# The computational complexity of the initial value problem for the three body problem

## TL;DR

This paper proves that solving the initial value problem for the three body problem cannot be done in polynomial time, using analysis of complex oscillatory solutions in the Sitnikov problem to demonstrate computational intractability.

## Contribution

It establishes the non-polynomial complexity of the three body problem's initial value problem through rigorous analysis of oscillatory solutions.

## Key findings

- The IVP for the three body problem is not solvable in polynomial time.
- Oscillatory solutions in the Sitnikov problem exhibit complex behavior.
- Polynomial-time algorithms for the three body problem are unlikely to exist.

## Abstract

The paper is concerned with the computational complexity of the initial value problem (IVP) for a system of ordinary dynamical equations. Formal problem statement is given, containing a Turing machine with an oracle for getting the initial values as real numbers. It is proven that the computational complexity of the IVP for the three body problem is not bounded by a polynomial. The proof is based on the analysis of oscillatory solutions of the Sitnikov problem that have complex dynamical behavior. These solutions contradict the existence of an algorithm that solves the IVP in polynomial time.