Symplectic vector space

About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.

Symplectic methods for the nonlinear Schrödinger equation

Abstract: In this paper, we show that the spatial discretization of the nonlinear Schrodinger equation leads to a Hamiltonian system, which can be simulated with symplectic numerical schemes. In particular, we apply two symplectic integrators to the nonlinear Schrodinger equation, and we demonstrate that they are able to produce accurate results and to preserve very well the invariants of the original system, such as the energy and charge.

Notes on monotone Lagrangian twist tori

Abstract: We construct monotone Lagrangian tori in the standard symplectic vector space, in the complex projective space and in products of spheres. We explain how to classify these Lagrangian tori up to symplectomorphism and Hamiltonian isotopy, and how to show that they are not displaceable by Hamiltonian isotopies.

An introduction to Lie groups and symplectic geometry

Abstract: A series of nine lectures on Lie groups and symplectic This is an unofficial version of the notes and was last modified on 19 February 2003. (Mainly to correct some very bad mistakes in Lecture 8 about Kähler and hyperKähler reduction that were pointed out to me by Eugene Lerman.) Please send any comments, corrections or bug reports to the above e-mail address. Introduction These are the lecture notes for a short course entitled " Introduction to Lie groups and symplectic geometry " which I gave at the 1991 Regional Geometry Institute at Park City, Utah starting on 24 June and ending on 11 July. The course 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. My purpose was not to provide an exhaustive treatment of either Lie groups, which would have been impossible even if I had had an entire year, or of symplectic manifolds, which has lately undergone something of a revolution. Instead, I tried to provide an introduction to what I regard as the basic concepts of the two subjects, with an emphasis on examples which drove the development of the theory. I deliberately tried to include a few topics which are not part of the mainstream subject, such as Lie's reduction of order for differential equations and its relation with the notion of a solvable group on the one hand and integration of ODE by quadrature on the other. I also tried, in the later lectures to introduce the reader to some of the global methods which are now becoming so important in symplectic geometry. However, a full treatment of these topics in the space of nine lectures beginning at the elementary level was beyond my abilities. After the lectures were over, I contemplated reworking these notes into a comprehensive introduction to modern symplectic geometry and, after some soul-searching, finally decided against this. Thus, I have contented myself with making only minor modifications and corrections, with the hope that an interested person could read these notes in a few weeks and get some sense of what the subject was about. An essential feature of the course was the exercise sets. Each set begins with elementary material and works up to more involved and delicate problems. My object …