Digression.For a general symplectic manifoldM, the symplectic two-formω
gives us an analog of Hamilton’s equations. This is the following equality of
one-forms, relating a Hamiltonian functionhand a vector fieldXhdetermining
time evolution of trajectories inM
iXhω=ω(Xh,·) =dh(hereiXis interior product with the vector fieldX). The Poisson bracket in
this context can be defined as
{f 1 ,f 2 }=ω(Xf 1 ,Xf 2 )Recall that a symplectic two-form is defined to be closed, satisfying the equa-
tiondω= 0, which is then a condition on a three-formdω. Standard differential
form computations allow one to expressdω(Xf 1 ,Xf 2 ,Xf 3 )in terms of Poisson
brackets of functionsf 1 ,f 2 ,f 3 , and one finds thatdω= 0is the Jacobi identity
for the Poisson bracket.
The theory of “prequantization” (see [52], [41]) enlarges the phase spaceM
to aU(1)bundle with connection, where the curvature of the connection is the
symplectic formω. Then the problem of lack of injectivity of the Lie algebra
homomorphism
f→−Xf
is resolved by instead using the map
f→−∇Xf+if (15.8)where∇Xis the covariant derivative with respect to the connection. For details
of this, see [52] or [41].
In our treatment of functions on phase spaceM, we have always been taking
such functions to be time-independent. Mcan be thought of as the space of
trajectories of a classical mechanical system, with coordinatesq,phaving the
interpretation of initial conditionsq(0),p(0) of the trajectories. The exponential
maps exp(tXh) give an action on the space of trajectories for the Hamiltonian
functionh, taking the trajectory with initial conditions given bym∈Mto the
time-translated one with initial conditions given by exp(tXh)(m). One should
really interpret the formula for Hamilton’s equations
df
dt={f,h}as meaning
d
dt
f(exp(tXh)(m))|t=0={f(m),h(m)}for eachm∈M.
Given a Hamiltonian vector fieldXf, the maps
exp(tXf) :M→M