Ensemble distribution on the double-pendulum energy surface

Determine the actual ensemble distribution on the constant-energy surface of the Hamiltonian double-pendulum system with parameters (m1, m2, l1, l2) at fixed total energy E, specifying the sampling density over the phase-space coordinates (θ1, ω1, θ2, ω2) for initial condition generation.

Background

In the paper’s double-pendulum experiments, initial conditions must be sampled on a constant-energy surface to study ergodicity within Hamiltonian dynamics. The authors provide an algorithm for sampling angles and angular velocities at a fixed energy.

However, they explicitly state that they do not know the actual ensemble distribution over the phase-space energy surface and therefore use uniform sampling as a practical choice. Identifying the correct ensemble distribution would inform principled, potentially non-uniform sampling of initial conditions.

References

Note, that since we do not know the actual ensemble distribution of the phase space, we do not have any reason to put more emphasis on some regions of the space than others, hence, we use a uniform distribution for sampling.

— Wasserstein-Based Test for Empirical Measure Convergence of Dependent Sequences  (2604.02700 - Yordanov et al., 3 Apr 2026) in Appendix A (Double Pendulum Simulation Initial Conditions Sampling Algorithm)