>>11264schizophrenia also known as "Geometric Measure theory". Unless you want to do explicit estimates in, say, General Relativity, you probably won't see that formalism ever again.
What he's basically saying is that there's one degree of freedom, letβs pick the arc-length angle ΞΈ (=angle that sweeps out the arc length), so β(t) = ββ β aΞΈ and the Lagrangian is Β½ mβΒ²ΞΈΜΒ² β mgβ sin ΞΈ.
Then he derives aΞΈ = s, i.e. ΞΈ = s/a but that is just the central angle due to the convention of denoting arclength per radius