Equilibrium Solutions of the Logarithmic Hamiltonian Leapfrog for the N-body Problem

Abstract

We prove that a second-order logarithmic Hamiltonian leapfrog for the classical general N-body problem (CGNBP) designed by Mikkola and Tanikawa and some higher-order logarithmic Hamiltonian methods based on symmetric multicompositions of the logarithmic algorithm exactly reproduce the orbits of elliptic relative equilibrium solutions in the original CGNBP. These methods are explicit symplectic methods. Before this proof, only some implicit discrete-time CGNBPs proposed by Minesaki had been analytically shown to trace the orbits of elliptic relative equilibrium solutions. The proof is therefore the first existence proof for explicit symplectic methods. Such logarithmic Hamiltonian methods with a variable time step can also precisely retain periodic orbits in the classical general three-body problem, which generic numerical methods with a constant time step cannot do.

Quasi-conservative Integration Method for Restricted Three-body Problem

“…However, SI8 does not yield elliptic triangular Lagrange point orbits of P 1 , P 2 , and P 3 , whereas ( ) H Log 2 and d-R3BP precisely maintain all of them. The numerical results for ( ) H Log 2 and d-R3BP in Figure 2 validate Theorem 3 in Minesaki (2018) for ( ) H Log 2 and indicate that the analytical results in Section 3.3.2 ensure the reproducibility of stable triangular Lagrange point orbits.…”

“…An energy-preserving method based on a d'Alembert-type scheme (Minesaki 2013a(Minesaki , 2013b(Minesaki , 2013c, ( ) H Log 2 (Mikkola & Tanikawa 1999a, 1999b, its higher-order versions (Minesaki 2018), and a TCI approach for the general three-body problem (Minesaki 2019), which are proven to retain equilibrium solutions of general or restricted N-body problems, can trace the equilibrium orbits with high accuracy. This indicates that d-R3BP (3)-( 4) precisely reproduces the triangular Lagrange point orbits of R3BP (1)-( 2), as we analytically clarified in Section 3.3.1 based on the existence of collinear Lagrange points in R3BP and in Section 3.3.2 based on the existence of triangular Lagrange points.…”

“…(a) methods that precisely hold some but not all of the values of the conserved quantities expressed by elementary functions (Mikkola & Tanikawa 1999a, 1999bMinesaki 2013aMinesaki , 2013bMinesaki , 2018 and (b) methods that accurately preserve all conserved quantities described by elementary functions (Kotovych & Bowman 2002;Minesaki 2019).…”