Lectures on symplectic geometry department of mathematics. The journal of geometry and physics is an international journal in mathematical physics. This book is a true introduction to symplectic geometry, assuming only a general background in analysis and familiarity with linear algebra. Explicit examples are discussed in section 2 below. An introduction to lie groups and symplectic geometry. It is, by the way, no coincidence that this looks like the.
Symplectic geometry is the mathematical apparatus of such areas of physics. The fight to fix symplectic geometry quanta magazine. The reason is that this one semester course was aiming for students at the beginning of their masters. Since their inception, the study of symplectic structures and the applications of symplectic techniques as well as their odddimensional contact geometric counterparts have benefited from a strong extraneous motivation. Symplectic geometry and geometric tization quan 1 matthias blau 2 nikhefh p. To specify the trajectory of the object, one requires both the position q and the momentum p, which form a point p, q in the euclidean plane. In particular symplectic vector spaces have even dimension. The canonical 1form and the symplectic 2form on the cotangent bundle.
Any complex manifold has a canonical almost complex structure. Department of mathematics, princeton university, princeton, nj 08544, usa. The roots of symplectic geometry are as old as the other two branches of geometry mentioned above, since it originated as the natural mathematical framework for classical mechanics. Starting from a classical mechanical system with symmetry group g, the corresponding. Quantitative symplectic geometry 3 symplectic geometry is the geometry underlying hamiltonian systems. Symplectic structures a new approach to geometry dusa mcduff introduction symplectic geometry is the geometry of a closed skewsymmetric form. It turns out to be very different from the riemannian geometry with which we are familiar. Symplectic geometry of rationally connected threefolds tian, zhiyu, duke mathematical journal, 2012. Least action principle and the hamiltonian mechanics 3 1. The topology of a subspace of the legendrian curves in a closed contact 3manifold. Symplectic geometry 2nd semester 201220 symplectic geometry studies the geometry of manifolds equipped with a nondegenerate, closed 2form.
Symplectic manifolds are necessarily evendimensional and orientable, since nondegeneracy says that the top exterior power of a symplectic form is a volume form. Lectures on symplectic geometry pdf 225p download book. Symplectic geometry curvature is a local invariant in riemannian geometry darbouxweinstein theorem let p be any point on a symplectic manifold of dimension 2n. Classical mechanics is a peculiar branch of physics with a long history.
An introduction to symplectic topology through sheaf theory. We start by describing symplectic manifolds and their transformations, and by explaining connections to topology and other geometries. Jun 17, 2009 pdf this is a survey on symplectic birational geometry. The authors cover two main methods for proving the \h\ principle. Since their inception, the study of symplectic structures and the applications of symplectic techniques as well as their odddimensional contact geometric counterparts have benefited from a. The book first offers information on the symplectic and discontinuous groups, symplectic metric, and hermitian forms. Symplectic linear algebra 1 symplectic forms and presymplectic forms 2 normal form theorem 3 weak and strong in. Box 41882, 1009 db dam mster a the netherlands 1 this is a preliminary ersion v not tended in for distribution outside this ho scol.
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds. Symplectic geometry let mbe a smooth manifold without boundary. Symplectic manifold u i has local coordinate q 1,q n,p 1,p n coordinate change is symplectic diffeomorphism. Many stretches of this text are rearrangements from the book lec tures on symplectic. A special emphasis in the book is made on applications to symplectic and contact geometry. Symplectic geometry is the mathematical apparatus of such areas of physics as classical mechanics, geometrical optics and thermodynamics. Quantitative symplectic geometry the library at msri.
Part of the encyclopaedia of mathematical sciences book series ems, volume 4. Symplectic geometry nicholas proudfoot department of mathematics, university of oregon, eugene, or 97403 these notes are written for a ten week graduate class on symplectic geometry. Symplectic geometry focuses on the processes, methodologies, and numerical approaches involved in symplectic geometry. The idea in question sometimes is called geometric quantization.
Symplectic geometry has its origins in the hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. Pdf this is a survey on symplectic birational geometry. Further, every complex submanifold of a k ahler manifold is a k ahler manifold. Second, position and momentum are \intertwined by a skewsymmetric nondegenerate bilinear form. Symplectic geometry is the geometry of symplectic manifolds. Introduction to symplectic and hamiltonian geometry impa. This book would be an excellent text for a graduate course or as a source for anyone who wishes to learn about symplectic geometry. Anintroductiontoliegroups andsymplecticgeometry a series of nine lectures on lie groups and symplectic geometry delivered at the regional geometry institute in park city, utah, 24 june20 july 1991. The list of questions on symplectic forms begins with those of existence and uniqueness on a given manifold. Introduction to symplectic and hamiltonian geometry by ana cannas da silva.
R is a skewsymmetric bilinear map and w p varies smoothly in p. Contact and homogeneous symplectic geometry 32 chapter 5. Numerical calculations are presented to show the values and transformations of these groups. The course note really was designed to be an introduction, aimed at an audience of students who were familiar with basic constructions in differential topology and rudimentary differential geometry, who wanted to get a feel for lie groups and symplectic geometry. Hamiltonian dynamics and symplectic geometry 1 chapter 1. This fact establishes a link from symplectic geometry to complex geometry and it is a point of departure for the modern techniques in symplectic geometry. The symplectic camel principle and semiclassical mechanics. Berndts presentation of symplectic geometry is a clear and concise introduction to the major methods and applications of the subject, and requires only a minimum of prerequisites. This book provides a self contained exposition of basic floer homology in both. Ergodic theory studies the properties of volume pre serving mappings. The following theorem by kobayashi clarifies the relations between the canonical.
Buy symplectic geometry and analytical mechanics mathematics and its applications no 35 on free shipping on qualified orders symplectic geometry and analytical mechanics mathematics and its applications no 35. Lectures on symplectic geometry pdf 225p this note contains on the following subtopics of symplectic geometry, symplectic manifolds, symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, kahler manifolds, hamiltonian mechanics, moment maps, symplectic reduction, moment maps revisited and symplectic toric manifolds. An introduction to symplectic geometry 3 we now turn to manifolds. The product of symplectic manifolds is still symplectic.
Introduction to symplectic and hamiltonian geometry. Our main sources have been the books symplectic techniques by guilleminsternberg and introduction to symplectic topology by mcduffsalamon. Since these forms are closed, they are locally exact, and hence up to some coverings of xand g, we may assume these form are exact. Notes on symplectic geometry 5 conversely, suppose that x is a symplectic manifold with a transitive. Symplectic geometry and analytical mechanics mathematics and. The reader will find that, with a few notable exceptions, most instances of the \h\ principle can be treated by the methods considered here.
It turns out that this geometric approach to hamiltonian systems is very fruitful. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. In the final lecture he describes the h principle and others ideas of gromov in symplectic geometry, like pseudoholomorphic curves. Whenever the equations of a theory can be gotten out of a variational principle, symplectic geometry clears up and systematizes the relations between the quantities. One important difference is that, although all its concepts are initially expressed in. There are several books on symplectic geometry, but i still took the trouble of writing up lecture notes. These \textit foliated open books offer a finer tool for. From a language for classical mechanics in the xviii century, symplectic geometry has matured since the 1960s to a rich and central branch of differential geometry and topology. We the construct semiclassical waveforms on lagrangian submanifolds using. Proofs and more information can be found in the books 50, 72. Jun, 2003 symplectic geometry originated as a mathematical outgrowth of hamiltonian mechanics and dynamical systems and their applications to the theory of elementary particles, oceanographic and atmospheric sciences, condensed matter, accelerator and plasma physics and other disciplines at the classical and quantum levels. Symplectic geometry lecture notes, university of toronto.
The proof is similar to the proof of the existence of an orthonormal basis in a euclidean vector space. This was partly because symplectic geometry began to run out of problems that could be solved independently of the foundational issues, but also because, in 2012, a pair of researchers dusa mcduff, a prominent symplectic geometer at barnard college and author of a pair of canonical textbooks in the field, and katrin wehrheim, a. Since s6 is compact and h2 dr s 6 0, it cannot be symplectic. In arbitrary dimension, this subject is centered around the notion of uniruledness. An introduction to symplectic geometry megumi harada, lecture notes by kaname hashimoto summer school, osaka, 2007 abstract the intent of this series of lectures is twofold. Whenever the equations of a theory can be gotten out of a variational principle, symplectic geometry clears up and systematizes the relations between the quantities entering into the theory. Mosers lemma and local triviality of symplectic differential geometry 17 2. For a time, people wondered whether every symplectic manifold was k ahler. This book provides a selfcontained exposition of basic floer homology in both. Important developments in the early 1980s led to the emergence of new questions about these structures. Gromovs famous book partial differential relations, which is devoted to the same subject, is an encyclopedia of the \h\ principle, written for experts, while the present book is the first broadly accessible exposition of the theory and its. Sg this paper introduces a new type of open book decomposition for a contact threemanifold with a specified characteristic foliation on its boundary. Symplectic geometry eckhard meinrenken lecture notes, university of toronto these are lecture notes for two courses, taught at the university of toronto in spring 1998 and in fall 2000.
For a more liegroup focused account, you can try robert bryants lectures on lie groups and symplectic geometry which are available online here. Our main sources have been the books symplectic techniques by guilleminsternberg and introduction to symplectic topology by mcdu. Mathematically maslovs theory gave rise to hormanders theory of. Look up the gauss lemma in a book on riemannian geometry. The course note really was designed to be an introduction, aimed at an audience of students who were familiar with basic constructions in differential topology and rudimentary differential geometry, who wanted to. Hence we recall the notion of complex manifolds and we show that.
Symplectic geometry is the geometry of manifolds equipped with a symplectic form, that is. Indeed, symplectic methods are key ingredients in the study of dynamical systems, differential equations, algebraic geometry, topology, mathematical physics and representations of lie groups. An introduction to symplectic geometry alessandro assef institute for theoretical physics university of cologne these notes are a short sum up about two talks that i gave in august and september 2015 an the university of cologne in my workgroup seminar by prof. Least action principle and the hamiltonian mechanics. Symplectic geometry arose from the study of classical mechanics and an example of a symplectic structure is the motion of an object in one dimension. Symplectic geometry and analytical mechanics mathematics. This text covers foundations of symplectic geometry in a modern language. Hamiltonian mechanics and symplectic geometry well now turn from the study of speci. Symplectic geometry 81 introduction this is an overview of symplectic geometrylthe geometry of symplectic manifolds.
Symplectic structures have their origin in the study of classical mechanics. Most of the material here is included in mich ele audins book torus actions on symplectic manifolds, which i used heavily in preparing these notes. Robert gompf ut, 19934 showed that there are many examples of nonk ahler symplectic manifolds. Symplectic structures in geometry, algebra and dynamics.