# Is there a skip ahead formula for simulating planetary motion using time step increments?

Is there a formula that would allow me to skip to a particular time step in a simulation involving planetary motion without having the simulation pass through every time step before?

For instance if I wanted to get to the trillionth time step of a simulation is there a formula that would allow me to get to the trillionth time step without passing through ever time step before the trillionth time step? Alternatively if I wanted to get to the millionth time step in the simulation could I use the same formula to skip to the millionth time step? I would like a formula that could also give the positions and velocities of all the bodies in the simulation at whatever time step I decide to skip to.

• Are you considering only, say, one planet orbiting a sun, or are you considering something more complicated, like 100 masses all interacting with each other? – knzhou Apr 15 '18 at 9:23
• This formula would just be a solution to the equations of motion, and if you have more than two bodies there is no such formula. – Javier Apr 15 '18 at 12:49
• To clarify what Javier said: what you're after is a closed form for the solution to the equations of motion, which can be taken to mean an expression involving only a finite number of basic terms, where a basic term includes various things like constants, addition, trig functions, roots, powers and so on. Such a thing does not exist for gravitating systems with more than two bodies in general. – tfb Apr 15 '18 at 15:53