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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 2011-03-31