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).
Highlights
One of the main continuous models is a dynamic system described by an autonomous system of ordinary differential equations, that is, a system of equations of the form: Received: 16 October 2021 dxi
Since we have at our disposal a large and well-studied family of schemes that preserve linear and quadratic integrals, i.e., the symplectic Runge–Kutta schemes, the simplest way to construct such schemes is to introduce additional variables with respect to which all algebraic integrals of the many-body problem are expressed in terms of linear and quadratic integrals
The major result of our work is Theorem 2: by introducing additional variables, the many-body problem can be reduced to a dynamical system with a rational right-hand side, all algebraic integrals of which are expressed in terms of quadratic and linear ones
Summary
One of the main continuous models is a dynamic system described by an autonomous system of ordinary differential equations, that is, a system of equations of the form: Received: 16 October 2021 dxi. The simplest way to construct such schemes is to introduce additional variables with respect to which all algebraic integrals of the many-body problem are expressed in terms of linear and quadratic integrals This assumption can be considered as a development of the invariant energy quadratization method. This result is formulated as two Theorems 1 and 2, published and proven here for the first time.
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