This appendix shows the derivation of Lagrange equations from
Newton's second law. Let us consider a system formed by
rigid bodies whose center of mass position vectors are:
,
, …,
. Thus, to
determine the configuration of the whole system at a given
instant, it is necessary to know the values of
coordinates, which could be a combination of distances and
angles.

The system is called holonomic, if some of those
are related,
making it possible to reduce the number of independent coordinates to
generalized
coordinates (
):

Each position vector
can depend on some of the generalized
coordinates and on time:

and the velocity vector of the
th body is then

Therefore,
depends also on time, the generalized
coordinates, and the generalized velocities
:

and the partial derivatives of
are then

(B.1)

The acceleration vector of the
th rigid body is:

(B.2)

If at a given instant the value of each coordinate
is changed to
, where the virtual displacements
are
consistent with the constraints, each position vector will undergo a
displacement:

(B.3)

Taking the dot product of each side of this equation times the sides
of equation B.2, we obtain

(B.4)

since the derivative of the product
is,

According to equations B.1, the derivative
and the term inside the parenthesis
on the right-hand side of the equation are the partial derivatives of
, with respect to
and
. Therefore, the
following result is obtained

Furthermore, notice that the partial derivatives of
with
respect to the generalized coordinates and velocities are:

substituting these two expressions in
equation B.5 and multiplying both sides of
the equation by the mass
of the
th body we get,

where
is the kinetic energy of the
th
body. According to Newton's second law,
is the total
force acting on the
th body. Using the
expression B.3 and summing over all the
th body, we obtain

which leads to the Lagrange equations:

(B.6)

where
is the total kinetic energy of the system and the
generalized force
has been defined as