On Application of Local Integrals of Motion in Simulations of Anisotropic Disk Systems
Abstract
A new class of local linear integrals of motion is introduced for two-dimensional rotating systems with the anisotropic effective mass and angular potential. In polar coordinates, a generalized integral analogous to the angular momentum, the existence of which is determined by a specific relationship between the mass function $$\mu (r,\theta )$$ and the potential $$U(r,\theta )$$ , is constructed. Analytical sufficient conditions for the conservation of the local integral at $$J = 0$$ , which link the effective mass $$\mu (r,\theta )$$ , the structural function $$\varphi (r,\theta )$$ , and the potential energy $$U(r,\theta )$$ , are obtained. The fulfillment of these conditions ensures the invariance of the integral in a local region of phase space and establishes criteria for the self-consistency of the dynamical model. We present three analytically solvable examples, demonstrating a wide range of orbital dynamics—from closed orbits to resonant and chaotic regimes. We consider systems with radial–angular modulation of mass, which model the density structures in protoplanetary and accretion disks. The proposed approach is distinguished by its analytical novelty and can serve as a tool for studying stability, identifying resonant structures, and constructing self-consistent models in stellar dynamics and astrophysics. The work is partially based on a report presented at the Modern Stellar Astronomy 2025 conference.