5.1 Conserved generalized momenta

Given a Lagrangian, \(L\), the generalized momenta are defined as:

\begin{align} p_i\equiv\frac{\partial L}{\partial \dot{q}_i} \tag{5.1}\end{align}

The Lagrange equations of motion are thus:

\begin{align} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right)-\frac{\partial L}{\partial q_i}&=0\\ \therefore \dot{p}_i=\frac{\partial L}{\partial q_i} \tag{5.2}\end{align}

Thus, if the Lagrangian does not depend explicitly on \(q_i\), the generalized momentum \(p_i\) is a constant (i.e. it does not change with time). The interesting point to note is that the conserved generalized momenta depend on the choice of generalized coordinates.

Example 5-1

Compare the conserved quantities for a particle moving in a potential that depends on the distance from the origin for cartesian and spherical coordinates

The Lagrangian in cartesian coordinates is given by:

\begin{align*} L=\frac{1}{2}m(\dot{x}^2+\dot{y}^2+\dot{z}^2)-V\left(\sqrt{x^2+y^2+z^2}\right) \end{align*}

and does not appear to have any cyclic coordinates. We might be led to think that there is no conserved quantity. In spherical coordinates, the Lagrangian is given by:

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

where the variable \(\phi\) is cyclic. The corresponding conserved quantity is:

\begin{align*} p_\phi &=\frac{\partial L}{\partial \dot{\phi}}\\ &=mr^2\sin^2\theta\dot{\phi} \end{align*}

The question then arises of how to determine if there are any conserved quantities or an optimal choice of generalized coordinates that uncover those conserved quantities. In this chapter, we will focus on uncovering the conserved quantities. In a later chapter, we will discuss “Canonical Transformations”, which is the method of discovering a set of generalized coordinates where all generalized momenta are conserved (at the expense that the coordinate transformations are not straightforward).