Newtonian gravity: a_i = sum_j G m_j (r_j - r_i)/|r_j - r_i|^3, with G = 2.959e-4 AU^3 / (solar mass day^2), integrated by the symplectic kick-drift-kick leapfrog scheme (second order, time-reversible, no secular energy drift).
Kepler's laws follow from the two-body problem: ellipses with the primary at one focus (r = a(1-e^2)/(1+e cos nu)), equal areas in equal times (angular momentum L = r x v is conserved), and T^2 = 4 pi^2 a^3 / (G (M+m)). Speed obeys the vis-viva relation v^2 = mu (2/r - 1/a).
Total energy E = sum 1/2 m v^2 - sum_{i<j} G m_i m_j / r_ij is tracked; its relative drift is graph 1. With many bodies and large time steps the drift grows; the step adapts to the fastest orbital period.
Simplifications: point masses, Newtonian (no relativity, so Mercury's 43"/century perihelion shift is absent), coplanar orbits, planetary elements at one epoch (osculating, rounded). Disk sizes are exaggerated.