In 1909–1912, Finnish mathematician Karl Sundman proved that the three-body problem admits an analytical solution in the form of convergent power series. His work represents the pinnacle of classical celestial mechanics, completing the search for an analytical solution begun by Newton.

Sundman's Theorem

Sundman showed that the trajectories of three bodies can be represented by convergent power series in a regularized time variable s. The key idea is a time transformation dt = Ω(r) ds with a function Ω that vanishes at collisions. This removes the singularities from the differential equations, and the series converge for all real t.

Collision Regularization

The main difficulty is the singularity at collisions, when the denominator r_ → 0. Levi-Civita proposed a coordinate transformation that unfolds the collision: a complex coordinate w = u + iv is introduced such that z = w², where z = x + iy is the relative position. This maps the collision point (z = 0) to a regular point (w = 0) and doubles angles — the collision becomes an ordinary passage through zero.

Levi-Civita transformation:

(u + iv)^2 = x + iy \quad \Longrightarrow \quad x = u^2 - v^2,\; y = 2uv

Along with the coordinate transformation, Sundman used a change of the time variable dt = r ds, where r is the distance between the colliding bodies. This transformation slows down time near the collision, making the trajectory smooth in s.

Time regularization:

dt = r \, ds \quad \Longrightarrow \quad \frac{d}{dt} = \frac{1}{r}\frac{d}{ds}

Analytic Continuation

In the regularized variables (w, s), the equations of motion have no singularities. The solution can be analytically continued through the collision point — the trajectory is holomorphic (complex-analytic) in the vicinity of w = 0. The solution expands into a convergent Taylor series in w and s throughout the domain, except for isolated singular points.

Power Series and Convergence

After regularization, the solution r_i(s) is represented by a power series with a nonzero radius of convergence R. The series converges on the entire real axis, but singular points in the complex plane limit the convergence rate. The coefficients behave as a_n ~ C·R⁻ⁿ·n^α. The radius R is so small that practical accuracy requires an astronomical number of terms.

Sundman's series:

\mathbf{r}_i(s) = \sum_{n=0}^\infty \mathbf{a}_n s^n, \quad |s| < R

The convergence rate is determined by the distance to the nearest singularity in the complex s-plane:

|\mathbf{a}_n| \sim C \cdot R^{-n} \cdot n^\alpha, \quad R \ll 1

Practical Uselessness

Computing one orbital period requires ~N = 10^(8·10⁶) terms. This number exceeds the number of atoms in the observable universe. Sundman's series is a landmark theoretical result, demonstrating the gap between mathematical solvability and practical computability.

Theoretical Significance

Sundman completed the classical stage of the three-body problem investigation. An analytical solution exists, but his work revealed fundamental limitations: even when a solution exists, obtaining it in practice may require prohibitive resources — foreshadowing ideas in computational complexity theory.

Example: 1D Free-Fall Regularization

Two bodies of equal mass m = 100 fall toward each other along the X axis with initial velocity v₀ = 0, initial distance d = 20.

Relative motion equation:

\ddot{x} = -\frac{G(m_1 + m_2)}{x^2} = -\frac{2Gm}{x^2}, \quad x(0) = 20,\; \dot{x}(0) = 0

A singularity occurs at x → 0. Apply 1D Levi-Civita regularization: x = u², dt = x ds. After substitution, we obtain the harmonic oscillator equation — regular at the collision point:

\frac{d^2 u}{ds^2} + \frac{Gm}{2} u = 0

The equation is regular at u = 0. Solution: u(s) = A sin(ωs + φ), x(s) = u²(s) is analytic everywhere. The Taylor series for x(s) converges in the vicinity of s_collision — the collision has become a regular point.

In regularized time s, the trajectory expands into a convergent series:

x(s) = a_0 + a_2 s^2 + a_4 s^4 + \cdots

Three-Body Simulation

Numerical integration (Velocity-Verlet) is a practical alternative to Sundman's prohibitively slow series.