# Yeomans (1966): Exposition of Sundman's Regularization of the Three Body Problem **Donald K. Yeomans** NASA TM X-55636, X-640-66-481. Goddard Space Flight Center, Greenbelt, Maryland. December 1966. A complete English-language exposition of Karl Sundman's regularization of the three-body problem, providing the first detailed vectorial treatment of Sundman's historic 1912 paper. The convergence of the resulting power series solutions is investigated and found to be computationally useless. > [!info] > This is not the original Sundman (1912) paper. It is Yeomans' 1966 NASA exposition, which provides the first complete English treatment of Sundman's work in vectorial form. The original Sundman paper is: "Memoire sur le probleme des trois corps," *Acta Mathematica*, 36, 1912, 105-179. ## The fundamental problem: equations break at collisions ![[Yeomans1966_Foreword.png]] From the foreword: > "The classical differential equations for the problem of three bodies remain valid only if there are no collisions or other discontinuities for real values of time. The equations of motion are not analytic when two or three of the bodies occupy coincident positions." > > -- Yeomans (1966), Foreword This is the starting point. Newton's equations of motion for three bodies contain terms like $1/r^3$ where $r$ is the distance between two bodies. When two bodies collide ($r \to 0$), these terms blow up to infinity. The equations of motion become singular. The differential equations literally stop being valid. ## Sundman's approach: change the independent variable Sundman's key insight was that the singularity at collision is not an essential feature of the physics but an artefact of using time $t$ as the independent variable. By introducing a new independent variable $u$ (related to $t$ through a regularizing transformation), the equations of motion become analytic even at collision points. The paper shows that: 1. At double collision (two bodies collide), the velocity and acceleration approach infinity, but the distance vector $\mathbf{r}$ approaches a well-defined limit direction 2. A new independent variable $u$ is introduced which removes the singularity in the equation of motion for double collision 3. The mutual distances and the original time variable can be expanded as convergent power series in $u$ 4. A second change of variable (from $u$ to $w$) handles the case of multiple successive double collisions The solution is a power series in $u$ (or $w$), **not in $t$**. This is critical: Sundman did not solve the three-body problem as a function of time. He solved it as a function of a regularized variable that is related to time through a non-trivial transformation. ## The convergence is useless ![[Yeomans1966_Abstract.png]] From the abstract: > "The convergence of the power series solutions is investigated and found to be extremely slow." > > -- Yeomans (1966), Abstract The paper derives lower limits for the radius of convergence of the series solutions. The strip of convergence in the $u$-plane is determined, and the relationship between $u$ and physical time $t$ means that computing any trajectory over a physically meaningful timescale requires on the order of $10^{8{,}000{,}000}$ terms of the series. > [!critical] > Sundman's "solution" requires a change of independent variable (from time $t$ to a new variable $u$) just to make the equations analytic at collision points. The solution is a power series in $u$, not in $t$. Even then, convergence is so slow ($\sim 10^{8{,}000{,}000}$ terms) that it is computationally useless. No computer that has existed, exists, or will exist can evaluate this many terms. The series is not a calculational tool for prediction future motion. ## The 10 integrals and the remaining gap Yeomans derives the 10 classical integrals of motion for the three-body problem explicitly (Equations 18, p. 4): 1. Energy integral (1 integral) 2. Angular momentum integral (3 integrals) 3. Centre of mass position (3 integrals) 4. Centre of mass velocity / linear momentum (3 integrals) He then notes: "It is evident that our problem consists of nine equations in 14 of second order and hence 18 degrees of freedom. With the corresponding 10 integrals of motion, there remains 8 integrals for a solution to the problem. These integrals are not known." This is the same gap identified by Bruns (1887): 18 first-order ODEs, 10 known integrals, 8 unknowns that cannot be determined by any algebraic conserved quantity. ## Relation to Bruns' theorem Sundman's result does not contradict Bruns ([[1887_Bruns_Vielkorper_Problem]]) or Poincare ([[1890_Poincare_Three_Body_Problem]]). Bruns proved there are no algebraic first integrals beyond the 10 classical ones. Poincare proved there are no analytic uniform integrals. Sundman's series is not a first integral. It is a representation of the solution in a regularized time variable. The series converges, but it does not close the system. It is the mathematical equivalent of saying "the solution exists" without being able to compute it. ## Relevance to the null hypothesis Sundman's theorem is relevant to the null hypothesis as follows: even the mathematical existence of a convergent series does not make the three-body problem solvable in practice. The "solution" requires: 1. Abandoning physical time as the independent variable 2. Introducing a regularized variable that removes collision singularities 3. Computing a power series with $\sim 10^{8{,}000{,}000}$ terms The only way to compute three-body trajectories remains stepwise numerical integration. The integrated acceleration calculated at each step to make up the difference reduces this to kinematics. Everything done here is kinematics. ## Cross-references - [[1887_Bruns_Vielkorper_Problem]] -- No algebraic integrals beyond the 10 classical ones - [[1890_Poincare_Three_Body_Problem]] -- No analytic uniform integrals; Sundman's series does not contradict this - [[Three_Body_Null]] -- The null hypothesis note where Sundman is discussed ## Citation Yeomans, D.K. (1966). "Exposition of Sundman's Regularization of the Three Body Problem." NASA TM X-55636 (X-640-66-481). Goddard Space Flight Center, Greenbelt, Maryland. Original result: Sundman, K.F. (1912). "Memoire sur le probleme des trois corps." *Acta Mathematica*, 36, 105-179.