Conserved quantities share a very special relationship with the symmetries of physical systems. The beautiful Noether’s Theorem from the physicist Emmy Noether says that every symmetry in our equations of motion will produce a conserved quantity. By symmetries we refer to when a system does not change, i.e. is invariant, when it undergoes some transformation. This could be a spatial translation (moving our system somewhere else), a rotational transformation, time translation (doing our experiment at a later date for instance), etc etc. When a system is invariant to that transformation, Noether’s Theorem tells us there will be an associated conserved quantity.
Some of the most important symmetries and their associated conserved quantities are:
The Lagrangian formalism of mechanics allows us to demonstrate why this theorem is valid by looking at transformations to our “right” path that we discussed in detail on the section on Lagrangians ( The Lagrangian ). Essentially, the right path is the one that follows Newton’s Second Law and this path minimizes the action, $\mathcal A$. Lets suppose we have determined this “right” path given by $\vec x(t)$ that follows Newton’s Laws and describes how our path will move between two times $t_1$ and $t_2$. Let’s now apply a transformation to the path with a function $\vec\eta(\vec x,t)$ which in general may be a function of our coordinates and possibly time:
$$ \vec x(t)\rightarrow \vec x(t)+\vec\eta(\vec x,t) $$
Some examples could be:
Spatial Translation: $\vec x(t)+\vec\epsilon$ where we add $\epsilon$ to each component of $\vec x$
Rotational Transforamation: $\theta+\theta_0$ where $\theta_0$ is some constant angle
In the 2D case we may rearrange our path vector $\vec x$ to be in polar coordinates:
$\vec x=x_r\hat r+x_\theta \hat\theta$ and thus our transformation function is $\vec\eta=\theta_0\hat\theta$
In general our transformations involve some change to our original path that we apply to the system
A separate special case we will treat later are Time Translations: $t+t_0$ where $t_0$ is some constant time we add.
Note: its important that we note that while these transformations look similar to the deviations in the path we used in proving that the principle of stationary action, our transformations are different, namely that in general they do not need to follow the boundary conditions that our path deviations had to follow previously. We think of these transformations more as a coordinate transformation of our path that causes a system to undergo some action, like a translation or a rotation.
In our analysis we will look at incremental changes in the Lagranigan. To do this we want our transformations to be incremental as well. So in general we will have
$$ \eta_i(\vec x,t)=f_i(\vec x,t)\delta $$
Where $\delta$ is an infinitesimal increment with a coefficient given by the function $f$. We can use the shorthand $\vec\eta(\vec x,t)=\vec f(\vec x,t)\delta$ to represent the incremental changes for each coordinate $x_1,\dots,x_n$.
This allows us to treat $\eta$ as a differential change and so
$$ \delta x_i = f_i(\vec x,t)\delta \\ \delta\dot x=\dot f_i(\vec x,t)\delta $$