Given a system, we look for a decomposition of the Hamiltonian in the form

where *H*_{0} is solvable. We assume that the Hamiltonian has no
explicit time dependence; this can be ensured by going to the extended
phase space if necessary. We also assume that a canonical transformation
has been made so that *H*_{0} depends solely on the momenta:

We carry out a Lie transformation and find the conditions that the Lie
generator *W* must satisfy to eliminate the order- terms
from the Hamiltonian.

The Lie transform and associated Lie series specify a canonical transformation:

where *Q* = *I*_{1} and *P* = *I*_{2} are the coordinate and momentum selectors and *I* is
the identity function.
Recall the definitions

with *L*_{W} *F* = { *F*, *W* }.

Applying the Lie transformation to *H* gives us

The first-order term in is zero if *W* satisfies the condition

which is a linear partial differential equation for *W*. The
transformed Hamiltonian is

where we have used condition (6.7) to simplify the
^{2} contribution.

This basic step of perturbation theory has eliminated terms of a
certain order (order ) from the Hamiltonian, but in doing so
has generated new terms of higher order (here ^{2} and higher).

At this point we can find an approximate solution by truncating
Hamiltonian (6.8) to *H*_{0}, which is
solvable.
The approximate solution for given initial conditions (*t*_{0}, *q*_{0},
*p*_{0}) is obtained by finding the corresponding (*t*_{0}, *q*'_{0}, *p*'_{0}) using the
inverse of transformation (6.4).
Then the system is
evolved to time
*t* using the solutions of the truncated Hamiltonian *H*_{0}, giving the
state
(*t*, *q*', *p*'). The phase-space coordinates
of the evolved point are transformed back to the
original variables using the
transformation (6.4) to state (*t*, *q*,
*p*). The approximate solution is

If the Lie transform *E*'_{, W} = *e*^{ LW} must be
evaluated by summing the series, then we must specify the order to
which the sum extends.

Assuming everything goes okay, we can imagine repeating this process to
eliminate the order-^{2} terms and so on, bringing the
transformed Hamiltonian as close as we like to *H*_{0}. Unfortunately,
there are complications. We can understand some of these
complications and how to deal with them by considering some
specific applications.