The general theory of generalized coordinates follows the same
pattern as the simple examples. We wish to consider a transformation
from Cartesian coordinates to a new, more natural set of coordinates
for the physical probem. Suppose that we have an N-particle system
in d spatial dimensions. We will also assume that the system
is subject to 
constraints so that the total number of degrees
of freedom is 
. Hence, we need 
coordinates
to describe the system. Let these coordinates be denoted
. These will be related to the original
Cartesian coordinates by a set of transformation equations
of the form
An important property of a transformation is that there be a well defined inverse transformation
What is the Lagrangian in terms of the qs? In order to derive the Lagrangian, we begin with the Lagrangian in Cartesian coordinates:
In order to transform the kinetic energy, we need the derivative of the transformation equations, which is given by the chain rule:
Thus,
Thus, the kinetic energy becomes
where we have introduced a matrix
which is a function of the new coordinates, .
The matrix G is an 
matrix known as the
mass-metric matrix (or mass-metric tensor).
In terms of G, the Lagrangian becomes
where the dependence of the potential on the generalized coordinates
occurs through the dependence on the Cartesian coordinates.
For each generalized coordinate, 
, there will be
an equation of motion given, generally, by the Euler-Lagrange equation:
The general form of the equations of motion may be rather complicated by the dependence of G on the coordinates. Nevertheless, the above equation provides a very general framework for computing the equations of motion of a system in any set of coordinates.
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Mark Tuckerman