Chapter 2 SYMPLECTIC GEOMETRY AND ANALYSIS 2.1 Flows 2.2 Symplectic structure on R2n 2.3 Symplectic mappings 2.4 Hamiltonian vector fields 2.5 Lagrangian submanifolds 2.6 Notes We provide in this chapter a quick discussion of the symplectic geometric structure on R2n = Rn × Rn and its interplay with Hamiltonian dynamics. These will be important for our later goal of understanding interrelationships between dynamics and PDE. The reader may wish to first review our basic notation and also the theory of differential forms, set forth, respectively, in Appendices A and B. 2.1. FLOWS Let V : RN RN denote a smooth vector field. Fix a point z RN and solve the ordinary differential equations (ODE) (2.1.1) ˙ = V (w) (t R) w(0) = z, 13
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