Towards investigation of evolution of dynamical systems with independence of time accuracy: more classes of systems

V G Gurzadyan,A A Kocharyan

doi:10.1140/epjc/s10052-015-3560-0

Abstract

The recently developed method (Paper 1) enabling one to investigate the evolution of dynamical systems with an accuracy not dependent on time is developed further. The classes of dynamical systems which can be studied by that method are much extended, now including systems that are; (1) non-Hamiltonian, conservative; (2) Hamiltonian with time-dependent perturbation; (3) non-conservative (with dissipation). These systems cover various types of N-body gravitating systems of astrophysical and cosmological interest, such as the orbital evolution of planets, minor planets, artificial satellites due to tidal, non-tidal perturbations and thermal thrust, evolving close binary stellar systems, and the dynamics of accretion disks.

Highlights

The typical procedure used in simulations of N-body gravitating systems [1] is the numerical iterative integration of the equations of motion, but this has a well-known difficulty: the inevitable accumulation of errors as the number of iterations increases and beginning from a distinct time further computations become meaningless
In our analysis we essentially use a result by Chernoff [10] concerning the properties of operators on a complete Riemannian manifold
According to Chernoff (Theorem 2.2 in [10]), if M is a complete Riemannian manifold and dr/c(r ) = +∞, where c(r ) = sup{c(x); x ∈ S(y, r )}, S(y, r ) is a ball of radius r with center y, the operator L defined on the space L2(M) with domain F0(M) is essentially self-adjoint

Summary

Introduction

The typical procedure used in simulations of N-body gravitating systems [1] is the numerical iterative integration of the equations of motion, but this has a well-known difficulty: the inevitable accumulation of errors as the number of iterations increases and beginning from a distinct time further computations become meaningless. The systems of N gravitating bodies involving various astrophysical problems possess chaotic properties Our initial aim is to investigate a rather general type of first order differential equations: xa = f a(x). The solution of this equation by means of the computation of the resolvent using computer algebraic analytical codes gives the evolution of the function u(t) in time. The functions xa(t), being determined by the function u(t) and describing the initial system, can be found (see [8])

The resolvent

Conclusions

Systems with perturbed Hamiltonian