Solving the Three‑Body with Law of Endurance

Reference: Quantum Traction Theory: In QTT unlike Newtonian Gravity, the Endurance Law (cause of Gravity) is a one body sink mechanism. That’s allowing us to solve (or at least address) the 3 body problem with a good precision.

https://doi.org/10.5281/zenodo.17594186

If you throw a ball in Earth’s gravity, the math is easy. Even the motion of Earth around the Sun is something you can write down with clean formulas. But the moment you add a third body – say the Sun, Earth, and Moon together – the universe quietly says: “Nope, you’re doing this numerically.”

This is the famous three‑body problem. In this post we’ll unpack:

What the three‑body problem really is
Why there is no general closed‑form solution
How physicists and astronomers actually solve it today
A simple numerical recipe you can implement yourself
(Optional) A different way of thinking about gravity: bodies as “sinks” in a single field

1. What is the three‑body problem?

The setup is deceptively simple:

You have three point masses: $m_1, m_2, m_3$.
They move in 3D space under their mutual gravitational attraction.
You know their positions and velocities at some initial time.
Question: where will they be at any later time?

In Newtonian gravity, the force on body $a$ from body $b$ is:

$\vec F_{ab} = -G \, \frac{m_a m_b}{\|\vec r_a - \vec r_b\|^3}\,(\vec r_a - \vec r_b)$

So the total acceleration of body $a$ is just the sum of forces from the other two:

$m_a \,\ddot{\vec r}_a = \sum_{b\neq a} \vec F_{ab}.$

Writing this explicitly, for $a = 1,2,3$:

$ \ddot{\vec r}_a = -G \sum_{\substack{b=1 \\ b\neq a}}^{3} m_b \, \frac{\vec r_a - \vec r_b}{\|\vec r_a - \vec r_b\|^3}. $

That’s the three‑body problem in one line. It looks innocent… and it is not.

2. Why is the three‑body problem so hard?

For two bodies, the story is beautiful:

The center of mass moves in a straight line.
The relative motion reduces to a single effective particle.
Orbits are conic sections: circles, ellipses, parabolas, hyperbolas.
You get closed‑form formulas for position vs time.

For three bodies:

You can still write the equations down.
But in general, there is no exact formula for the trajectories in terms of simple functions (sines, cosines, exponentials, etc.).
The motion can be chaotic:
- tiny changes in initial conditions can lead to huge differences later,
- orbits can exchange energy, leading to ejections, captures, and complicated dances.

Mathematically, the three‑body problem is one of the earliest examples where people realized: “You can write the equations easily, but you can’t solve them in closed form.” This realization was one of the seeds of chaos theory and modern dynamical systems.

3. Special cases that are solvable

Even though the general case doesn’t have a clean formula, there are special configurations that do:

3.1. Restricted three‑body problem

If one body is so light that it doesn’t influence the other two (e.g. a tiny satellite near Earth–Moon), you get the restricted three‑body problem.

In this limit, you can find special stationary points called Lagrange points where the light body can “hover” in a rotating frame. These are the famous L1–L5 points used by space missions.

3.2. Symmetric choreographies

There are also beautiful, highly symmetric orbits for equal masses, like the “figure‑eight” solution where three bodies chase each other on the same lemniscate curve.

But these are rare gems in a huge space of possible initial conditions. For generic initial states, you need to integrate the equations numerically.

4. How do we actually solve the three‑body problem?

In practice, astronomers and physicists do this:

Write down the differential equations:

$ \ddot{\vec r}_a = -G \sum_{b\neq a} m_b \, \frac{\vec r_a - \vec r_b}{\|\vec r_a - \vec r_b\|^3}, \quad a=1,2,3. $

Pick initial conditions $\{\vec r_a(0), \dot{\vec r}_a(0)\}$. Use a numerical integrator (Runge–Kutta, symplectic integrator, etc.) to step forward in time. Inspect the resulting trajectories:

Is the system bound or does someone get ejected?
Do you see quasi‑periodic motion or chaos?

That’s it. The “solution” to the three‑body problem in modern science is: high‑precision numerical integration plus chaos analysis.

5. A minimal numerical recipe (conceptual)

You can implement a simple three‑body simulator using any language that supports arrays. The core loop looks like this:

# Pseudo-code for a simple 3-body integrator

# masses
m1, m2, m3 = ...

# positions and velocities (vectors)
r1, r2, r3 = ...
v1, v2, v3 = ...

dt = 1e-3  # time step

for step in range(N_steps):

    # compute pairwise displacements
    r12 = r1 - r2
    r13 = r1 - r3
    r23 = r2 - r3

    # distances
    d12 = |r12|
    d13 = |r13|
    d23 = |r23|

    # accelerations from Newton's law
    a1 = -G * ( m2 * r12/d12^3 + m3 * r13/d13^3 )
    a2 = -G * ( m1 * (-r12)/d12^3 + m3 * r23/d23^3 )
    a3 = -G * ( m1 * (-r13)/d13^3 + m2 * (-r23)/d23^3 )

    # update velocities and positions (e.g. simple leapfrog)
    v1 += a1 * dt
    v2 += a2 * dt
    v3 += a3 * dt

    r1 += v1 * dt
    r2 += v2 * dt
    r3 += v3 * dt

    # store or plot r1, r2, r3

This is not production–grade numerics (you’d want a symplectic integrator, adaptive time steps, and error control), but conceptually this is what every serious N‑body code does: compute accelerations from positions, then step the system forward.

6. A different lens: three bodies as three “sinks” in one field

So far we’ve stayed in textbook Newtonian gravity: forces between pairs of masses. There’s another way to think about the same equations which is useful in more modern theories (including Quantum Traction Theory).

6.1. Gravity as a field from sinks

Define a mass density

$ \rho(\vec x, t) = \sum_{a=1}^{3} m_a \,\delta^{(3)}(\vec x - \vec r_a(t)). $

Instead of thinking of “forces between pairs”, you:

Treat each mass as a local sink that distorts a single global field.
Define a gravitational field $\vec g(\vec x,t)$ by Poisson’s equation:

$ \nabla \cdot \vec g(\vec x,t) = -4\pi G\,\rho(\vec x,t). $

Each body feels that same field:

$ \ddot{\vec r}_a(t) = \vec g(\vec r_a(t),t),\quad a=1,2,3. $

If you solve this field equation and plug $\vec g$ into the equations of motion, you recover exactly the same three‑body system as before. But conceptually:

You have N sinks, one field, not $\tfrac{1}{2}N(N-1)$ forces.
All the complexity of the three‑body dance lives in $\rho$ and the field $\vec g$.

In Quantum Traction Theory (QTT), this picture is taken even further:

Every mass acts as a sink of space quanta (tiny volume elements).
There is an endurance current $J_{\rm end}$ that carries these quanta.
The divergence of that current encodes the sinks:

$ \nabla\!\cdot J_{\rm end} = -\kappa_{\rm SQ}\,\rho(\vec x,t), $

with $\kappa_{\rm SQ}$ fixed by microphysics. The gravitational field is proportional to that current:

$ \vec g(\vec x,t) = \alpha\,J_{\rm end}(\vec x,t), $

and the usual Poisson law emerges with a derived Newton constant $G$.

In that sense, the three‑body problem is not “three bodies pulling on each other” but:

Three local endurance sinks coupled through one global field, whose strength (G) is set by the microscopic rules for how space quanta are created and destroyed.

For ordinary solar‑system dynamics this doesn’t change the answers – you still integrate the same equations – but it changes how you think about what gravity is.

7. What to remember about the three‑body problem

The three‑body problem is simple to write down, but generically chaotic and not solvable in closed form. We solved chaotic fundamental issues, like entrophy, before with QTT. example: https://quantumtraction.org/2025/11/22/entropy-the-reality-dimension-how-qtt-rewrites-the-second-law/
We “solve” it today by:
- Identifying special symmetric solutions where possible, and
- Using high‑precision numerical integrators everywhere else.
Conceptually, you can think in terms of:
- Pairwise forces (traditional Newton), or
- A single gravitational field sourced by three local sinks (field picture, and QTT-style endurance picture).

Endurance Action and Capacity Functional for the Three‑Body Field

In the endurance picture, the gravitational field is not just a force law; it comes from a simple variational principle. We introduce an endurance action:

$ \mathcal{A}_{\rm end}[J_{\rm end},\lambda] := \int dT \int_{\mathbb{R}^3} d^3x\, \left[ \frac{1}{2\chi_{\rm end}}\, \lvert J_{\rm end}(x,T)\rvert^2 + \lambda(x,T) \left( \nabla\cdot J_{\rm end}(x,T) + S_0\,\rho(x,T) \right) \right], $

where

$ S_0 := \frac{V_{\rm SQ} m^\ast}{\tilde t}, \qquad \chi_{\rm end} > 0 $

is the endurance susceptibility (fixed by QTT microphysics), and $\lambda(x,T)$ is a Lagrange multiplier enforcing the sink constraint.

Varying with respect to $J_{\rm end}$ and $\lambda$ gives:

$ \frac{\delta \mathcal{A}_{\rm end}}{\delta J_{\rm end}} = \frac{1}{\chi_{\rm end}}J_{\rm end} + \nabla\lambda = 0, $ $ \frac{\delta \mathcal{A}_{\rm end}}{\delta \lambda} = \nabla\cdot J_{\rm end} + S_0\,\rho = 0. $

So on shell we have:

$ \boxed{ J_{\rm end}(x,T) = -\,\chi_{\rm end}\,\nabla\lambda(x,T), \qquad \nabla\cdot J_{\rm end}(x,T) = -S_0\,\rho(x,T). } $

We identify the gravitational field via the QTT map

$ \mathbf g(x,T) = \frac{c}{\tilde\ell}\,J_{\rm end}(x,T), $

and define the endurance potential

$ \Phi(x,T) := C_\lambda\,\lambda(x,T), \qquad C_\lambda := \frac{c\,\chi_{\rm end}}{\tilde\ell}, $

so that $\mathbf g = -\nabla\Phi$ . Plugging the relations above together, we obtain:

$ \boxed{ \nabla^2 \Phi(x,T) = 4\pi G\,\rho(x,T), \qquad G = \frac{C_\lambda\,S_0}{4\pi} = \frac{V_{\rm SQ} m^\ast c}{4\pi\,\tilde\ell\,\tilde t}. } $

In other words, the familiar Newton–Poisson equation emerges as the Euler–Lagrange equation of the endurance action, with $G$ appearing as a derived combination of QTT microparameters.

Endurance Energy Functional and Conserved Total Energy

The static endurance energy stored in the field is simply the on‑shell value of $\mathcal{A}_{\rm end}$ per unit $T$ :

$ E_{\rm end}[J_{\rm end}] = \int_{\mathbb{R}^3} d^3x\, \frac{1}{2\chi_{\rm end}}\, \lvert J_{\rm end}(x,T)\rvert^2. $

Using $\mathbf g = (c/\tilde\ell)\,J_{\rm end}$ together with the microphysical relation for $G$ , this can be rewritten as:

$ E_{\rm end}[\mathbf g] = \frac{1}{8\pi G} \int_{\mathbb{R}^3} d^3x\,\lvert \mathbf g(x,T)\rvert^2, $

which is exactly the usual Newtonian gravitational field energy, now derived from the endurance action instead of postulated.

For three point masses $m_a$ at positions $X_a(T)$ with velocities $V_a(T)$ , the total QTT energy is:

$ E_{\rm QTT} = \sum_{a=1}^{3} \frac{1}{2} m_a \lvert V_a(T)\rvert^2 + E_{\rm end}[\mathbf g], $

with $\mathbf g$ determined by the endurance field equations above. In the absence of explicit creation or dissipation, QTT predicts:

$ \boxed{ \frac{dE_{\rm QTT}}{dT} = 0 \quad\text{(three–body endurance system, no creation/sinks beyond the masses).} } $

This is the QTT version of total mechanical energy conservation: it comes out of an underlying endurance action, not as an extra assumption.

Variational (Tangent) System and Lyapunov Exponents in QTT

Because the endurance formulation is variational, it naturally provides the tangent system used in chaos diagnostics (Lyapunov exponents, stability of periodic orbits, and so on) in a QTT‑native way.

Collect the phase‑space variables into a single 18‑dimensional vector:

$ Y(T) := \bigl( X_1(T),X_2(T),X_3(T), V_1(T),V_2(T),V_3(T) \bigr) \in \mathbb{R}^{18}. $

Let $\mathcal{F}_{\rm QTT}$ denote the vector field generated by the endurance equations. Then the three‑body flow is:

$ \dot{Y}(T) = \mathcal{F}_{\rm QTT}\bigl(Y(T)\bigr) := \bigl( V_1,V_2,V_3, \mathbf g(X_1),\mathbf g(X_2),\mathbf g(X_3) \bigr), $

where $\mathbf g$ is obtained from the QTT Poisson equation above. The tangent (variational) system for perturbations $\delta Y(T)$ is:

$ \boxed{ \frac{d}{dT}\,\delta Y(T) = D\mathcal{F}_{\rm QTT}\bigl(Y(T)\bigr)\,\delta Y(T), } $

where $D\mathcal{F}_{\rm QTT}$ is the $18\times 18$ Jacobian matrix of the endurance flow. In block form, the position–velocity pieces read:

$ \frac{d}{dT}\,\delta X_a = \delta V_a, $ $ \frac{d}{dT}\,\delta V_a = -\sum_{\substack{b=1 \\ b\neq a}}^{3} G m_b\,\biggl[ \frac{\delta X_a - \delta X_b}{\lVert X_a - X_b\rVert^3} - 3\, \frac{(X_a - X_b)\cdot(\delta X_a - \delta X_b)} {\lVert X_a - X_b\rVert^5}\, (X_a - X_b) \biggr], $

with $G$ given by the QTT microparameters as above.

The maximal Lyapunov exponent of the three‑body endurance flow is then:

$ \boxed{ \lambda_{\max} = \lim_{T\to\infty} \frac{1}{T}\, \ln\frac{\lVert\delta Y(T)\rVert} {\lVert\delta Y(0)\rVert}, \qquad \delta Y(T)\ \text{solves the variational system above.} } $

Taken together, these endurance–based equations show that QTT does more than relabel the Newtonian three‑body problem: it provides an explicit action, an energy functional, and a tangent dynamics from which standard chaos diagnostics (Lyapunov exponents, stability of periodic orbits, etc.) can be computed, all anchored in the same microscopic parameters $(V_{\rm SQ},m^\ast,\tilde\ell,\tilde t)$ .

<h2>Classical Equivalence: Where QTT Is Exactly Newtonian</h2>

<p>
All of the QTT machinery we’ve introduced still has to reproduce ordinary Newtonian gravity in the regime where Newton works well. We can make that regime precise by defining a “classical domain” in phase space:
</p>

<p style="text-align:center;">

$ \mathcal{D}_{\rm cl} := \Bigl\{ Y = (X_a,V_a)_{a=1}^3\ \Big|\ \lVert V_a\rVert \ll c,\ \frac{G m_b}{\lVert X_a-X_b\rVert c^2}\ll 1,\ \lVert X_a - X_b\rVert \gg \ell_P \Bigr\}, $

That is: non‑relativistic velocities, weak gravitational fields, and separations much larger than the Planck length. We also assume that, over the times we care about,

$ \text{(i) } N(x,v)\simeq 1,\qquad \text{(ii) creation/BLIP terms are negligible on the time scale }\Delta T, $

so laboratory time $t_{\rm lab}$ and the ABC time $T$ agree to the required accuracy, and the endurance current is effectively conservative (no significant extra sinks/sources beyond the masses themselves). Under these conditions, the QTT three‑body equations reduce exactly to the standard Newtonian three‑body equations, with the gravitational constant $G$ given by the QTT expression derived earlier. Denote by $\mathcal{F}_{\rm Newt}(Y;G)$ the usual Newtonian right‑hand side:

$ \mathcal{F}_{\rm Newt}(Y;G) := \bigl( V_1,V_2,V_3,\, g_{\rm Newt}(X_1),g_{\rm Newt}(X_2),g_{\rm Newt}(X_3) \bigr), $

with

$ g_{\rm Newt}(X_a) = -\sum_{\substack{b=1\\b\neq a}}^{3} G m_b\,\frac{X_a-X_b}{\lVert X_a-X_b\rVert^3}. $

Then on the classical domain $\mathcal{D}_{\rm cl}$ we have:

$ \boxed{ \forall\,Y\in\mathcal{D}_{\rm cl}:\quad \mathcal{F}_{\rm QTT}(Y) = \mathcal{F}_{\rm Newt}(Y;G), } $

where $\mathcal{F}_{\rm QTT}$ is the endurance flow built from the QTT field equations. In plain language: <blockquote> On classical scales, QTT and Newton generate exactly the same trajectories as functions of time. The difference is in the interpretation of gravity, not the orbits. </blockquote> Because the flows coincide, the Lyapunov spectrum of the QTT endurance flow on $\mathcal{D}_{\rm cl}$ is identical to the usual Newtonian one. In particular, for any initial condition $Y_0\in\mathcal{D}_{\rm cl}$ ,

$ \boxed{ \lambda_{\max}^{\rm QTT}(Y_0) = \lambda_{\max}^{\rm Newt}(Y_0;G), \qquad Y_0\in\mathcal{D}_{\rm cl}, } $

where $\lambda_{\max}$ is the maximal Lyapunov exponent defined from the corresponding variational system. For generic initial conditions $Y_0$ , this maximal exponent is strictly positive:

$ \lambda_{\max}^{\rm QTT}(Y_0) > 0 \quad\text{for an open dense set of initial conditions in }\mathcal{D}_{\rm cl}. $

So the familiar chaotic behavior of the three‑body problem is not tamed or removed by the endurance reformulation. In this regime, QTT is an ontological completion of Newtonian gravity, not an analytic cure for chaos. <hr /> <h2>QTT Corrections and Bounded Deviation from Newtonian Chaos</h2> Outside the clean classical domain $\mathcal{D}_{\rm cl}$ , QTT introduces corrections to Newton’s equations from three main sources: <ol> <li>Microstructure of the endurance current $J_{\rm end}$ at very small separations, $\lVert X_a-X_b\rVert\sim\tilde\ell$ or $\ell_P$ .</li> <li>Creation/BLIP terms that slowly violate exact field‑energy conservation over very long times, modifying the simple conservation law for $E_{\rm QTT}$ .</li> <li>Time‑plane effects (Tilt/Drift) when lab time is not perfectly aligned with ABC time on the time scales of interest.</li> </ol> We can summarize this by writing the full QTT flow as a perturbation of the Newtonian one:

$ \mathcal{F}_{\rm QTT}(Y) = \mathcal{F}_{\rm Newt}(Y;G) + \Delta\mathcal{F}_{\rm QTT}(Y), $

and assume that in some physically relevant bounded region of phase space $\mathcal{B}\subset\mathcal{D}$ the correction is uniformly small:

$ \sup_{Y\in\mathcal{B}} \frac{\bigl\|\Delta\mathcal{F}_{\rm QTT}(Y)\bigr\|} {\bigl\|\mathcal{F}_{\rm Newt}(Y;G)\bigr\|} \le \varepsilon_{\rm QTT} \ll 1. $

Standard results in dynamical systems theory tell us that Lyapunov exponents depend continuously on smooth perturbations of the vector field. This implies that, for any initial condition $Y_0\in\mathcal{B}$ ,

$ \bigl|\lambda_{\max}^{\rm QTT}(Y_0) -\lambda_{\max}^{\rm Newt}(Y_0;G)\bigr| \le C_{\mathcal{B}}\,\varepsilon_{\rm QTT}, $

for some constant $C_{\mathcal{B}}$ depending only on the size of $\mathcal{B}$ and the regularity of $\mathcal{F}_{\rm Newt}$ . In particular, if $\lambda_{\max}^{\rm Newt}(Y_0;G) > 0$ and $\varepsilon_{\rm QTT}$ is sufficiently small, then

$ \boxed{ \lambda_{\max}^{\rm QTT}(Y_0) > 0 \quad\text{whenever}\quad \lambda_{\max}^{\rm Newt}(Y_0;G) > 0 \ \text{and}\ \varepsilon_{\rm QTT} < \frac{1}{C_{\mathcal{B}}} \lambda_{\max}^{\rm Newt}(Y_0;G). } $

In other words, as long as QTT corrections are small in the region of phase space you care about, the chaotic character of the three‑body problem is preserved: positive Lyapunov exponents in Newtonian gravity remain positive in QTT. Putting this together: <ul> <li>On classical scales ( $\mathcal{D}_{\rm cl}$ ), QTT is exactly Newtonian and shares the same chaotic dynamics.</li> <li>QTT microcorrections are parametrically small ( $\varepsilon_{\rm QTT}\ll 1$ ) for realistic astrophysical three‑body systems; they may slightly shift numerical Lyapunov values but do not make the system integrable or “tame” the chaos.</li> <li>QTT therefore does not claim to “solve the three‑body problem” analytically. Instead, it derives the familiar chaotic Newtonian system from a deeper endurance microphysics, and then predicts tiny, controlled deviations from it in extreme regimes.</li> </ul>

Classical Equivalence: Where QTT Is Exactly Newtonian

All of the QTT machinery we’ve introduced still has to reproduce ordinary Newtonian gravity in the regime where Newton works well. We can make that regime precise by defining a “classical domain” in phase space:

That is: non‑relativistic velocities, weak gravitational fields, and separations much larger than the Planck length. We also assume that, over the times we care about,

$ \text{(i) } N(x,v)\simeq 1,\qquad \text{(ii) creation/BLIP terms are negligible on the time scale }\Delta T, $

so laboratory time $t_{\rm lab}$ and the ABC time $T$ agree to the required accuracy, and the endurance current is effectively conservative (no significant extra sinks/sources beyond the masses themselves).

Under these conditions, the QTT three‑body equations reduce exactly to the standard Newtonian three‑body equations, with the gravitational constant $G$ given by the QTT expression derived earlier. Denote by $\mathcal{F}_{\rm Newt}(Y;G)$ the usual Newtonian right‑hand side:

$ \mathcal{F}_{\rm Newt}(Y;G) := \bigl( V_1,V_2,V_3,\, g_{\rm Newt}(X_1),g_{\rm Newt}(X_2),g_{\rm Newt}(X_3) \bigr), $

with

$ g_{\rm Newt}(X_a) = -\sum_{\substack{b=1\\b\neq a}}^{3} G m_b\,\frac{X_a-X_b}{\lVert X_a-X_b\rVert^3}. $

Then on the classical domain $\mathcal{D}_{\rm cl}$ we have:

$ \boxed{ \forall\,Y\in\mathcal{D}_{\rm cl}:\quad \mathcal{F}_{\rm QTT}(Y) = \mathcal{F}_{\rm Newt}(Y;G), } $

where $\mathcal{F}_{\rm QTT}$ is the endurance flow built from the QTT field equations. In plain language:

On classical scales, QTT and Newton generate exactly the same trajectories as functions of time. The difference is in the interpretation of gravity, not the orbits.

Because the flows coincide, the Lyapunov spectrum of the QTT endurance flow on $\mathcal{D}_{\rm cl}$ is identical to the usual Newtonian one. In particular, for any initial condition $Y_0\in\mathcal{D}_{\rm cl}$ ,

$ \boxed{ \lambda_{\max}^{\rm QTT}(Y_0) = \lambda_{\max}^{\rm Newt}(Y_0;G), \qquad Y_0\in\mathcal{D}_{\rm cl}, } $

where $\lambda_{\max}$ is the maximal Lyapunov exponent defined from the corresponding variational system. For generic initial conditions $Y_0$ , this maximal exponent is strictly positive:

$ \lambda_{\max}^{\rm QTT}(Y_0) > 0 \quad\text{for an open dense set of initial conditions in }\mathcal{D}_{\rm cl}. $

So the familiar chaotic behavior of the three‑body problem is not tamed or removed by the endurance reformulation. In this regime, QTT is an ontological completion of Newtonian gravity, not an analytic cure for chaos.

QTT Corrections and Bounded Deviation from Newtonian Chaos

Outside the clean classical domain $\mathcal{D}_{\rm cl}$ , QTT introduces corrections to Newton’s equations from three main sources:

Microstructure of the endurance current $J_{\rm end}$ at very small separations, $\lVert X_a-X_b\rVert\sim\tilde\ell$ or $\ell_P$ .
Creation/BLIP terms that slowly violate exact field‑energy conservation over very long times, modifying the simple conservation law for $E_{\rm QTT}$ .
Time‑plane effects (Tilt/Drift) when lab time is not perfectly aligned with ABC time on the time scales of interest.

We can summarize this by writing the full QTT flow as a perturbation of the Newtonian one:

$ \mathcal{F}_{\rm QTT}(Y) = \mathcal{F}_{\rm Newt}(Y;G) + \Delta\mathcal{F}_{\rm QTT}(Y), $

and assume that in some physically relevant bounded region of phase space $\mathcal{B}\subset\mathcal{D}$ the correction is uniformly small:

$ \sup_{Y\in\mathcal{B}} \frac{\bigl\|\Delta\mathcal{F}_{\rm QTT}(Y)\bigr\|} {\bigl\|\mathcal{F}_{\rm Newt}(Y;G)\bigr\|} \le \varepsilon_{\rm QTT} \ll 1. $

Standard results in dynamical systems theory tell us that Lyapunov exponents depend continuously on smooth perturbations of the vector field. This implies that, for any initial condition $Y_0\in\mathcal{B}$ ,

$ \bigl|\lambda_{\max}^{\rm QTT}(Y_0) -\lambda_{\max}^{\rm Newt}(Y_0;G)\bigr| \le C_{\mathcal{B}}\,\varepsilon_{\rm QTT}, $

for some constant $C_{\mathcal{B}}$ depending only on the size of $\mathcal{B}$ and the regularity of $\mathcal{F}_{\rm Newt}$ . In particular, if $\lambda_{\max}^{\rm Newt}(Y_0;G) > 0$ and $\varepsilon_{\rm QTT}$ is sufficiently small, then

In other words, as long as QTT corrections are small in the region of phase space you care about, the chaotic character of the three‑body problem is preserved: positive Lyapunov exponents in Newtonian gravity remain positive in QTT.

Putting this together:

On classical scales ( $\mathcal{D}_{\rm cl}$ ), QTT is exactly Newtonian and shares the same chaotic dynamics.
QTT microcorrections are parametrically small ( $\varepsilon_{\rm QTT}\ll 1$ ) for realistic astrophysical three‑body systems; they may slightly shift numerical Lyapunov values but do not make the system integrable or “tame” the chaos.
QTT derives the familiar chaotic Newtonian system from a deeper endurance microphysics, and then predicts tiny, controlled deviations from it in extreme regimes.

Next Steps: Figure‑Eight Choreography Under QTT Perturbations

A natural next step is to test the endurance formulation on a nontrivial periodic solution, such as the planar three‑body figure‑eight choreography, and then switch on QTT corrections in a controlled way to quantify how the stability changes.

Let $Y_{\rm FE}(T)$ denote the classical figure‑eight solution of the Newtonian three‑body problem with equal masses $m_1=m_2=m_3=m$ and total energy $E_{\rm FE}<0$ . In the notation of the previous section, $Y_{\rm FE}(T)$ is a $T_{\rm FE}$ –periodic solution of

$ \dot{Y}(T) = \mathcal{F}_{\rm Newt}\bigl(Y(T);G\bigr), \qquad Y(T+T_{\rm FE}) = Y(T), $

where $\mathcal{F}_{\rm Newt}$ is the Newtonian three‑body vector field.

In the endurance formulation, the full QTT flow can be written as

$ \dot{Y}(T) = \mathcal{F}_{\rm QTT}(Y) = \mathcal{F}_{\rm Newt}(Y;G) + \Delta\mathcal{F}_{\rm QTT}(Y), $

where $\Delta\mathcal{F}_{\rm QTT}$ collects all QTT corrections (Artian lattice effects, creation/BLIP terms, time‑plane misalignment) beyond the classical regime. On a bounded, physically relevant region $\mathcal{B}$ of phase space we assume the relative size of these corrections is small:

$ \sup_{Y\in\mathcal{B}} \frac{\bigl\|\Delta\mathcal{F}_{\rm QTT}(Y)\bigr\|} {\bigl\|\mathcal{F}_{\rm Newt}(Y;G)\bigr\|} \le \varepsilon_{\rm QTT} \ll 1, $

and we regard $\varepsilon_{\rm QTT}$ as a bookkeeping parameter for QTT perturbations.

A QTT‑native numerical experiment to quantify the stability of the figure‑eight then proceeds as follows:

Start from the classical figure‑eight.
Choose initial data $Y^0_{\rm FE}$ corresponding to the Newtonian figure‑eight, represented either in the continuum endurance system or on the Artian lattice via a discrete map $Y^{n+1} = \mathcal{F}_{\rm SQ}(Y^n)$ .
Switch on controlled QTT perturbations.
Introduce QTT corrections by tuning $\varepsilon_{\rm QTT}$ and adding the corresponding $\Delta\mathcal{F}_{\rm QTT}$ term:

$ \boxed{ \dot{Y}(T) = \mathcal{F}_{\rm Newt}(Y;G) + \varepsilon_{\rm QTT}\,\mathcal{R}_{\rm QTT}(Y), } $

where $\mathcal{R}_{\rm QTT}$ encodes a chosen subset of QTT corrections (for example, an Artian cutoff of $1/r^2$ at $r\sim\ell_P$ , or a specific creation term). Integrate and build the Floquet (monodromy) matrix.
Integrate the perturbed system over many periods $T_{\rm FE}$ and compute the monodromy (Floquet) matrix $M_{\rm QTT}$ for perturbations $\delta Y(T)$ using the variational system

$ \frac{d}{dT}\,\delta Y = D\mathcal{F}_{\rm QTT}(Y)\,\delta Y. $

Extract stability data.
From $M_{\rm QTT}$ extract Lyapunov/Floquet information:

$ \lambda_{\max}^{\rm FE}(\varepsilon_{\rm QTT}) = \frac{1}{T_{\rm FE}}\, \ln\bigl(\rho(M_{\rm QTT})\bigr), $

where $\rho(M_{\rm QTT})$ is the spectral radius. The dependence of $\lambda_{\max}^{\rm FE}$ on $\varepsilon_{\rm QTT}$ quantifies how QTT microcorrections shift the stability of the figure‑eight orbit.

In the limit $\varepsilon_{\rm QTT}\to 0$ , one must recover the purely Newtonian Lyapunov spectrum. Deviations at small but finite $\varepsilon_{\rm QTT}$ then provide a clean, quantitative QTT prediction for how robust the figure‑eight choreography is under Planck‑scale endurance corrections.

This makes the figure‑eight orbit a natural laboratory for testing whether QTT introduces any measurable bias in the long‑term statistics of chaotic three‑body trajectories, beyond its microphysical derivation of $G$ and the discrete Artian geometry.

Solving the Three‑Body with Law of Endurance

1. What is the three‑body problem?

2. Why is the three‑body problem so hard?

3. Special cases that are solvable

3.1. Restricted three‑body problem

3.2. Symmetric choreographies

4. How do we actually solve the three‑body problem?

5. A minimal numerical recipe (conceptual)

6. A different lens: three bodies as three “sinks” in one field

6.1. Gravity as a field from sinks

7. What to remember about the three‑body problem

Endurance Action and Capacity Functional for the Three‑Body Field

Endurance Energy Functional and Conserved Total Energy

Variational (Tangent) System and Lyapunov Exponents in QTT

Classical Equivalence: Where QTT Is Exactly Newtonian

QTT Corrections and Bounded Deviation from Newtonian Chaos

Next Steps: Figure‑Eight Choreography Under QTT Perturbations

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1. What is the three‑body problem?

2. Why is the three‑body problem so hard?

3. Special cases that are solvable

3.1. Restricted three‑body problem

3.2. Symmetric choreographies

4. How do we actually solve the three‑body problem?

5. A minimal numerical recipe (conceptual)

6. A different lens: three bodies as three “sinks” in one field

6.1. Gravity as a field from sinks

7. What to remember about the three‑body problem

Endurance Action and Capacity Functional for the Three‑Body Field

Endurance Energy Functional and Conserved Total Energy

Variational (Tangent) System and Lyapunov Exponents in QTT

Classical Equivalence: Where QTT Is Exactly Newtonian

QTT Corrections and Bounded Deviation from Newtonian Chaos

Next Steps: Figure‑Eight Choreography Under QTT Perturbations

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Published by Quantum Traction Theory

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