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Generalized Momenta
Consider the motion of a single particle moving in one dimension. The
kinetic energy is

(649) 
where is the mass of the particle, and its displacement.
Now, the particle's linear momentum is . However,
this can also be written

(650) 
since , and the potential energy is independent of .
Consider a dynamical system described by generalized coordinates
, for . By analogy with the above expression, we can
define generalized momenta of the form

(651) 
for . Here, is sometimes called the momentum conjugate to the coordinate . Hence, Lagrange's equation (613) can be written

(652) 
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 (652) that

(653) 
Hence,

(654) 
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, in Section 9.5, the Lagrangian (615) for a
particle moving in a central potential is independent of the angular
coordinate . Thus, is an ignorable coordinate,
and

(655) 
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.
Again, in Section 9.7, the Lagrangian (642) for a mass
sliding down a sliding slope is independent
of the Cartesian coordinate . It follows that is an ignorable coordinate,
and

(656) 
is a constant of the motion. Of course, is the total linear momentum in the direction. This is conserved because there is no external force acting on
the system in the direction.
Next: Spherical Pendulum
Up: Lagrangian Dynamics
Previous: Sliding down a Sliding
Richard Fitzpatrick
20110331