Generalized Momenta

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

$$K = \frac{1}{2}\,m\,\dot{x}^{\,2},$$ (649)

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

$$p = \frac{\partial K}{\partial \dot{x}}= \frac{\partial L}{\partial\dot{x}},$$ (650)

since $L=K-U$, and the potential energy $U$ is independent of $\dot{x}$.

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

$$p_i = \frac{\partial L}{\partial\dot{q}_i},$$ (651)

for $i=1,{\cal F}$. Here, $p_i$ is sometimes called the momentum conjugate to the coordinate $q_i$. Hence, Lagrange's equation (613) can be written

$$\frac{d p_i}{dt} = \frac{\partial L}{\partial q_i},$$ (652)

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

Suppose that the Lagrangian $L$ does not depend explicitly on some coordinate $q_k$. It follows from Equation (652) that

$$\frac{d p_k}{dt} = \frac{\partial L}{\partial q_k}=0.$$ (653)

Hence,

$$p_k = {\rm const.}$$ (654)

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

$$p_\theta = \frac{\partial L}{\partial\dot{\theta}} = m\,r^2\,\dot{\theta}$$ (655)

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.

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

$$p_x = \frac{\partial L}{\partial \dot{x}} = M\,\dot{x} + m\,(\dot{x}+\dot{x}'\,\cos\theta)$$ (656)

is a constant of the motion. Of course, $p_x$ is the total linear momentum in the $x$-direction. This is conserved because there is no external force acting on the system in the $x$-direction.