2.4: Other Conservation Laws

Looking at the Lagrange equation (19), we immediately see that if \(L \equiv T-U\) is independent of some generalized coordinate \(q_{j}, \partial L / \partial q_{j}=0,{ }^{15}\) then the corresponding generalized momentum is an integral of motion: 16 \[p_{j} \equiv \frac{\partial L}{\partial \dot{q}_{j}}=\text { const. }\] For example, for a 1D particle with the Lagrangian (21), the momentum \(p_{x}\) is conserved if the potential energy is constant (and hence the \(x\)-component of force is zero) - of course. As a less obvious example, let us consider a 2D motion of a particle in the field of central forces. If we use polar coordinates \(r\) and \(\varphi\) in the role of generalized coordinates, the Lagrangian function, \({ }^{17}\) \[L \equiv T-U=\frac{m}{2}\left(\dot{r}^{2}+r^{2} \dot{\varphi}^{2}\right)-U(r),\] is independent of \(\varphi\) and hence the corresponding generalized momentum, \[p_{\varphi} \equiv \frac{\partial L}{\partial \dot{\varphi}}=m r^{2} \dot{\varphi},\] is conserved. This is just a particular (2D) case of the angular momentum conservation - see Eq. (1.24). Indeed, for the 2 \(\mathrm{D}\) motion within the \([x, y]\) plane, the angular momentum vector, \[\mathbf{L} \equiv \mathbf{r} \times \mathbf{p}=\left|\begin{array}{ccc} \mathbf{n}_{x} & \mathbf{n}_{y} & \mathbf{n}_{z} \\ x & y & z \\ m \dot{x} & m \dot{y} & m \dot{z} \end{array}\right|,\] has only one component different from zero, namely the component normal to the motion plane: \[L_{z}=x(m \dot{y})-y(m \dot{x}) \text {. }\] Differentiating the well-known relations between the polar and Cartesian coordinates, \[x=r \cos \varphi, \quad y=r \sin \varphi,\] over time, and plugging the result into Eq. (52), we see that \[L_{z}=m r^{2} \dot{\varphi} \equiv p_{\varphi} .\] Thus the Lagrangian formalism provides a powerful way of searching for non-evident integrals of motion. On the other hand, if such a conserved quantity is evident or known a priori, it is helpful for the selection of the most appropriate generalized coordinates, giving the simplest Lagrange equations. For example, in the last problem, if we knew in advance that \(p_{\varphi}\) had to be conserved, this could provide a motivation for using the angle \(\varphi\) as one of generalized coordinates.

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\({ }^{16}\) This fact may be considered a particular case of a more general mathematical statement called the Noether theorem - named after its author, Emmy Nöther, sometimes called the "greatest woman mathematician ever lived". Unfortunately, because of time/space restrictions, for its discussion I have to refer the interested reader elsewhere - for example to Sec. \(13.7\) in H. Goldstein et al., Classical Mechanics, \(3^{\text {rd }}\) ed. Addison Wesley, 2002

\({ }^{17}\) Note that here \(\dot{r}^{2}\) is the square of the scalar derivative \(\dot{r}\), rather than the square of the vector \(\dot{\mathbf{r}}=\mathbf{v}\).