OPHobserver · patch · holography

Technical Appendix

Three-Body Problem via OPH.

The full derivation. Reduced Hamiltonian, patch-net construction, mismatch functional, normal-form claim, and pointers to the OPH paper for the underlying theorems.

1. Equations of motion.

mi r¨i  =  ∑j≠i G mimj rj−ri∣rj−ri∣3,i=1,2,3m_i\,\ddot{\mathbf r}_i \;=\; \sum_{j\neq i}\, G\,m_i m_j\,\frac{\mathbf r_j-\mathbf r_i}{|\mathbf r_j-\mathbf r_i|^3},\qquad i=1,2,3

The pairwise potential

V  =  −Gm1m2r12−Gm2m3r23−Gm3m1r31V \;=\; -\frac{Gm_1m_2}{r_{12}}-\frac{Gm_2m_3}{r_{23}}-\frac{Gm_3m_1}{r_{31}}

2. Reduction to Jacobi coordinates.

Remove the center-of-mass and total-momentum degrees of freedom. Define

ρ=r2−r1,λ=r3−m1r1+m2r2m1+m2\boldsymbol\rho = \mathbf r_2 - \mathbf r_1,\qquad \boldsymbol\lambda = \mathbf r_3 - \frac{m_1\mathbf r_1+m_2\mathbf r_2}{m_1+m_2}

with reduced masses

μρ=m1m2m1+m2,μλ=m3(m1+m2)M,M=m1+m2+m3.\mu_\rho=\frac{m_1m_2}{m_1+m_2},\qquad \mu_\lambda=\frac{m_3(m_1+m_2)}{M},\quad M=m_1+m_2+m_3.

The exact reduced Hamiltonian:

H=∣pρ∣22μρ+∣pλ∣22μλ−Gm1m2∣ρ∣−Gm1m3∣λ+m2m1+m2ρ∣−Gm2m3∣λ−m1m1+m2ρ∣H = \frac{|\mathbf p_\rho|^2}{2\mu_\rho} + \frac{|\mathbf p_\lambda|^2}{2\mu_\lambda} - \frac{Gm_1m_2}{|\boldsymbol\rho|} - \frac{Gm_1m_3}{\bigl|\boldsymbol\lambda+\frac{m_2}{m_1+m_2}\boldsymbol\rho\bigr|} - \frac{Gm_2m_3}{\bigl|\boldsymbol\lambda-\frac{m_1}{m_1+m_2}\boldsymbol\rho\bigr|}

Hamilton's equations:

ρ˙=∂pρH,λ˙=∂pλH,p˙ρ=−∂ρH,p˙λ=−∂λH.\dot{\boldsymbol\rho}=\partial_{\mathbf p_\rho}H,\quad \dot{\boldsymbol\lambda}=\partial_{\mathbf p_\lambda}H,\quad \dot{\mathbf p}_\rho=-\partial_{\boldsymbol\rho}H,\quad \dot{\mathbf p}_\lambda=-\partial_{\boldsymbol\lambda}H.

3. OPH patch net.

Discretize time t_n = n·Δt. At each slice introduce:

  • three body patches Bin carrying (rin, pin);
  • three pair patches Pijn carrying (ri, rj, Vij, Fij);
  • time-collar patches linking n → n+1;
  • a loop collar around the triangle 1→2→3→1.

Pair force assertion:

Fij  =  G mimj rj−ri∣rj−ri∣3\mathbf F_{ij} \;=\; G\,m_i m_j\,\frac{\mathbf r_j-\mathbf r_i}{|\mathbf r_j-\mathbf r_i|^3}

Discrete Hamiltonian evolution (leapfrog):

rin+1−rinΔt=pin+1/2mi,pin+1−pinΔt=∑j≠iFijn+1/2.\frac{\mathbf r_i^{n+1}-\mathbf r_i^n}{\Delta t} = \frac{\mathbf p_i^{n+1/2}}{m_i},\qquad \frac{\mathbf p_i^{n+1}-\mathbf p_i^n}{\Delta t} = \sum_{j\neq i}\mathbf F_{ij}^{n+1/2}.

4. Mismatch functional.

ΦN  =  ∑n,i∣pin+1−pinΔt−∑j≠iFijn+1/2∣2+∑n,i∣rin+1−rinΔt−pin+1/2mi∣2\Phi_N \;=\; \sum_{n,i}\Bigl|\tfrac{\mathbf p_i^{n+1}-\mathbf p_i^n}{\Delta t}-\sum_{j\neq i}\mathbf F_{ij}^{n+1/2}\Bigr|^2 + \sum_{n,i}\Bigl|\tfrac{\mathbf r_i^{n+1}-\mathbf r_i^n}{\Delta t}-\tfrac{\mathbf p_i^{n+1/2}}{m_i}\Bigr|^2
+  ∑collars∣shared-data mismatch∣2  +  ∑γ∣1−eiΔSγ/ϵ∣2\hspace{2em}+\;\sum_{\text{collars}}\bigl|\text{shared-data mismatch}\bigr|^2 \;+\; \sum_{\gamma}\bigl|1-e^{i\Delta S_\gamma/\epsilon}\bigr|^2

The last sum is the loop term: action accumulated around the interaction triangle must glue. On the regular noncollision branch, ΦN = 0 ⇔ the discrete equations are exactly satisfied.

5. Normal-form claim.

qOPH(t)  =  lim⁡N→∞  nf⁡N(mi,ri(0),pi(0))q_{\text{OPH}}(t) \;=\; \lim_{N\to\infty}\;\operatorname{nf}_N\bigl(m_i,\mathbf r_i(0),\mathbf p_i(0)\bigr)

where nfN is the schedule-independent quotient normal form on the finite patch net of resolution N. Existence and schedule-independence rest on the OPH finite theorem stack:

  • Accepted repairs strictly lower a Lyapunov functional.
  • Termination on finite state spaces.
  • Local confluence + repair completeness ⇒ schedule-independent normal form.
  • Physical observables are unique on the gauge quotient.

See OPH paper, Consensus & Repair sections.

6. Why generic orbits look 'unsolvable'.

The two-body problem has no nontrivial interaction loop after reduction. The three-body problem has a triangle loop. That loop carries action holonomy. Small initial-data changes change the repair path and loop phase, surfacing macroscopically as chaos. Special orbits are special precisely because the loop holonomy closes cleanly:

  • Euler collinear — loop collapses to 1-D.
  • Lagrange equilateral — symmetric triangle fixed point.
  • Figure-eight (Chenciner–Montgomery, 2000) — periodic holonomy cancellation.

7. Aharonov–Bohm in the same language.

Two electron path patches with B = 0 locally; their overlap at the detector carries the phase

U(γ)  =  exp⁡ ⁣(i qℏ∮γA⋅dx)  =  exp⁡ ⁣(i qℏ ΦB).U(\gamma) \;=\; \exp\!\Bigl(i\,\tfrac{q}{\hbar}\oint_\gamma \mathbf A\cdot d\mathbf x\Bigr) \;=\; \exp\!\Bigl(i\,\tfrac{q}{\hbar}\,\Phi_B\Bigr).

Same machinery: gauge transformations alter local representatives but preserve overlap data; physical uniqueness lives on the gauge quotient. The interference shift is the observer-facing record of the loop holonomy.

8. References.

  • FloatingPragma, Observer-Patch Holography, paper folder. github.com/FloatingPragma/observer-patch-holography/tree/main/paper
  • FloatingPragma, From Observers to Gravity (Book 1). PDF
  • FloatingPragma, From Observers to the Standard Model (Book 2). PDF
  • Reading site: oph-book.floatingpragma.io
  • Aharonov & Bohm, Significance of Electromagnetic Potentials in the Quantum Theory, Phys. Rev. 115, 485 (1959).
  • Tonomura et al., Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave, Phys. Rev. Lett. 56, 792 (1986).
  • Chenciner & Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. Math. 152, 881 (2000).