Mathematics of War and Peace

The Three-Body-Problem of US-Russia-China

SUMMARY - Open Letter to World Leaders on Christmas of 2O24

With just a few days left for the Biden-administration, I propose a mathematical guidance for a balanced "Kissinger 2.0 Security System" for US-Russia-China in our nearly unmanageable multi-polar World. Geopolitics might do better with numerical solutions of the "3-body problem" in our time, preferably by neural network algorithms (Breen et al, 2019) that rely on high power computing.

It is well known since Newton-Euler-Lagrange-Lyapunov (see summary e.g. by Koon et al.) that "N-body systems" where N>2 are beyond closed-form mathematical solution. Having suffered the failure of Vietnam war, astoundingly, Kissinger mustered an intuitive feel, without apparent mathematics, to peacefully stabilize the trilateral US-Soviet-Chinese geopolitics. There was, however, a 1997 Senate debate of NATO-expansion that contrasted Kissinger and Brzezinski, where the conclusion promulgated by Biden straight-jacketed the already more than bipolar World into the classic "2-body problem" of the Cold War era. Unaware of basic mathematics, Biden thus created a Russia-China-India-Iran block, pitched against the USA.

While in 2018 Shijun Liao and Xiaoming Li applied a new strategy of numerical solution for chaotic systems called the clean numerical simulation, with the use of a national supercomputer, the geostrategic solution is especially difficult, since our present World is already at least a 4-party-, but indeed a multi-party problem, with an uncertain Europe plus India and Iran, at the least. Notably, leaders in Europe are struggling with "N-body problems" where N=27 in the EU and N=32 in NATO. As unstable multi-body systems are chaotic sets of difficult to manage 3-body subsets each with a predilection for chaos on their own, our World might call for advanced studies based on nonlinear dynamics of chaos and fractals (see fundamentals e.g. by Feldman). This year, a special class of such algorithms was awarded by Nobel Prize, see “neural networks” of Hopfield.

VISUAL METAPHORS

Fig.1. Three-Body-Problem visualized by a simulation as a stable solution on the left, where the 3 bodies of equal weight are rotating in a perfectly symmetrical manner. On the right, with perturbations unseen by naked eye, the system deteriorates in a chaotic manner. In the “geostrategic equivalent”, with the weights of the 3 bodies growing at different rates, there is no stable solution possible, but a constant sophisticated management must be computed to manage symmetrical equilibrium that prevents a chaotic break down.

Perfectly Symmetric vs. Chaotic | The Three-Body Problem | Physics Simulations

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Fig. 2. Tracked Three-Body Problem Simulation with 3 Free Masses | Gravity | Physics Simulations

Tracked chaotic swirl of a Three-Body System shows the frequent change of entanglement-pairs of Green-Red, later Blue-Red, Green-Red and again Blue-Red “battles”, not unlike rapidly changing warring pairs in the beginning of WWII.

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Fig. 3. Four-Body-Problem visualized by a simulation

N-Body Problem Simulation with 4 Free Masses - Gravity - Physics Simulations

For multi-body problems (where n>3) even a numerical iteration for balancing is not available to compute long-term stable pattern, thus the danger of chaotic deterioration is perilous. In such cases the complex N-system might be decomposed into linked "three bodies", but the complexity of such cases grows far beyond the limitation of the reach of this essay.

CONCLUSION

To translate the above chaos-theoretical considerations into geostrategy and politics of our time is extremely timely, as evidenced e.g. by Flournoy (2021) Institute for National Strategic Studies, National Defense University. However, she missed any and all chaos-theoretical aspects of nonlinear dynamics. It is a highly demanding and complex task, since e.g. the roots of neural net theory go back to the functioning of the brain (see Pellionisz 1989 and Rosenfeld et al., 1990), while practical aspects how to successfully navigate N-systems also call both for non-trivial mathematics as well as extensive team work (see e.g. Koon et al, 2011). At the same time the job of applying nonlinear dynamics of chaos theory to our multipolar World may become abundantly fruitful since maneuvering present realities has our lives at stake, not only in space, but also on the ground. Therefore, calling for much more than mere intuitions and rules of thumb for strategy seems well justified.

REFERENCES

Breen, P.G, Foley C.N., Boekholt T., Zwart, S.P. (2019) Newton vs the machine: solving the chaotic three-body problem using deep neural networks. In Astrophysics of Galaxies, https://doi.org/10.48550/arXiv.1910.07291

Brzezinski, Z. (1997) The Grand Chessboard. Perseus Basic Books

Feldman, D.P. (2012) Chaos and Fractals. Oxford University Press

Flournoy, M. (2021) The Three-Body Problem; The U.S., China, and Russia In. Institute for National Strategic Studies, National Defense University

Hopfield, J.J. (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci, 79(8):2554–2558. doi: 10.1073/pnas.79.8.2554 (Nobel Prize, 2024)

Lorenz, E.,N. (1963) Deterministic non-periodic flow. Journal of the Atmospheric Sciences. 20 (2): 130–141

Lorenz, E.,N. (1972) Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set off a Tornado in Texas. American Association for the Advancement of Science.

Kissinger, H. (1994) Diplomacy. Simon and Schuster

Koon, W.S., Lo, M.W., Marsden, J.E. (2011) Dynamical Systems, the Three-Body Problem and Space Mission Design, California Institute of Technology, KoLoMaRo_DMissionBook_2011-04-25.pdf

Li, X., S Liao, S. (2018) Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems. In: Applied Mathematics and Mechanics, Springer

Pellionisz, A. (1989) Neural Geometry; Towards a Fractal Model of Neurons, Cambridge University Press

Rosenfeld, E., Pellionisz, A., Anderson, J. (1990) Neurocomputing-2, MIT Press

US Senate Foreign Relations Committee Hearings. (1997, October 7.-November 5.)The Debate on NATO Enlargement. CHRG-105shrg46832.pdf

ELABORATION (separately)