Despite that terms came from physics, this question is purely about building math model. So I'll try to explain what I want.
1) First of all, I am building a N-body PC simulation program. So I am using numeric Verlet integration scheme, where velocity is replaced by position delta. It could express new position, after small timestep: $x_{new} = x_{old} + (x_{new}-x_{old}) + a\cdot dt^2$. Lets call it (1)
2) As there are multiple bodies, according to Newton's gravity law, for each pair of bodies, attractive force may be calculated $F_1 = m_1 \cdot a_1 = {G m_1 m_2\over r^2}$, where distance between them $r = x_2-x_1$. This can be transormed to finally useful $a_1 = { G m_2\over r^2 }$, lets refer to it as (2)
Plugging (2) into (1) one may get the idea of simulation of objects motion in space.
Now there is third idea, there exists higher order position derivatives, called jerk and jounce. Trying to extend verlet integration manner to acceleration, I cannot use formula 2 anymore, but have to calculate it another way: $a_{new} = a_{old} + (a_{new}-a_{old}) + jounce\cdot dt^2$. Jerk, like velocity, is of no interest here. Please, correct me if, I am wrong.
Question is, how to "upgrade" newtons gravity law, or how to upgrade formula (2), in order to express jounce, (to be able to calculate acceleration within a process of verlet numeric integration)?
Bonus. If you feel this question simple, can you also offer modification of (2) for "pounce"(6-th derivative of position, with respect to time)? Constraints are same: you cannot "touch" acceleration and jounce, but only may calculate them within a process of numeric integration.