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Generalized momenta

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

$\displaystyle K = \frac{1}{2}\,m\,\skew{3}\dot{x}^{\,2},$ (7.33)

where $ m$ is the mass of the particle and $ x$ its displacement. The particle's linear momentum is $ p=m\,\skew{3}\dot{x}$ . However, this can also be written

$\displaystyle p = \frac{\partial K}{\partial \skew{3}\dot{x}}= \frac{\partial {\cal L}}{\partial\skew{3}\dot{x}},$ (7.34)

because $ {\cal L}=K-U$ and the potential energy $ U$ is independent of $ \skew{3}\dot{x}$ .

Consider a dynamical system described by $ {\cal F}$ generalized coordinates $ q_i$ , for $ i=1,{\cal F}$ . By analogy with the preceding expression, we can define generalized momenta of the form

$\displaystyle p_i = \frac{\partial {\cal L}}{\partial\skew{3}\dot{q}_i},$ (7.35)

for $ i=1,{\cal F}$ . Here, $ p_i$ is sometimes called the momentum conjugate to the coordinate $ q_i$ . Hence, Lagrange's equation (7.22) can be written

$\displaystyle \frac{d p_i}{dt} = \frac{\partial {\cal L}}{\partial q_i},$ (7.36)

for $ i=1,{\cal F}$ . Note that a generalized momentum does not necessarily have the dimensions of linear momentum.

Suppose that the Lagrangian $ {\cal L}$ does not depend explicitly on some coordinate $ q_k$ . It follows from Equation (7.36) that

$\displaystyle \frac{d p_k}{dt} = \frac{\partial {\cal L}}{\partial q_k}=0.$ (7.37)

Hence,

$\displaystyle p_k = {\rm const.}$ (7.38)

The coordinate $ q_k$ 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 $ \theta $ . Thus, $ \theta $ is an ignorable coordinate, and

$\displaystyle p_\theta = \frac{\partial {\cal L}}{\partial\skew{5}\dot{\theta}} = m\,r^{\,2}\,\skew{5}\dot{\theta}$ (7.39)

is a constant of the motion. Of course, $ p_\theta$ is the angular momentum about the origin; this is conserved because a central force exerts no torque about the origin.


next up previous
Next: Exercises Up: Lagrangian mechanics Previous: Lagrange's equation
Richard Fitzpatrick 2016-03-31