Definition(Hamiltonian vector field). A vector field onM=R^2 given by
∂f
∂p
∂
∂q
−
∂f
∂q
∂
∂p
=−{f,·}
for some functionfonM=R^2 is called a Hamiltonian vector field and will be
denoted byXf. In higher dimensions, Hamiltonian vector fields will be those of
the form
Xf=
∑d
j=1
(
∂f
∂pj
∂
∂qj
−
∂f
∂qj
∂
∂pj
)
=−{f,·} (15.2)
for some functionfonM=R^2 d.
The simplest non-zero Hamiltonian vector fields are those forfa linear
function. Forcq,cpconstants, if
f=cqq+cpp
then
Xf=cp
∂
∂q
−cq
∂
∂p
and the map
f→Xf
is the isomorphism ofMandMof equation 14.6.
For example, takingf=p, we haveXp=∂q∂. The exponential map for this
vector field satisfies
q(exp(tXp)(m)) =q(m) +t, p(exp(tXp)(m)) =p(m) (15.3)
Similarly, forf=qone hasXq=−∂p∂ and
q(exp(tXq)(m)) =q(m), p(exp(tXq)(m)) =p(m)−t (15.4)
Quadratic functionsfgive vector fieldsXfwith components linear in the
coordinates. An important example is the case of the quadratic function
h=
1
2
(q^2 +p^2 )
which is the Hamiltonian function for a harmonic oscillator, a system that will
be treated in much more detail beginning in chapter 22. The Hamiltonian vector
field for this function is
Xh=p
∂
∂q
−q
∂
∂p
The trajectories satisfy
dq
dt
=p,
dp
dt
=−q