Generalized Momenta

(B.23) |

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

(B.24) |

because , and the potential energy, , is independent of .

Consider a dynamical system described by
generalized coordinates,
, for
. By analogy with the previous expression, we can
define *generalized momenta* of the form

(B.25) |

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 (B.26) that

(B.27) |

Hence,

(B.28) |

The coordinate is said to be