I came up with the idea for a game involving messing with circuits a couple weeks back, so I searched for rope physics. I found this post and a couple others talking about Verlet integration. The basic gist is that Verlet integration stores only the current position and the previous position and calculates velocity from that, as opposed to Euler integration which uses a stored position and velocity. This has a number of benefits including more accuracy, stability, and simpler constraint code. It’s apparently used in ragdoll physics.
In any case, the wires are just an array of points using Verlet integration, constrained to have a certain distance, and colliding against the terrain (I just use 1x1 AABBs to represent each point). If you want to know more details the code is up on github (the relevant details are in src/main.zig), though the code is fairly messy: https://github.com/desttinghim/wired