On the quadratization of the integrals for the many-body problem

A new approach to the construction of difference schemes of any order for the many-body problem that preserves all its algebraic integrals is proposed herein. We introduced additional variables, namely distances and reciprocal distances between bodies, and wrote down a system of differential equations with respect to the coordinates, velocities, and the additional variables. In this case, the system lost its Hamiltonian form, but all the classical integrals of motion of the many-body problem under consideration, as well as new integrals describing the relationship between the coordinates of the bodies and the additional variables are described by linear or quadratic polynomials in these new variables. Therefore, any symplectic Runge–Kutta scheme preserves these integrals exactly. The evidence for the proposed approach is given. To illustrate the theory, the results of numerical experiments for the three-body problem on a plane are presented with the choice of initial data corresponding to the motion of the bodies along a figure of eight (choreographic test). © 2021 by the authors. Licensee MDPI, Basel, Switzerland.

Authors
Journal
Publisher
MDPI AG
Number of issue
24
Language
English
Status
Published
Number
3208
Volume
9
Year
2021
Organizations
  • 1 School of Science, KaiLi University, Kaili, 556011, China
  • 2 Department of Applied Probability and Informatics, RUDN University, Moscow, 117198, Russian Federation
  • 3 Joint Institute for Nuclear Research, Dubna, 141980, Russian Federation
Keywords
Algebraic integrals of motion; Dynamical system; Finite difference method; Midpoint scheme; Quadratization of energy; Symplectic Runge–Kutta scheme; Three-body problem
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