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University of Crete Department of Mathematics
 

Summary: University of Crete
Department of Mathematics
Notes on Symplectic Geometry
Konstantin Athanassopoulos
Iraklion, 2007
ii
Contents
1 Introduction 3
1.1 Newtonian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 The equations of Hamilton . . . . . . . . . . . . . . . . . . . . . . . 12
2 Symplectic spaces 15
2.1 Symplectic vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Local description of symplectic manifolds . . . . . . . . . . . . . . . 24
2.4 Hamiltonian vector fields and Poisson bracket . . . . . . . . . . . . . 29
2.5 Coadjoint orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Homogeneous symplectic manifolds . . . . . . . . . . . . . . . . . . . 40
2.7 Poisson manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Symmetries and integrability 49

  

Source: Athanassopoulos, Konstantin - Department of Mathematics, University of Crete

 

Collections: Mathematics