Generalized momenta

(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

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

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.