4.4 The generalized momentum

We saw in example 4-8, that in the case of a simple Lagrangian, the quantities \(p_i\equiv\frac{\partial L}{\partial \dot{q}_i}\) are related to the linear momentum of a particle. For a more general system, the \(p_i\), are called the generalized momenta of the system. We also saw that in the case where the Lagrangian does not explicitly depend on a coordinate, the corresponding generalized momentum is a constant of motion. We call the generalized momentum \(p_i\) the “conjugate momentum” of coordinate \(q_i\). The generalized momenta are not necessarily related to the linear momentum of the system.

Inserting this into the Lagrange equations:

\begin{align} p_i&\equiv\frac{\partial L}{\partial \dot{q}_i}\tag{4.38}\\ \frac{dp_i}{dt}-\frac{\partial L}{\partial q_i}&=0 \tag{4.39}\end{align}

we see that we recover Newton’s Second Law in the case where the \(p_i\) are linear momenta and the forces monogenic (thus given by the second term).

Example 4-9

Determine the generalized momenta of a free particle described in polar coordinates

The Lagrangian is given by:

\begin{align*} L=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2) \end{align*}

The generalize momenta are given by:

\begin{align*} p_r&=\frac{\partial L}{\partial \dot{r}}=m\dot{r}\\ p_\theta&=\frac{\partial L}{\partial \dot{\theta}}=mr^2\dot{\theta}\\ p_z&=\frac{\partial L}{\partial \dot{z}}=m\dot{z}\\ \end{align*}

\(p_\theta\) is easily identified with the angular momentum, which is conserved, since the Lagrangian does not depend on \(\theta\) explicitly. \(p_r\) cannot be identified with any of the usual quantities and is not conserved, since \(L\) depends explicitly on \(r\). \(p_z\) is the linear momentum in \(z\) and is also conserved, since \(L\) does not depend explicitly on \(z\). It is interesting to note that the (arbitrary) choice of coordinates highlighted which conserved quantities are relevant. Had we chosen cartesian coordinates, as we did in the previous example, we would have seen that the linear momenta are conserved.