Every body attracts every other: a_i = Σ G m_j (r_j − r_i) / |r_j − r_i|³. Units here are AU, days and solar masses, so the Sun has GM☉ = k² with Gauss's constant k = 0.01720209895. Integration is the symplectic velocity-Verlet (leapfrog) scheme, step 0.2 day, which keeps energy error bounded rather than drifting.
Initial planet states come from J2000 orbital elements (a, e, ϖ, L) placed in the ecliptic plane; velocities follow the vis-viva relation v² = GM(2/r − 1/a). Inclinations are ignored (2D). The Sun moves around the barycentre.
Conserved checks: total energy E = Σ ½mv² − Σ Gm_im_j/r_ij and angular momentum L = Σ m (x v_y − y v_x). Kepler's third law P² = 4π² a³ / GM predicts the period of a two-body orbit. A body launched at the circular speed √(GM/r) stays nearly circular.
Simplifications: planar, Newtonian point masses (no relativity, no oblateness, no moons, no asteroid belt), fixed 0.2 d step so very close encounters are inaccurate. Planet masses are listed in Earth masses. The simulated positions are not an ephemeris for real dates.