Symmetry & the Least Action Principle
A route from Fermat’s principle to Euler–Lagrange equations, and from invariance to Noether’s theorem.
This note begins with an apparently narrow question: why does light choose the path that takes the least time? Following that question far enough leads beyond optics and into a different grammar for all of mechanics.
The useful shift is to stop asking what force acts at every instant and instead ask which complete history makes a quantity stationary. The result is not merely a new method of calculation. It changes what counts as an explanation.
The full illustrated note will live here. This first edition establishes the archive, its notation, and the thread connecting symmetry to conservation.
This is a representative first edition. Replace it with the complete work whenever it is ready; the address and design will stay the same.