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)
˙ w = V (w) (t R)
w(0) = z,
13
http://dx.doi.org/10.1090/gsm/138/02
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