In this appendix, the problem of treating an N-particle system in
a set of generalized coordinate that explicitly contains the reaction
coordinate is considered. From a dynamical point of view, if one
begins with the Lagrangian in Cartesian coordinates, 
then, under a transformation to generalized coordinates 
the Lagrangian becomes
where 
denotes the full set of generalized coordinates,
and 
is given by
is the mass-metric tensor. By performing the Legendre transform, the Hamiltonian can be shown to be
where
is the inverse of 
. The introduction of coordinate-dependent
mass-metric factors into the Lagrangian/Hamiltonian of a system via a transformation
to generalized coordinates leads to a considerable increase in complexity in the
description of the true dynamics of the system.
Although the AFED method requires that generalized coordinates be used, the dynamics only needs to generate the correct configurational averages, and, hence, it is not necessary to work with correct conjugate momenta and the true adiabatic dynamics. This fact leads to a large simplification in the analysis and implementation of the method. In order to show this, consider the canonical partition function of the system. In Cartesian coordinates, this is given by
If the coordinate integration only is transformed according to Eq. (29), leaving the momentum integrals unchanged, the partition function picks up a Jacobian factor according to
Defining an effective potential according to
the partition function becomes
It is, thus, clear that the phase space distribution function in Eq. (37) can be generated via molecular dynamics with a Hamiltonian of the form
where the 3N Cartesian momenta treated as ``conjugate'' to the qs even though
they are not the true conjugate momenta. Similarly, if 
is a reaction
coordinate of interest, then the AFED method can be implemented by introducing
a temperature 
for this variable and writing the Hamiltonian in the
form
where 
, 
, in order to ensure an adiabatic separation.
Under these conditions, the analysis of Sec. 2 can be applied to a general
N-particle system in generalized coordinates with no modification except
for a simple replacement of the N-particle potential V with the
modified potential 
. It should be noted that, under certain conditions,
it may be straightforward to work in terms of a full canonical set of
generalized coordinates, such as in the example of Sec. 3.2.
