Vaisman LOCALLY LAGRANGIAN SYMPLECTIC AND POISSON MANIFOLDS

نویسنده

  • I. Vaisman
چکیده

We discuss symplectic manifolds where, locally, the structure is that encountered in Lagrangian dynamics. Examples and characteristic properties are given. Then, we refer to the computation of the Maslov classes of a Lagrangian submanifold. Finally, we indicate the generalization of this type of symplectic structures to Poisson manifolds. The paper is the text of a lecture presented at the Conference “Poisson 2000” held at CIRM, Luminy, France, between June 26 and June 30, 2000. It reviews results contained in the author’s papers [9, 12, 13] as well as in papers by other authors. 1. Locally Lagrangian Symplectic Manifolds The present paper is the text of a lecture presented at the Conference “Poisson 2000” held at CIRM, Luminy, France, between June 26 and June 30, 2000, and it reviews results contained in the author’s papers [9, 12, 13] as well as in papers by other authors. The notion of a locally Lagrangian Poisson manifold is defined for the first time here. The symplectic structures used in Lagrangian dynamics are defined on a tangent manifold TN , and consist of symplectic forms of the type (1) ωL = 1 2 ( ∂L ∂qi∂uj − ∂ L ∂qj∂ui ) dq ∧ dq + ∂ L ∂ui∂uj du ∧ dq . In (1), (q)i=1 (n = dimN) are local coordinates in the configuration space N , (u ) are the corresponding natural coordinates in the fibers of TN , and L is the non degenerate Lagrangian function on TN (i.e., L ∈ C∞(TN), rank(∂L/∂u∂u) = n). (In this paper, everything is C∞.) The most known geometric description of ωL is that it is the pullback of the canonical symplectic form of T ∗N by the Legendre transformation defined by L. But, geometrically, it is more significant that ωL is related with the tangent structure of the manifold TN . The latter is the bundle morphism S : TTN → TTN defined by (2) SX ∈ TV, (SX)(< απ(u), u >) =< απ(u), π∗Xu >, where V is the foliation of TN by fibers (the vertical foliation), u ∈ TN , X ∈ ΓTTN , α ∈ ΓT ∗N , and π : TN → N is the natural projection. (Γ denotes spaces of global cross sections.) Formulas (2) define the action of SX on q, u, and one has

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

J an 2 00 3 Kähler - Nijenhuis Manifolds by Izu Vaisman

A Kähler-Nijenhuis manifold is a Kähler manifold M , with metric g, complex structure J and Kähler form Ω, endowed with a Nijenhuis tensor field A that is compatible with the Poisson structure defined by Ω in the sense of the theory of Poisson-Nijenhuis structures. If this happens, and if AJ = ±JA, M is foliated by im A into non degenerate Kähler-Nijenhuis submanifolds. If A is a non degenerate...

متن کامل

Symplectic Microgeometry Iii: Monoids

We show that the category of Poisson manifolds and Poisson maps, the category of symplectic microgroupoids and lagrangian submicrogroupoids (as morphisms), and the category of monoids and monoid morphisms in the microsymplectic category are equivalent symmetric monoidal categories.

متن کامل

Geometric Structures on Spaces of Weighted Submanifolds

In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on “convenient” vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold (M,ω), we construct a weak symplectic structure on each leaf Iw of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree fo...

متن کامل

Poisson Manifolds of Compact Types (pmct 1)

This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like the d...

متن کامل

Gauge fields and Sternberg-Weinstein approximation of Poisson manifolds

The motion of a classical particle in a gravitational and a Yang-Mills field was described by S. Sternberg and A. Weinstein by a particular Hamiltonian system on a Poisson manifold known under the name of Sternberg-Weinstein phase space. This system leads to the generalization of the Lorentz equation of motion first discovered by Wong. The aim of this work is to show that inversely, a Hamiltoni...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2003