Generalized momenta
Consider the motion of a single particle moving in one dimension. The
kinetic energy is
 |
(7.33) |
where
is the mass of the particle and
its displacement.
The particle's linear momentum is
. However,
this can also be written
 |
(7.34) |
because
and the potential energy
is independent of
.
Consider a dynamical system described by
generalized coordinates
, for
. By analogy with the preceding expression, we can
define generalized momenta of the form
 |
(7.35) |
for
. Here,
is sometimes called the momentum conjugate to the coordinate
. Hence, Lagrange's equation (7.22) can be written
 |
(7.36) |
for
. Note that a generalized momentum does not necessarily have
the dimensions of linear momentum.
Suppose that the Lagrangian
does not depend explicitly on some coordinate
. It follows from Equation (7.36) that
 |
(7.37) |
Hence,
 |
(7.38) |
The coordinate
is said to be ignorable in this case.
Thus, we conclude that the generalized momentum associated with
an ignorable coordinate is a constant of the motion.
For example, the Lagrangian [Equation (7.24)] for a
particle moving in a central potential is independent of the angular
coordinate
. Thus,
is an ignorable coordinate,
and
 |
(7.39) |
is a constant of the motion. Of course,
is the angular momentum
about the origin; this is conserved because a central force exerts no torque
about the origin.