PT -symmetric classical mechanics

This paper reports the results of an ongoing in-depth analysis of the classical trajectories of the class of non-Hermitian PT -symmetric Hamiltonians H = p2 + x2 (ix) ε (ε ⩾ 0). A variety of phenomena, heretofore overlooked, have been discovered such as the existence of infinitely many separatrix trajectories, sequences of critical initial values associated with limiting classical orbits, regions of broken PT -symmetric classical trajectories, and a remarkable topological transition at ε = 2. This investigation is a work in progress and it is not complete; many features of complex trajectories are still under study.

Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits

We consider a family of explicit Runge–Kutta pairs of orders six and five without any additional property (reduced truncation errors, Hamiltonian preservation, symplecticness, etc.). This family offers five parameters that someone chooses freely. Then, we train them in order for the presented method to furnish the best results on a couple of Kepler orbits, a certain interval and tolerance. Consequently, we observe an efficient performance on a wide range of orbital problems (i.e., Kepler for a variety of eccentricities, perturbed Kepler with various disturbances, Arenstorf and Pleiades). About 1.8 digits of accuracy is gained on average over conventional pairs, which is truly remarkable for methods coming from the same family and order.