In the previous efforts, we constructed N-Body systems based on non-temporal and nonlinear extension of Lorentz transformation. In this construction, we rely only on two parameters, nonlinear degree and normalized momentum to characterize the systems. We then explored root computation via iteration in the context of dynamical systems. The solution sets demonstrate various forms similar to canonical distributions. In a related application, we also explored this construction for modeling gravitational lens limitations (image number and brightness), and observed counter examples of Fundamental Theorem of Algebra. These efforts were recently further advanced for simulating the pre-chaotic distributions, and we observed that hierarchical structures are formed, which we attribute to the nonlinear coupling of momentum and angular momentum.
In this paper, we propose a new duality, momentum-angular-momentum duality from the efforts for two purposes. One is for unifying internal and external symmetry, and the other is for providing an alternative candidate for the phenomena currently claimed to be induced by dark matter and dark energy. This duality together with electromagnetic duality forms the dual-duality theory, which is expected to clarify some unsolved problems in GUT.