First, via a local approximation of the poisson tensor, we split the hamiltonian dynamics into an almost symplectic part and the trivial evolution of the casimir invariants. R3 with the standard symplectic form the system has the hamiltonian 1. 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. The modern theory of dynamical systems, as well as symplectic geometry, have their origin with poincare as one field with integrated ideas. The main result asserts the existence of noncontractible periodic orbits for compactly supported time dependent hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating hamiltonian is sufficiently large over the zero section. In this particular case, proposition 1 reduces to the wellknown fact from linear algebra that there exist linear and thus global. Symplectic and contact geometry and hamiltonian dynamics mikhail b. Gromovwitten invariants count holomorphic maps satisfying certain requirements from riemann surfaces into symplectic manifolds m. All fanomanifolds and calabiyau manifolds are special examples of semipositive symplectic manifolds. The floer memorial volume, progress in mathematics 3, birkh auser 1995 55.
Symplectic mobius integrators for lq optimal control. Surprising rigidity phenomena demonstrate that the nature of symplectic map pings is very different from that of volume preserving mappings which raised new questions, many of them still unanswered. Pdf symplectic integration of hamiltonian wave equations. 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. The normal form has action variables, j, which are formal invariants when the rotation vector, v, of the elliptic orbit is nonresonant. We present a very general and brief account of the prehistory of the. We nd a complete set of local invariants of singular symplectic forms with the structurally smooth martinet hypersurface on a 2ndimensional manifold. I ceremade, universit6de parisdauphine, place du m. Given the progress in these fields one can make a good argument why the time is ripe to bring them closer together around the core area of hamiltonian.
A fascinating feature of symplectic geometry is that it lies at the crossroad of many other mathematical disciplines. One of the fundamental questions in hamiltonian dynamics is to study how many closed 1periodic orbits of this hamiltonian. Sorrentino this course is an introduction to the theory of hamiltonian systems of di erential equations. An intrinsic hamiltonian formulation of the dynamics of lc.
An example of how things can go wrong is shown in figure 1. Consider a threemanifoldm equippedwithacontactformthereebvector. 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. In this section we mention a few examples of such interactions. As noted before, any skewsymmetric constant m x m matrix defines a poisson bracket on r. 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 with e. This allows a first glimpse of the fast developing new field of symplectic topology. 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. Symplectic and contact geometry and hamiltonian dynamics. Hamiltonian dynamics on convex symplectic manifolds. I small divisor problems in classical and celestial mechanics. One of the links is provided by a special class of symplectic invariants discovered by i.
Attempts to push this to cover general symplectic manifolds so far have failed. One of the proposals is extending symplectic hamiltonian dynamics to contact hamiltonian dynamics by adding an extra dimension in a natural way. Quantitative symplectic geometry the library at msri. They are symplectic invariants attached to hamiltonian systems which have a lot of dynamical applications. Dynamic moment invariants for nonlinear hamiltonian systems. Symplectic topology and hamiltonian dynamics with i. When the appropriate number of constraints is imposed there are only. 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 first order elliptic systems, and finally a survey on floer homology and symplectic homology. In the recent years, considerable attention has been paid to preserving structures and.
The proof is based on floer homology and on the notion of a relative symplectic capacity. Download symplectic invariants and hamiltonian dynamics. Dynamic moment invariants for linear hamiltonians have already been. The dynamics in the neighborhood of a linearly stable periodic orbit of a hamiltonian. Second, canonically symplectic reduced basis techniques are applied to the nontrivial component of the dynamics, preserving the local poisson tensor kernel exactly. Symplectic integration of nonlinear hamiltonian systems. All proposed methods preserve symmetry, positivity and quadratic invariants for the riccati equations, and. Symplectic invariants and hamiltonian dynamics springerlink. The topological invariants associated with the boundary states are interpreted as winding numbers for windings around noncontractible loops on a riemann sheet constructed using the algebraic structure of the transfer matrices, as well as with a maslov index on a symplectic group manifold, which is the space of transfer matrices. On the other hand, analysis of an old variational principle in classical mechanics has established global periodic phenomena in hamiltonian systems. This failure of wellknown methods in mimicking hamiltonian dynamics motivated the consideration of. Hamiltonian dynamics and the canonical symplectic form.
For the implicit euler rule, the correspondingintegrationloses area and givesrise to a family of smaller and smaller circles that spiral toward the origin. Symplectic theory of completely integrable hamiltonian systems. Symplectic maps to projective spaces and symplectic invariants. Local symplectic invariants for curves niky kamran1, peter olver2 and keti tenenblat3 1 department of mathematics and statistics, mcgill university, montreal, qc, h3a 2k6, canada. Holm, and cesare tronci for henry mckean, on the occasion of his 75th birthday abstract. Symplectic aspects of aubrymather theory internet archive. The vlasov equation for the collisionless evolution of the singleparticle probability distribution function pdf is a wellknown example of coadjoint motion. There is another piece of data associated to namely the contact structure. Symplectic invariants and hamiltonian dynamics modern. Relations between global invariants of convex contact manifolds and. There is a mysterious relation between rigidity phenomena of symplectic geometry and global periodic solutions of hamiltonian dynamics. A pseudoholomorphic curve can be thought as a map u. Symplectic invariants and hamiltonian dynamics mathematical.
Usually one wants the domain to have a given form and the image to pass through. Publication list helmut hofer ias school of mathematics. Symplectic theory of completely integrable hamiltonian systems in memory of professor j. In order to do so, we introduce a barrier in phase space, and propose definitions of aubry and mane sets for nonconvex hamiltonian systems. The download symplectic invariants and hamiltonian dynamics between the run trimester order and the structures object of the advantage started 88 industry for important details and especially 43 course for aware workers. Hamiltonian dynamics on convex symplectic manifolds urs frauenfelder1 and felix schlenk2 abstract. We call functions of moments that remain invariant only under the action of a particular hamiltonian system or the equivalent symplectic map. Again, such a stable focus is incompatible with hamiltonian dynamics. The nonlocal symplectic vortex equations and gauged. Maschke et al intrinsic hamiltonlug formulation of the dynamics of lccircuits remark. Structurepreserving reduced basis methods for hamiltonian.
Symplectic invariants and hamiltonian dynamics helmut hofer. This is an introduction to the contributions by the lecturers at the minisymposium on symplectic and contact geometry. Symplectic maps to projective spaces and symplectic invariants denisauroux abstract. We will also sketch the authors more recent proof of equivalence of symplectic and algebraic gwinvariants for projective manifolds. Propagation in hamiltonian dynamics and relative symplectic homology.
The goal of these lectures was to present a family of invariants called \action selectors or \spectral invariants. This is not only a matter of was to free classical mechanics from the constraints of specific coordinate systems and to. From these symplectic invariants one derives surprising symplectic rigidity phenomena. Hamiltonian dynamics on convex symplectic manifolds article in israel journal of mathematics 1591. Would it for instance provide any advantage to studying hamiltonian dynamic. The key role of the symplectic invariants in such a determination is pointed out. Symplectic invariants and hamiltonian dynamics helmut. Properties of pseudoholomorphic curves in symplectisations i. What can symplectic geometry tell us about hamiltonian.
Weinstein therefore sought a condition for y to carry a closed orbit that is invariant under symplectic transformations. Symplectic integration of hamiltonian wave equations. New approaches are needed in order to get a hold on the gwinvariants of general varieties or symplectic manifolds. In 1985, gromovs work 12 on pseudoholomorphic curves revolutionized the eld of symplectic geometry. The purpose of the present paper is to supplement behrends contribution to this volume by the symplectic point of view. 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. There is an intimate relation between the periodic orbits of hamiltonian systems and a class of symplectic invariants called symplectic capacities. The origins of symplectic topology lie in classical dynamics, and the search for periodic orbits of hamiltonian systems. Introduction my research focuses on the study of symplectic geometry and its relationships with mirror symmetry and hamiltonian dynamics. Since then these fields developed quite independently. Propagation in hamiltonian dynamics and relative symplectic. Regular articles twist singularities for symplectic maps. What can symplectic geometry tell us about hamiltonian dynamics.