"Cyclic" Variables This post may belong in the physics StackExchange site, but I'd to hear an answer from a mathematician rather than a physicist. I did find a few answers for this question there, but none made a whole lot of sense.  I'm not sure how to directly ask this question, but any response that would help me better understand this phenomenon would be greatly appreciated. 
In Lagrangian formalism, when $\partial \mathcal L / \partial q = 0$, the variable $q$ is called "cyclic" and implies there is a conserved quantity. Why is this called cyclic? 
 A: I've never understood why the physicists used the term "cyclic" there, either. To my mind, a better term would have been "symmetric". So, for example, if $\dfrac{\partial\mathcal{L}}{\partial x}=0,$ then $\mathcal{L}$ is symmetric in $x$. Physically, you can imagine taking the whole system described by $\mathcal{L}$ and shifting it in the $x$ direction, either positive or negative: doing so doesn't change any results, and it doesn't change the physics of the situation. When that happens, as you've mentioned, the corresponding conjugate momentum, $p_x,$ is conserved, as proven by the Euler-Lagrange equation (Noether's Theorem). This is the mantra of theoretical physicists today: symmetries lead to conservation laws. And they mean, at this deep level, what I've just described. It's the most important concept in all of physics. 
In summary: the term "cyclic" is just a term. The term "symmetric," in the same context, means the same thing, and is much more physically descriptive. So I would recommend using that term instead!
