Symplectic invariants and hamiltonian dynamics springer. One of the links is provided by a special class of symplectic invariants discovered by i. Hamiltonian dynamics and the canonical symplectic form. 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. Section 3 expresses the hamiltonian dynamics in its historical 2. Symplectic and contact structure, lagrangian submanifold. What can symplectic geometry tell us about hamiltonian. Introduction to symplectic and hamiltonian geometry. Differential invariants for symplectic lie algebras realized. We show how these series are related to the singular. In this paper we present an attempt to better understand the space of all symplectic capacities, and discuss some further general.
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. 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. 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. I ceremade, universit6de parisdauphine, place du m. Salamon, propagation in hamiltonian dynamics and relative symplectic homology, duke math. 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. Symplectic invariants and hamiltonian dynamics helmut hofer. The flows are symplectomorphisms and hence obey liouvilles theorem. We start by describing symplectic manifolds and their transformations, and by explaining connections to topology and other geometries. 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 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 structure perturbations and continuity of. Gromovwitten invariants of symplectic quotients and adiabatic limits gaio, ana rita pires and salamon, dietmar a.
What can symplectic geometry tell us about hamiltonian dynamics. Symplectic invariants and hamiltonian dynamics modern. Symplectic invariants and hamiltonian dynamics symplectic invariants and hamiltonian dynamics modern birkh. Spectral invariants in rabinowitzfloer homology and global hamiltonian perturbations. The first volume covered the basic materials of hamiltonian dynamics and symplectic geometry and the analytic foundations of gromovs pseudoholomorphic curve theory. Symplectic invariants and hamiltonian dynamics reprint of the 1994 edition helmut. Floer homology in symplectic geometry and in mirror symmetry.
Symplectic invariants and hamiltonian dynamics mathematical. 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. This is not only a matter of was to free classical mechanics from the constraints of specific coordinate systems and to. Hamiltonian dynamics on the symplectic extended phase. We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable hamiltonian systems with two degrees of freedom. There is a mysterious relation between rigidity phenomena of symplectic geometry and global periodic solutions of hamiltonian dynamics. These symplectic invariants include spectral invariants, boundary depth, and partial symplectic quasistates. It is now understood to arise naturally in algebraic geometry, in lowdimensional topology, in representation theory and in string theory. 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. Recall that hamiltonian mechanics is based upon the flows generated by a smooth hamiltonian over a symplectic manifold. Symplectic topology of integrable hamiltonian systems, ii. Hamiltonian dynamics on convex symplectic manifolds urs frauenfelder1 and felix schlenk2 abstract. Would it for instance provide any advantage to studying hamiltonian dynamic.
Phase spaces and equations of motion are abstract symplectic manifolds and hamiltonian vector fields respectively. Happily, it is very well written and sports a lot of very useful commentary by the authors. Symplectic topology and floer homology by yonggeun oh. One of the links is a class of symplectic invariants, called symplectic capacities. They are defined through an elementarylooking variational problem involving poisson brackets.
This raises new questions, many of them still unanswered. Symplectic invariants near hyperbolichyperbolic points. B2r zeclz symplectic vectorspaces v, 09, and w, cow the product is defined by. In hamiltonian dynamical system, any time evolution is defined by hamiltonian equations and expressed by canonical transformations or symplectic diffeomorphisms on phase spaces. 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. 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. Symplectic invariants for parabolic orbits and cusp. Bayesian inference from symplectic geometric viewpoint. On the other hand, analysis of an old variational principle in classical mechanics has established global periodic phenomena in hamiltonian systems. Symplectic maps to projective spaces and symplectic invariants.
The proof of the nontriviality of these invariants involves various flavors of floer theory. Download symplectic invariants and hamiltonian dynamics. 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. Periodic orbits for symplectic twist maps of t n x ir n. Symplectic and contact geometry and hamiltonian dynamics mikhail b. Symplectic and contact geometry and hamiltonian dynamics. Symplectic twist maps advanced series in nonlinear dynamics. Download fulltext pdf download fulltext pdf download fulltext pdf on symplectic dynamics article pdf available in differential geometry and its applications 202. This is an introduction to the contributions by the lecturers at the minisymposium on symplectic and contact geometry. Differential invariants for symplectic lie algebras. Prominent among them are the socalled symplectic capacities. I v ijr potential function i qi, pi positions and momenta of atoms i m i atomic mass of ith atom in molecular dynamics.
In this article, we study the behavior of the ohschwarz spectral invariants under c 0small perturbations of the hamiltonian flow. The nonlocal symplectic vortex equations and gauged. Hamiltonian system 1 isnt necessary to be symplectic, and not all symplectic integrator can preserve the quadratic invariants of hamiltonian system 1 6, 16. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. The symplectic differential invariants are obtained, both in a closed form when n. Denote by 2 v the power set of v, and by bzr the euclidean ball of radius r in c, i. We consider an explicitly timedependent hamiltonian h that is defined on a finitedimensional contact manifold with its closed, generally degenerate contact 2form. This theorem shows that the symplectic invariants called symplectic capac.
This text covers foundations of symplectic geometry in a modern language. As it turns out, these seemingly differ ent phenomena are mysteriously related. While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Symplectic invariants and hamiltonian dynamics helmut hofer, eduard zehnder auth. 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. We present a very general and brief account of the prehistory of the. We present applications to approximation theory on symplectic manifolds and to hamiltonian dynamics. Zehnder, symplectic invariants and hamiltonian dynamics birkhauser, 1995. 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. 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. On an exact symplectic manifold, there exists a 1form. Symplectic invariants and hamiltonian dynamics core. Symplectic invariants and hamiltonian dynamics helmut.
The origins of symplectic topology lie in classical dynamics, and the search for periodic orbits of hamiltonian systems. Symplectic maps to projective spaces and symplectic invariants denisauroux abstract. Hamiltonian dynamics on convex symplectic manifolds, israel. 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. Bisgaard, invariants of lagrangian cobordisms via spectral numbers, journal of topology and analysis, 11 2019, 205231. However, in general, we cannot assume that these coordinates x, y. 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. This was soon generalized to flows generated by a hamiltonian over a poisson manifold. In mathematics, nambu mechanics is a generalization of hamiltonian mechanics involving multiple hamiltonians. The nonlocal symplectic vortex equations and gauged gromov. 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. Symplectic invariants and hamiltonian dynamics pdf free. In this paper we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of.
Introduction to symplectic and hamiltonian geometry by ana cannas da silva. 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 topology and floer homology is a comprehensive resource suitable for experts and newcomers alike. For some handson experience of the standard map, download meiss simulation code 4. We know that elliptic and hyperbolic orbits have no symplectic. Surprising rigidity phenomena demonstrate that the nature of symplectic mappings is very different from that of volume preserving mappings. Dec 18, 2007 we construct symplectic invariants for hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolichyperbolic type. 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 springerlink.