# Three-Body Problem: No Dynamic Solution ## The question Can gravitational dynamics (Newton's $F = GMm/r^2$, Einstein's field equations) produce a general closed-form solution for the motion of three or more gravitating bodies? ## Null hypothesis **H$_1$:** Gravitational dynamics provide a general solution for N-body systems. The force law $F = GMm/r^2$ is general, therefore the solutions derived from it should also be general -- valid for arbitrary initial conditions and arbitrary numbers of bodies. **H$_0$:** Gravitational dynamics do not provide a general solution for N-body systems. The force law is general but the solutions are not. Every dynamic "solution" for N-bodies is particular (tied to specific initial conditions) or special-case (restricted configurations). The kinematic formulas $\psi = 6\pi(v/c)^2/(1-e^2)$ and $\dot{T} \propto (v/c)^5$ are general solutions that work for any orbit in any system. ## Companion to [[Cosmological_Dynamics_Null]] The first null hypothesis showed that for the **two-body problem**, every GR prediction (precession, orbital decay, deflection, redshift, Schwarzschild radius) reduces to kinematics when $GM$ is replaced by $4\pi^2 a^3/T^2$. The dynamic variables cancel. No dynamic solution diverges from the kinematic one. This second null hypothesis addresses a different failure mode. For three or more bodies, the dynamic framework does not merely reduce to kinematics. It fails to produce a solution at all. --- ## Newton: no general solution ### Bruns (1887) Heinrich Bruns proved that the Newtonian three-body problem admits no general algebraic first integrals beyond the ten classical ones (conservation of energy, linear momentum, angular momentum, and centre of mass motion). This means: you cannot write a formula $\mathbf{r}(t) = f(t, \text{initial conditions})$ that gives the positions of three gravitating bodies at arbitrary future time $t$. The ten conserved quantities are not enough to close the system. The equations of motion are six second-order ODEs (18 first-order), and ten integrals leave eight unknowns. No further integrals exist. [[1887_Bruns_Vielkorper_Problem]] ### Poincare (1890) Henri Poincare proved that the restricted three-body problem (two massive bodies, one test particle) exhibits sensitive dependence on initial conditions. Small changes in starting position or velocity lead to exponentially divergent trajectories. The system is chaotic. This is not a computational limitation. It is a mathematical property of the equations. Newton's law $F = GMm/r^2$ applied to three bodies generates deterministic chaos. The equations have a unique solution for any given initial condition, but that solution cannot be expressed in closed form, and nearby solutions diverge exponentially. Long-term prediction requires tracking the trajectory step by step. [[1890_Poincare_Three_Body_Problem]] ### Sundman (1912) Karl Sundman proved that a convergent power series solution to the three-body problem exists. This is sometimes cited as a "solution." It is not a practical one. Sundman's series converges so slowly that computing any useful trajectory would require on the order of $10^{8{,}000{,}000}$ terms. No computer that has existed or will exist can evaluate it. The series is a mathematical existence proof, not a calculational tool. [[1912_Sundman_Three_Body_Series]] ### Summary of Newton | Claim | Status | |-------|--------| | General closed-form solution exists | **No** (Bruns 1887, Poincare 1890) | | Convergent series solution exists | Yes, but requires $\sim 10^{8{,}000{,}000}$ terms (Sundman 1912) | | System is deterministic | Yes (unique solution for given initial conditions) | | System is predictable long-term | **No** (chaotic, exponential divergence of nearby trajectories) | | Practical solutions exist | Only via numerical step-by-step integration (kinematics) | Newton's law tells you the instantaneous acceleration of each body at each moment. It does not tell you where the bodies will be at a future time. To find that, you must integrate the acceleration kinematically, step by step. --- ## The general vs particular distinction The force law $F = GMm/r^2$ is **general**: it applies to any two masses at any separation. But a general force law does not guarantee a general solution. The distinction between a general law and a general solution is the crux of the three-body problem. ### Two bodies: the 11th integral closes the system The two-body problem in three dimensions gives 12 first-order ODEs (6 per body). The 10 classical integrals (energy, 3 linear momentum, 3 angular momentum, 3 centre of mass) reduce this, but 2 unknowns remain. The system is still not closed. What closes it is an 11th conserved quantity: the **Laplace vector** (also called the Runge-Lenz vector). This is a vector that points along the major axis of the orbit and whose magnitude equals the eccentricity. It is conserved for every two-body orbit. It does not follow from any symmetry of space or time. It is specific to the $1/r^2$ force law. With this 11th integral, the system is closed. The result is the Kepler ellipse: a general solution that works for any two masses, any initial separation, any initial velocity. The orbit is always a conic section. No numerical integration required. Bruns' theorem ([[1887_Bruns_Vielkorper_Problem]]) identifies this as the structural reason the two-body problem is solvable. Julliard-Tosel (2000, p. 245, case 1) states it explicitly: for $n = 1$ (two bodies) with Newtonian potential, the Laplace vector $L$ provides an additional algebraic first integral independent from the classical ones. ### Three or more bodies: the 11th integral does not exist For three or more bodies, Bruns proved there are **no** additional algebraic first integrals beyond the 10 classical ones. The Laplace vector does not generalize to 3+ bodies. The 11th integral that closed the two-body system simply does not exist. The counting for three bodies in three dimensions: - 18 first-order ODEs (6 per body) - 10 classical integrals - 8 remaining unknowns - 0 additional integrals available (Bruns 1887) The system cannot be closed. The gap between known integrals and required integrals is 8, and Bruns proved no algebraic quantity can fill it. Poincare (1890) extended this to analytic integrals. The result holds for any positive masses, any configuration. Julliard-Tosel (2000) provides the definitive rigorous proof: "The strength of this theorem is that no other hypothesis is made on masses than positivity." For the Solar System (9 major bodies): 54 ODEs, 10 integrals, 44 unknowns. The gap grows linearly with the number of bodies. The 10 integrals are the same 10 regardless of how many bodies you add. The problem gets worse, not better. ### Every N-body "solution" is particular or special-case Without closure, every method of handling 3+ bodies falls into one of two categories: 1. **Particular solutions** from fixed initial conditions. Numerical integration (REBOUND, MERCURY, GADGET) computes one trajectory at a time, starting from one specific configuration. Change the initial conditions and you must recompute from scratch. That fixed starting point is forever preserved in the solution and can be traced back by unwinding all the motion. Laskar (2009) ran 2,501 separate numerical integrations of the Solar System, each with slightly different initial conditions, getting different outcomes. 2. **Special-case solutions** restricted to specific configurations. Lagrange points (L1-L5) require one body to have negligible mass. Euler's collinear solutions require all three bodies to remain on a line. The restricted three-body problem assumes one mass is negligible. These are not general. They work only for specific geometric arrangements that avoid the 8-unknown closure gap by imposing additional constraints. Sundman's convergent series ([[1912_Sundman_Three_Body_Series]]) is formally a "solution" but requires $\sim 10^{8{,}000{,}000}$ terms, a change of independent variable from time to a regularized parameter (to handle collisions where the equations become non-analytic), and converges so slowly that it might appear to provide a real solution if it could be computed. It cannot. l The numerical approach works by injecting accelerations (force kicks) at each timestep to correct the deviations from independent two-body orbits ([[1991_Wisdom_Holman_Symplectic_Maps]]). Each body coasts on its own Kepler ellipse, then receives a velocity correction $\Delta v = (F/m)\Delta t$ for the influence of every other body. The three-body interaction is never solved. It is patched in as a series of discrete nudges. ### The kinematic formulas ARE general solutions In contrast, the kinematic precession formula: $\psi = \frac{6\pi (v/c)^2}{1 - e^2}$ is a **general solution**. It takes three observables ($a$, $T$, $e$) from ANY orbit and returns the precession. No initial conditions needed. No integration needed. No restriction on the configuration. It works for: - Mercury orbiting the Sun - S2 orbiting Sgr A* - The Moon orbiting Earth - PSR B1913+16 (Hulse-Taylor pulsar) All from the same formula. The inputs are purely kinematic: semi-major axis, period, eccentricity. The output is the precession per orbit. Similarly, the kinematic orbital decay formula: $\dot{T} = -\frac{192\pi}{5} \eta \frac{(v/c)^5 f(e)}{(1-e^2)^{7/2}}$ is a general solution for orbital decay. It works for any binary system. The inputs are kinematic observables. No $G$, no $M$ required (they enter only through $v/c = 2\pi a / (cT)$, a kinematic ratio). ### The asymmetry Dynamics give you a **general force law** but no **general solution** for 3+ bodies. Every dynamic prediction for 3+ bodies requires starting over from specific initial conditions and integrating step by step. Kinematics give you **general solutions** (for precession, for orbital decay) that work for any system. You plug in the observables and get the answer. No initial conditions. No integration. No restriction on configuration. This asymmetry is what H$_0$ captures: the dynamic framework produces the instantaneous acceleration, but it cannot go further. The kinematic framework produces general formulas that go directly from observables to predictions. --- ## The central body as absolute reference ### Kinematic equations reference a central body The kinematic orbital formulas work because every measurement is made relative to a central body. Mercury's orbit is measured relative to the Sun. The Moon's orbit is measured relative to Earth. S2's orbit is measured relative to Sgr A*. The orbital parameters $a$, $T$, $e$ are all defined with respect to this centre. The central body provides an absolute spatial reference for the orbit. Without it, "semi-major axis" and "period" have no meaning. ### N-body simulations use the same anchoring N-body simulations also require a central reference. Every position $\mathbf{r}_i$ and velocity $\mathbf{v}_i$ is computed relative to a fixed point -- either the barycenter of the system or a dominant central mass. The entire integration proceeds relative to this reference. The "N-body solution" is a chain of relative measurements anchored to a fixed spatial point. Remove the anchor and the positions become undefined. ### The force law requires absolute space To assert $F = GMm/r^2$ as a dynamic law requires a notion of absolute space. The quantity $r$ is a distance in absolute space between two bodies. Acceleration $\mathbf{a} = \mathbf{F}/m$ is the second time derivative of position in absolute space. Newton was explicit about this: the Scholium to the Definitions in the *Principia* introduces absolute space and absolute time as the framework within which the laws of motion operate. Without an absolute frame, "distance" and "acceleration" have no operational meaning. The force law is not merely a relationship between masses -- it is a statement about how masses move through absolute space. ### Initial conditions are given, not derived Modern cosmology assumes that planetary systems form from accretion disk ejections -- material already in dynamic motion creates new orbital configurations through gravitational collapse and angular momentum transfer. But the N-body framework cannot model this process. It propagates existing orbits from fixed initial conditions. It cannot create orbits from nothing. The initial conditions ($\mathbf{r}_i$, $\mathbf{v}_i$ at $t = 0$) are always given, never derived from the dynamics. They are the fixed starting point that is forever preserved and can be traced back by unwinding all the motion. Every N-body integration begins with a snapshot of the system handed in from outside the formalism. ### Kinematics avoids the origin problem The kinematic approach sidesteps the question of where orbits came from. It does not ask how Mercury arrived at its current orbit. It takes the observed orbital parameters ($a$, $T$, $e$) as given and computes the precession or decay directly. The central body serves as the absolute reference that makes these measurements meaningful. The formula $\psi = 6\pi(v/c)^2/(1 - e^2)$ does not need to know the history of the orbit. It needs only its current geometry, measured relative to the central body. --- ## General Relativity: worse, not better ### The two-body problem is already unsolved In Newtonian gravity, the two-body problem has an exact closed-form solution (Kepler orbits). In GR, it does not. The only exact solutions in GR are for a test particle (negligible mass) orbiting a single central mass (Schwarzschild 1916, Kerr 1963). When both bodies have comparable mass, there is no exact analytical solution. The equations must be solved numerically. The field of **numerical relativity** exists precisely because GR cannot solve the two-body problem analytically for comparable masses. The first successful numerical simulation of two merging black holes (Pretorius 2005) required decades of computational development after the field equations were written. [[2005_Pretorius_Binary_Black_Hole]] ### The EIH equations of motion Einstein, Infeld, and Hoffmann (1938) derived equations of motion for $N$ bodies in GR as a perturbation series. The EIH equations are not exact. They are a post-Newtonian expansion: $\mathbf{a}_i = \mathbf{a}_i^{\text{Newton}} + \frac{1}{c^2}\mathbf{a}_i^{\text{1PN}} + \frac{1}{c^4}\mathbf{a}_i^{\text{2PN}} + \cdots$ Each order in $1/c^2$ adds relativistic corrections. The series is asymptotic, not convergent. For strong fields (compact binaries, black hole mergers), the expansion breaks down and full numerical relativity is required. The EIH equations for the three-body problem in GR inherit all the Newtonian chaos (Poincare) plus additional nonlinear coupling from the relativistic corrections. No general closed solution exists. [[1938_Einstein_Infeld_Hoffmann_Equations_of_Motion]] Infeld himself, Einstein's collaborator, later wrote: > "We can annihilate the radiation by a proper choice of the laws of motion." > Infeld, L. and Plebanski, J. (1960). *Motion and Relativity*, p. 166. [[1960_Infeld_Plebanski_Motion_and_Relativity]] If the radiation term in the equations of motion can be zeroed by coordinate choice, the "dynamic" content of GR's N-body equations is coordinate-dependent, not physical. ### Summary of GR | Claim | Status | |-------|--------| | Exact 2-body solution (comparable mass) | **No** (requires numerical relativity) | | Exact 3-body solution | **No** | | Perturbative N-body solution (EIH) | Yes, but asymptotic series, not convergent | | Strong-field N-body | Numerical only (Pretorius 2005 onward) | | Radiation term is coordinate-invariant | **No** (Infeld and Plebanski 1960) | GR does not solve the three-body problem. It does not even solve the two-body problem for comparable masses. Every working GR prediction for multi-body systems comes from numerical integration, which is kinematic computation. --- ## The Solar System is a chaotic N-body system The Solar System contains 8 major planets, dwarf planets, moons, and asteroids. It is an N-body system. If gravitational dynamics provided a complete solution, the long-term future of the Solar System would be predictable from Newton's law (or GR's field equations) and the current positions and velocities. It is not. ### Laskar (1989, 2009) Jacques Laskar showed through numerical integration that the inner Solar System is chaotic on timescales longer than about 5 million years. The Lyapunov time (the timescale on which nearby trajectories diverge by a factor of $e$) is approximately 5 million years. Laskar (2009) demonstrated that in about 1% of numerical simulations, Mercury's eccentricity increases enough to allow collisions between Mercury and Venus, or Mercury and Earth, within the next 5 billion years. These results were obtained by numerical integration of Newton's equations (with relativistic corrections). The dynamics were known exactly ($G$, all masses, all initial conditions to high precision). The system is still unpredictable beyond 5 million years because the chaos is intrinsic to the equations, not a limitation of measurement precision. [[2009_Laskar_Collisional_Trajectories]] Laskar, J. (1989). "A numerical experiment on the chaotic behaviour of the Solar System." *Nature*, 338, 237-238. ### What this means The dynamic framework ($F = GMm/r^2$ for every pair of bodies) is fully specified. $G$ is known. Every mass is known. Every position and velocity is known to high precision. And yet the system cannot be predicted beyond $\sim 5$ Myr. The dynamic variables do not determine the future. They must be integrated kinematically, step by step, and even then the result diverges chaotically. If $GM$ were truly an independent dynamical quantity carrying information beyond kinematics, the three-body problem should be solvable. The masses should constrain the motion. They do not. The only thing that constrains the motion is the instantaneous kinematic state (positions, velocities) at each timestep, updated by the instantaneous acceleration. The "mass" enters only through the acceleration at each step, and that acceleration is $\mu/r^2 = 4\pi^2 a^3/(T^2 r^2)$, the same Kepler ratio from [[Cosmological_Dynamics_Null]]. --- ## What numerical N-body simulations actually do Every working N-body code (REBOUND, MERCURY, GADGET, etc.) operates the same way: 1. Start with positions $\mathbf{r}_i$ and velocities $\mathbf{v}_i$ at time $t$ 2. Compute the acceleration $\mathbf{a}_i = \sum_{j \neq i} \mu_j (\mathbf{r}_j - \mathbf{r}_i)/|\mathbf{r}_j - \mathbf{r}_i|^3$ 3. Update velocities: $\mathbf{v}_i \leftarrow \mathbf{v}_i + \mathbf{a}_i \, \Delta t$ 4. Update positions: $\mathbf{r}_i \leftarrow \mathbf{r}_i + \mathbf{v}_i \, \Delta t$ 5. Repeat This is kinematics. At no point does the code "solve the dynamics." It evaluates the acceleration from the current configuration and steps forward. The gravitational parameter $\mu_j = GM_j = 4\pi^2 a_j^3/T_j^2$ enters only as a coefficient in the acceleration, derived from each body's observed orbit. Symplectic integrators (Wisdom and Holman 1991) improve long-term energy conservation by splitting the Hamiltonian into Keplerian and interaction terms. The Keplerian part is solved analytically (the two-body solution). The interaction terms are applied as kinematic kicks. The "analytical" step is Kepler's solution, which is kinematics. The dynamics gave us the acceleration law. Everything after that is kinematics. And for three or more bodies, the dynamics cannot go further than the instantaneous acceleration. They cannot give you the trajectory. --- ## The pattern | Bodies | Newton | GR | |--------|--------|----| | 1 (free) | Trivial (straight line) | Trivial (geodesic) | | 2 | Closed solution (Kepler) | No closed solution for comparable mass | | 3 | No closed solution (Poincare) | No closed solution | | N | Chaotic, numerical only | Chaotic, numerical only | Newton's dynamics solve exactly one non-trivial case: two bodies, because the 11th integral (the Laplace vector) closes the system. GR's dynamics solve exactly zero non-trivial cases for comparable masses. Every other case requires numerical integration, which is kinematic computation step by step. --- ## Conclusion H$_1$ claims that gravitational dynamics provide a general solution for N-body systems. The force law $F = GMm/r^2$ is general. But a general law does not guarantee a general solution. The two-body problem is solvable because it has an 11th conserved quantity (the Laplace vector) that closes the system of equations. This 11th integral is specific to the $1/r^2$ force law and to the two-body case. For three or more bodies, Bruns (1887; rigorous proof: Julliard-Tosel 2000) proved that no such additional integral exists. The system has 18 equations, 10 integrals, and 8 unknowns that nothing can determine without stepping through the motion from a fixed starting point. The only alternatives are: - Numerical integration from particular initial conditions, injecting force corrections (acceleration kicks) at each timestep to offset the deviations from independent Kepler orbits. The fixed starting point is forever preserved and can be traced back by unwinding all the motion. - Sundman's convergent power series, which diverges so slowly ($\sim 10^{8{,}000{,}000}$ terms) that it might appear to provide a real solution if it could be computed. It cannot. GR makes the situation worse: it cannot produce a general solution even for two comparable-mass bodies (Pretorius 2005). The Solar System, despite having all dynamic parameters known to high precision, is chaotically unpredictable beyond approximately 5 million years (Laskar 1989, 2009). Meanwhile, the kinematic formulas $\psi = 6\pi(v/c)^2/(1-e^2)$ and $\dot{T} \propto (v/c)^5$ are general solutions. They work for any orbit in any system, with no initial conditions, no integration, and no restriction on configuration. The central body provides the absolute reference. The orbital parameters ($a$, $T$, $e$) are measured relative to that centre. The formula returns the answer. H$_0$ is not refuted. The dynamic framework gives a general force law but not a general solution. The kinematic framework gives general solutions. Meanwhile, the kinematic formulas are general solutions: - $\psi = 6\pi(v/c)^2/(1-e^2)$ gives the precession for ANY orbit (Mercury, S2, Moon, pulsars) - $\dot{T} \propto (v/c)^5 f(e)/(1-e^2)^{7/2}$ gives the orbital decay for ANY binary - No initial conditions required. No integration required. No restriction on configuration. Combined with the first null hypothesis ([[Cosmological_Dynamics_Null]]): - For two bodies, dynamics reduce to kinematics (all dynamic variables cancel) - For three or more bodies, dynamics fail to produce a general solution at all - In both cases, the kinematic formulas provide general solutions that work for any system H$_0$ is not refuted. The dynamic framework gives a general force law but not a general solution. The kinematic framework gives general solutions. . --- ## Sources Bruns, H. (1887). "Uber die Integrale des Vielkorper-Problems." *Acta Mathematica*, 11, 25-96. Rigorous proof: Julliard-Tosel, E. (2000). "Bruns' Theorem: The Proof and Some Generalizations." *Celestial Mechanics and Dynamical Astronomy*, 76, 241-281. [[1887_Bruns_Vielkorper_Problem]] Poincare, H. (1890). "Sur le probleme des trois corps et les equations de la dynamique." *Acta Mathematica*, 13, 1-270. [[1890_Poincare_Three_Body_Problem]] Sundman, K.F. (1912). "Memoire sur le probleme des trois corps." *Acta Mathematica*, 36, 105-179. Exposition: Yeomans, D.K. (1966). "Exposition of Sundman's Regularization of the Three Body Problem." NASA TM X-55636. [[1912_Sundman_Three_Body_Series]] Einstein, A., Infeld, L., and Hoffmann, B. (1938). "The Gravitational Equations and the Problem of Motion." *Annals of Mathematics*, 39(1), 65-100. [[1938_Einstein_Infeld_Hoffmann_Equations_of_Motion]] Infeld, L. and Plebanski, J. (1960). *Motion and Relativity*. Pergamon Press. [[1960_Infeld_Plebanski_Motion_and_Relativity]] Pretorius, F. (2005). "Evolution of Binary Black-Hole Spacetimes." *Physical Review Letters*, 95, 121101. [[2005_Pretorius_Binary_Black_Hole]] Laskar, J. (1989). "A numerical experiment on the chaotic behaviour of the Solar System." *Nature*, 338, 237-238. Laskar, J. (2009). "Existence of collisional trajectories of Mercury, Mars and Venus with the Earth." *Nature*, 459, 817-819. [[2009_Laskar_Collisional_Trajectories]] Wisdom, J. and Holman, M. (1991). "Symplectic maps for the N-body problem." *The Astronomical Journal*, 102, 1528-1538. [[1991_Wisdom_Holman_Symplectic_Maps]] Wayne, R. (2015). "Explanation of the Perihelion Motion of Mercury." *Turkish Journal of Physics*, 39, 183-192. [[2015_Wayne_Perihelion_Velocity_Correction]] Peters, P.C. and Mathews, J. (1963). "Gravitational Radiation from Point Masses in a Keplerian Orbit." *Physical Review*, 131(1), 435-440. [[1963_Peters_Mathews_Gravitational_Radiation]] --- ## See Also - [[00_Null_Hypothesis_Index]] — Master null hypothesis index - [[Cosmological_Dynamics_Null]] — Two-body case: dynamics reduce to kinematics - [[Perihelion_Precession_Null]] — The precession formula is a general kinematic solution