Perfect Fowl Whenever I read science journalists waxing lyrical about Noether’s Theorem, symmetries and conserved quantities, I feel a compulsion to make them reproduce (or at least read) a detailed account of how this pans out for the Laplace-Runge-Lenz vector: www.gregegan.net/SCIENCE/LRL/LRL.html#NOETHER In this case, the transformation of the Lagrangian that you need depends on the particle’s velocity as well as its coordinates, and it is not even a symmetry of the Lagrangian! Rather, it adds a total time derivative to the Lagrangian, and you need to do some further work to find the associated conserved quantity. The simplest cases are very nice, but if people take them too much to heart they can end up being hamstrung by an intuitive sense that every conserved quantity in a Lagrangian theory ought to be as simple as conservation of energy and angular momentum.