A numerical method that conserves the Runge-Lenz vector †*

An exhaustive discussion is carried out on isolating integrals and the trapezoidal formula which can conserve the Runge-Lenz vector. An isolating integral is an invariant that restricts the region of particle motion. The autonomous integrable Hamiltonian system with n degrees of freedom has only n mutually involutive independent isolating integrals, and the existence of other isolating integrals is meaningful to the particle motion. In the Kepler two-body system there exist the energy integral, the angular momentum integral and the Runge-Lenz vector. These correspond to 3 independent isolating integrals in the case of plane motion, and to 5 in the case of space motion. In the former, the integrals makes up the symmetry group SO (3) of the system, which can be transformed into the symmetry group of the two-dimensional isotropic harmonic oscillator through the Levi-Civita transformation, which is accurately conserved by the trapezoidal formula. On the other hand, in the case of space motion, the strict conservation of the energy and angular momentum inegrals and the Runge-Lenz vector by the trapezoidal formula is manifested in the 5 Kepler orbital elements a, e, i,and ω.