Consider the free energy difference between two values of the
reaction coordinate 
and 
, which can be written as
The integrand can be written as
Now, we introduce a coordinate transformation from Cartesian coordinates to a set of generalized coordinates that contains q:
where n=3N-1. In addition, a corresponding transformation is made on the conjugate momenta according to:
so that, in the measure, no overall Jacobian appears:
Thus, we have
Next, the derivative with respect to 
is changed to a derivative
with respect to q:
Then, an integration by parts is performed, which yields
where the final average is an ensemble average conditional upon
the restriction that the reaction coordinate q is equal to .
Generally,
Thus, the free energy difference becomes:
Note that the average in the above expression can also be performed with respect to Cartesian positions and momenta, in which case the derivative can be carried out via the chain rule. If
then
The quantity 
is the generalized force
on the
generalized coordinate q. Thus, let the conditional average
of this force be 
. Then,
and
Thus, the free energy difference can be seen to be equal to the
work performed against the averaged generalized force in bringing
the system from to , irrespective of the values of the
other degrees of freedom. Such an integration is called
thermodynamic integration.
Another useful expression for the free energy derivative can be obtained by integrating out the momenta before performing a transformation. We begin with
Now, noting that the 
-function condition is independent
of momenta, we can integrate out the Cartesian momenta to yield:
Next, the coordinate transformation to generalized coordinates is performed:
associated with which there is a Jacobian given by
Introducing the coordinate transformation, we obtain
or
