IAMAP, but n-body problems result in non-linear partial differential equations, to which only a few special cases are known, and even that case its do to simplifications, e.g., treating planets as point masses and ignoring tidal forces, or ignoring the pull of the planets on the Sun, (or on each other).
One such case where a solution is known is the Lagrange point of the Earth-Moon-Sun system (and similiarly for other points) https://en.wikipedia.org/wiki/Lagrange_point
But in reality, they exist only as an approximation. They aren't truly stable.
My understanding the way to calculate spacecraft, asteroid, etc. trajectories is just through a discrete simulation.
Like f you don't know how to solve the antiderivative of a given function, you can still calculate the integral since you know the value of the function.
One such case where a solution is known is the Lagrange point of the Earth-Moon-Sun system (and similiarly for other points) https://en.wikipedia.org/wiki/Lagrange_point But in reality, they exist only as an approximation. They aren't truly stable.
My understanding the way to calculate spacecraft, asteroid, etc. trajectories is just through a discrete simulation.
Like f you don't know how to solve the antiderivative of a given function, you can still calculate the integral since you know the value of the function.