13.4.2 General Lagrangian Expressions
As more complicated mechanics problems are considered, it is
convenient to express the differential constraints in a general form.
For example, evaluating (13.130) for a kinematic chain of
bodies leads to very complicated expressions. The terms of these
expressions, however, can be organized into standard forms that appear
simpler and give some intuitive meanings to the components.
Suppose that the kinetic energy is expressed using (13.126),
and let denote an entry of . Suppose that the
potential energy is . By performing the derivatives expressed
in (13.136), the EulerLagrange equation can be expressed as
scalar equations of the form [856]

(13.140) 
in which

(13.141) 
There is one equation for each from to . The components of
(13.140) have physical interpretations. The
coefficients represent the inertia with respect to . The
represent the affect on of accelerating . The
terms represent the centrifugal effect induced on
by the velocity of . The
terms
represent the Coriolis effect induced on by the velocities of
and . The term usually arises from gravity.
An alternative to (13.140) is often given in terms of
matrices. It can be shown that the EulerLagrange equation
reduces to

(13.142) 
which represents scalar equations. This introduces
,
which is an
Coriolis matrix. It turns out that
many possible Coriolis matrices may produce equivalent different
constraints. With respect to (13.140), the Coriolis
matrix must be chosen so that

(13.143) 
Using (13.141),

(13.144) 
A standard way to determine
is by computing Christoffel symbols. By subtracting
from the inside of
the nested sums in (13.144), the equation can be rewritten
as

(13.145) 
This enables an element of
to be written as

(13.146) 
in which

(13.147) 
This is called a Christoffel symbol, and it is obtained from
(13.145). Note that
. Christoffel
symbols arise in the study of affine connections in differential
geometry and are usually denoted as
. Affine
connections provide a way to express acceleration without coordinates,
in the same way that the tangent space was expressed without
coordinates in Section 8.3.2. For affine connections in
differential geometry, see [133]; for their application to
mechanics, see [156].
Subsections
Steven M LaValle
20120420