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 $\dot{\mathbf{r}}=\mathbf{v}$.