# Poincare (1890): Sur le probleme des trois corps et les equations de la dynamique **Henri Poincare** *Acta Mathematica*, Vol. 13, 1890, pp. 1-270. The revised King Oscar II prize memoir. Poincare proves that the three-body problem admits no analytical uniform integral beyond the 10 classical ones, discovers what we now call chaos (sensitive dependence on initial conditions), and identifies periodic, asymptotic, and doubly asymptotic solutions. Published 135 years ago. The general three-body problem is still not solved. ## Overview This is one of the most important papers in the history of mathematics and physics. Poincare submitted an earlier version for the prize competition held by King Oscar II of Sweden in 1888. That version contained an error (it claimed stability where there was instability). Phragmen spotted the issue, and Poincare revised the memoir extensively, discovering in the process the phenomenon of homoclinic tangles: deterministic systems whose trajectories are so sensitive to initial conditions that long-term prediction is impossible. The revised memoir is the foundation of dynamical systems theory, topology, and chaos theory. ## Introduction (pp. 5-6): Translation from the French ![[Poincare1890_Intro_p5.png]] ### Original (p. 5) > "Le travail qui va suivre et qui a pour objet l'etude du probleme des trois corps est un remaniement du memoire que j'avais presente au Concours pour le prix institue par Sa Majeste le Roi de Suede." **Translation:** The work that follows, which has as its object the study of the three-body problem, is a revision of the memoir that I had submitted to the competition for the prize established by His Majesty the King of Sweden. > [!note] > The revision was necessary because the original memoir contained an error about orbital stability. Poincare paid for the recall and reprinting out of his own pocket. The corrected version is more important than the original, because the error led Poincare to discover chaos. ### Original (p. 6) > "Je suis bien loin d'avoir completement resolu le probleme que j'ai aborde. Je me suis borne a demontrer l'existence de certaines solutions particulieres remarquables que j'appelle solutions periodiques, solutions asymptotiques, et solutions doublement asymptotiques." **Translation:** I am far from having completely solved the problem I have undertaken. I have limited myself to demonstrating the existence of certain remarkable particular solutions that I call periodic solutions, asymptotic solutions, and doubly asymptotic solutions. > [!note] > Poincare, one of the greatest mathematicians in history, admits he is "far from having completely solved" the three-body problem. He could only identify certain special classes of solutions. The general solution eluded him. It still eludes everyone. ### The negative results (p. 6) ![[Poincare1890_NoIntegrals_p6.png]] > "Mais j'attirerai surtout l'attention du lecteur sur les resultats negatifs qui sont developpes a la fin du memoire. J'etablis par exemple que le probleme des trois corps ne comporte, en dehors des integrales connues, aucune integrale analytique et uniforme." **Translation:** But I will especially draw the reader's attention to the negative results that are developed at the end of the memoir. I establish for example that the three-body problem admits, beyond the known integrals, no analytical and uniform integral. > [!critical] > This is the key result. The "known integrals" are the 10 classical first integrals: energy (1), linear momentum (3), angular momentum (3), and centre of mass (3). Poincare proves there are no others. This confirms and extends Bruns' (1887) algebraic result to the analytic case. See [[1887_Bruns_Vielkorper_Problem]]. The system of equations cannot be closed. No hidden conservation law exists. ### The impossibility of a complete solution (p. 6) > "Bien d'autres circonstances nous font prevoir que la solution complete, si jamais on peut la decouvrir, exigera des instruments analytiques absolument differents de ceux que nous possedons et infiniment plus compliques." **Translation:** Other circumstances lead us to foresee that the complete solution, if one can ever discover it, will require analytical instruments absolutely different from those we possess and infinitely more complicated. > [!critical] > Poincare is saying that the entire mathematical toolkit available in 1890 (and, as it turns out, the entire mathematical toolkit available in 2026) is insufficient to solve the general three-body problem. The tools needed, if they exist at all, are "absolutely different" from anything known. This is not pessimism. It is a mathematical assessment based on the structure of the problem. 135 years later, the complete solution has not been found. ### The divergence of series (p. 6) > "J'ai fait voir egalement que la plupart des series employees en mecanique celeste et en particulier celles de M. Lindstedt qui sont les plus simples, ne sont pas convergentes." **Translation:** I have also shown that most of the series employed in celestial mechanics, and in particular those of Mr. Lindstedt which are the simplest, are not convergent. > [!warning] > The standard perturbation series used in celestial mechanics are divergent. They do not converge to the true solution. They are asymptotic approximations that work well for short times but fail for long-term prediction. This applies directly to the EIH post-Newtonian expansion ([[1938_Einstein_Infeld_Hoffmann_Equations_of_Motion]]), which is an asymptotic series in $1/c^2$. ## What Poincare proved (summary) | Result | Significance | |--------|-------------| | No uniform analytic integrals beyond the 10 classical ones | The system cannot be solved in closed form | | Standard perturbation series diverge | Long-term analytical predictions are impossible | | Sensitive dependence on initial conditions (homoclinic tangles) | Nearby trajectories diverge exponentially | | Periodic, asymptotic, and doubly asymptotic solutions exist | Special solutions exist but do not cover the general case | ## The Sundman (1912) convergent series Karl Sundman later proved that a convergent power series solution to the three-body problem exists. This is sometimes cited as contradicting Poincare. It does not. Sundman's series converges so slowly that computing any useful trajectory requires on the order of $10^{8{,}000{,}000}$ terms. The series is a mathematical existence proof, not a calculational tool. > [!note] > Sundman's PDF (Sundman, K.F. (1912). "Memoire sur le probleme des trois corps." *Acta Mathematica*, 36, 105-179) was not obtainable for this vault. It is behind the Springer/Project Euclid paywall. See [[1912_Sundman_Three_Body_Series]] for a minimal stub. ## Relevance to the null hypothesis Poincare's results establish that Newton's dynamics for 3+ bodies: 1. Cannot produce a general closed-form solution (no uniform analytic integrals) 2. Cannot produce reliable long-term predictions (series diverge, trajectories diverge) 3. Generate deterministic chaos (the future is uniquely determined but unpredictable) The dynamic variables ($G$, $M_1$, $M_2$, $M_3$) do not contain enough information to determine the trajectory analytically. All practical computation requires step-by-step numerical integration, which is kinematics. GR inherits all of these problems and adds nonlinear coupling that makes them worse. ## Cross-references - [[1887_Bruns_Vielkorper_Problem]] -- Proved the algebraic version of the non-integrability result; Poincare extended it to the analytic case - [[1938_Einstein_Infeld_Hoffmann_Equations_of_Motion]] -- The EIH equations inherit Poincare's chaos plus GR nonlinear coupling - [[2009_Laskar_Collisional_Trajectories]] -- Numerical confirmation of Poincare's chaos in the real Solar System - [[1991_Wisdom_Holman_Symplectic_Maps]] -- Modern numerical methods for the problem Poincare proved cannot be solved analytically - [[1912_Sundman_Three_Body_Series]] -- The convergent but impractical series solution - [[Three_Body_Null]] -- The null hypothesis note this source supports ## Citation Poincare, H. (1890). "Sur le probleme des trois corps et les equations de la dynamique." *Acta Mathematica*, 13, 1-270.