I'm curious to what extent people have studied "curvature flows" on PL closed curves in the plane.

There's a paper by Gage and Hamilton from 1986 that describes the long-term behaviour of smooth curves in the plane under the curvature flow.

- MR0840401 (87m:53003) Gage, M.(1-RCT); Hamilton, R. S.(1-UCSD) The heat equation shrinking convex plane curves. J. Differential Geom. 23 (1986), no. 1, 69–96. 53A04 (35K05 52A40 58E99 58G11)

A discrete version of this flow could go like this. The ambient space for this will be the space of PL 1-dimensional compact connected submanifolds of $\mathbb R^2$ and for the sake of argument, let's fix the length of the intervals. So the curve consists of $n$ straight line segments, and the $i$-th interval has length $l_i$, and the set $\{l_1,l_2,\cdots,l_n\} \subset (0,\infty)$ is the data that describes this space of closed curves. The "curvature flow" would be the dynamical system given by placing a spring at each vertex of your curve (the spring acts on the angle), and you make the spring's "natural angle" to be $\pi$. Is this flow complete like the Gage-Hamilton flow?

More generally, has there been much study of finitary analogues to the Gage-Hamilton flow, in the spirit of my initial question? I imagine there has, I'm not sure which terms to search for on MathSciNet.