Using states of a double pendulum as a one-way function
It's an interesting idea, but I don't think it offers any advantages over current one-way function candidates and may be worse. For one thing, specifying the initial data of the simulation specifies a lot of properties of the dynamical system that will remain invariant (conservation of energy, horizontal component of momentum, et cetera). There are also constraints on the phase space of the coupled pendulums that never change over time either. I'd want the scheme to not have lots of internal constraints and symmetries for an adversary to exploit.

With a dynamical system like this, assuming the adversary can just look at the pendulums, she could just "look at the internals" of your scheme at any time, set up her own Lagrangian or Hamiltonian. By just observing the motion, the state of your one-way function is "leaked".



Also, since floating-point numbers are involved in the numerical solution of the system of equations, if you and I use slightly different numerical integrators, even for the same initial state, we may get different final states / "public keys" due to rounding or Runge-Kutta methods. In fact, you will have to specify the initial position and momentum extremely well, since any small changes in this data will cause wildly different subsequent states: chaos in a nutshell.