3 Hamiltonian vector fields

Symplectic geometry was first studied by physicists, who modeled their systems

by a symplectic manifold. The Hamiltonian function

H ∈ C

∞

(

M

), which returns

the energy of a particular configuration, generates a vector field on

M

which is

the equation of motion of the system. In this chapter, we will discuss how this

process works, and further study what happens when we have families of such

Hamiltonian functions, giving rise to Lie group actions.