• Corpus ID: 244478298

Newton's method for computing periodic orbits of the planar three-body problem

@article{Hristov2021NewtonsMF,
  title={Newton's method for computing periodic orbits of the planar three-body problem},
  author={Iavor Varbanov Hristov and Radoslava Hristova and Igor V. Puzynin and Taisia P. Puzynina and Zarif Sharipov and Zafar Tukhliev},
  journal={ArXiv},
  year={2021},
  volume={abs/2111.10839}
}
In this paper we present in detail Newton’s method and its modification, based on the Continuous analog of Newton’s method for computing periodic orbits of the planar three-body problem. The linear system at each step of the method is formed by solving a system of ODEs with the multiple precision Taylor series method. We consider zero angular momentum symmetric initial configuration with parallel velocities, bodies with equal masses and relatively short periods. Taking candidates for the… 

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References

SHOWING 1-10 OF 12 REFERENCES
The N-body problem, the braid group, and action-minimizing periodic solutions
A reduced periodic orbit is one which is periodic modulo a rigid motion. If such an orbit for the planar N-body problem is collision free then it represents a conjugacy class in the projective
More than six hundred new families of Newtonian periodic planar collisionless three-body orbits
The famous three-body problem can be traced back to Isaac Newton in the 1680s. In the 300 years since this “three-body problem” was first recognized, only three families of periodic solutions had
Three classes of newtonian three-body planar periodic orbits.
TLDR
A topological method is used to classify periodic three-body orbits into families, which fall into four classes, with all three previously known families belonging to one class.
A guide to hunting periodic three-body orbits
The recent discovery of thirteen new and distinct three-body periodic planar orbits suggests that many more such orbits remain undiscovered. Searches in two-dimensional subspaces of the full
Computing periodic orbits with arbitrary precision.
TLDR
The method is shown to be quadratically convergent and some numerical tests for the paradigmatic Lorenz model and the Hénon-Heiles Hamiltonian are presented, giving periodic orbits up to 1000 digits.
A Software Package for the Numerical Integration of ODEs by Means of High-Order Taylor Methods
This paper revisits the Taylor method for the numerical integration of initial value problems of Ordinary Differential Equations (ODEs). The main goal is to present a computer program that outputs a
The generalized continuous analog of Newton’s method for the numerical study of some nonlinear quantum-field models
A numerical method for studying nonlinear problems arising in mathematical models of physics is systematically described in this review. The unified basis for the development of numerical schemes is
On the clean numerical simulation (CNS) of chaotic dynamic systems
On the reliability of computed chaotic solutions of non-linear differential equations
A new concept, namely the critical predictable time Tc, is introduced to give a more precise description of computed chaotic solutions of non-linear differential equations: it is suggested that
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