Numerical comparisons

The scripts under experiments/ compare coordinate and integration choices on selected cart-pendulum cases. They are reproducible examples, not benchmarks of every physical system.

• E1 compares angle RK4, ordinary Cartesian \((c,s)\) RK4, and the exponential-map update. See experiments/e1_integration.py.
• E2-A compares angle and complex-coordinate swing-up programs from matched seeds. E2-B uses a fixed total solve budget. See experiments/e2_fair_benchmark.py.
• E3 tries a free-winding angle formulation without a prescribed final turn count. See experiments/e3_free_winding_phase.py.

Saved results under experiments/results/ record the selected inputs and outcomes. NLP solvers can settle in different local solutions. A successful nominal path still needs closed-loop validation with the chosen actuator, limits, and delay.