On the Quadratization of the Integrals for the Many-Body Problem

Yu Ying, Ali Baddour, Vladimir P. Gerdt, Mikhail Malykh and Leonid Sevastianov
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Yu Ying: School of Science, KaiLi University, Kaili 556011, China
Ali Baddour: Department of Applied Probability and Informatics, RUDN University, Moscow 117198, Russia
Vladimir P. Gerdt: Joint Institute for Nuclear Research, Dubna 141980, Russia
Mikhail Malykh: Department of Applied Probability and Informatics, RUDN University, Moscow 117198, Russia
Leonid Sevastianov: Department of Applied Probability and Informatics, RUDN University, Moscow 117198, Russia

Mathematics, 2021, vol. 9, issue 24, 1-12

Abstract: 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).

Keywords: finite difference method; algebraic integrals of motion; dynamical system; symplectic Runge–Kutta scheme; midpoint scheme; three-body problem; quadratization of energy (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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