a symplectic manifold. 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.
Introduction to symplectic and hamiltonian geometry. As it turns out, these seemingly differ ent phenomena are mysteriously related. 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. Hamiltonian dynamics on convex symplectic manifolds urs frauenfelder1 and felix schlenk2 abstract. 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. Salamon, propagation in hamiltonian dynamics and relative symplectic homology, duke math. Jun 08, 2007 hamiltonian dynamics on convex symplectic manifolds frauenfelder, urs. Symplectic maps to projective spaces and symplectic invariants. Hamiltonian dynamics and the canonical symplectic form. Differential invariants for symplectic lie algebras realized.
These symplectic invariants include spectral invariants, boundary depth, and partial symplectic quasistates. Hamiltonian system 1 isnt necessary to be symplectic, and not all symplectic integrator can preserve the quadratic invariants of hamiltonian system 1 6, 16. Would it for instance provide any advantage to studying hamiltonian dynamic. Symplectic invariants and hamiltonian dynamics pdf free.
Floer homology in symplectic geometry and in mirror symmetry. Symplectic invariants near hyperbolichyperbolic points. Surprising rigidity phenomena demonstrate that the nature of symplectic mappings is very different from that of volume preserving mappings. Symplectic structure perturbations and continuity of. In this paper we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of. The first volume covered the basic materials of hamiltonian dynamics and symplectic geometry and the analytic foundations of gromovs pseudoholomorphic curve theory. Symplectic topology of integrable hamiltonian systems, ii. Symplectic invariants and hamiltonian dynamics reprint of the 1994 edition helmut. Symplectic invariants and hamiltonian dynamics symplectic invariants and hamiltonian dynamics modern birkh.
Symplectic invariants and hamiltonian dynamics helmut hofer, eduard zehnder auth. Introduction to symplectic and hamiltonian geometry by ana cannas da silva. 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. We present applications to approximation theory on symplectic manifolds and to hamiltonian dynamics. Symplectic invariants and hamiltonian dynamics helmut. We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable hamiltonian systems with two degrees of freedom. Symplectic twist maps advanced series in nonlinear dynamics. However, in general, we cannot assume that these coordinates x, y. Symplectic topology and floer homology by yonggeun oh.
The flows are symplectomorphisms and hence obey liouvilles theorem. B2r zeclz symplectic vectorspaces v, 09, and w, cow the product is defined by. The nonlocal symplectic vortex equations and gauged gromov. Differential invariants for symplectic lie algebras. This raises new questions, many of them still unanswered. Symplectic and contact geometry and hamiltonian dynamics mikhail b.
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 and contact structure, lagrangian submanifold. There is a mysterious relation between rigidity phenomena of symplectic geometry and global periodic solutions of hamiltonian dynamics. Gromovwitten invariants of symplectic quotients and adiabatic limits gaio, ana rita pires and salamon, dietmar a.
We start by describing symplectic manifolds and their transformations, and by explaining connections to topology and other geometries. Symplectic invariants and hamiltonian dynamics modern. 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. 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 symplectic differential invariants are obtained, both in a closed form when n. Download fulltext pdf download fulltext pdf download fulltext pdf on symplectic dynamics article pdf available in differential geometry and its applications 202.
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. 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. This text covers foundations of symplectic geometry in a modern language. The proof of the nontriviality of these invariants involves various flavors of floer theory. Symplectic invariants and hamiltonian dynamics mathematical. It is now understood to arise naturally in algebraic geometry, in lowdimensional topology, in representation theory and in string theory. Symplectic invariants and hamiltonian dynamics springerlink. What can symplectic geometry tell us about hamiltonian dynamics.
1320 413 258 1123 990 144 887 215 1611 1384 317 421 684 1692 1419 1461 303 1207 1540 290 1573 1623 459 400 710 1657 1282 1501 916 577 321 1224 1347 535 970 928 1373 55 164 153