Making new sense of the three-body problem

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In summary, Maryam Mirzakhani just won the Fields Medal, becoming the first woman ever to do so. This was due to her contributions in the fields of essential topology, general relativity, and the Hamilton-Jacobi equation. Her discoveries showed that even seemingly chaotic systems, such as the three-body problem, can follow geometric laws. However, to fully understand her work, knowledge of moduli spaces and ergodic theory is necessary. For beginners in these areas, it is recommended to start with lecture series or resources such as "Ergodic Notes" or "Moduli Spaces Mini Course."
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victorvmotti
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You must have heard the recent news that the first woman ever, Maryam Mirzakhani, just won the Fields Medal.

Unfortunately I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in her math contributions. However, I am well conversant in essential topology, general relativity, Hamilton–Jacobi equation, and see why in integrable systems the motion takes place on an N torus and in general fills the surface. Also, know that even the reduced three-body problem could have a chaotic, highly irregular dynamical evolution.

Referring to what is reported in the news you can see that specifically one of her discoveries show that even the potentially chaotic dynamic of the three-body problem follows some deeply geometric laws. What we consider unpredictable, in the interaction of Sun, Moon, and Earth, is still to some extent predictable.

Is it possible for you to explain this restriction on phase space using what I now know and understand, I mean without moduli spaces and ergodic theory?

In addition I greatly appreciate it if you please suggest good and accessible resources for a beginner in moduli spaces and ergodic theory assuming understanding of the essential topology used in general relativity.
 
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Is it possible for you to explain this restriction on phase space using what I now know and understand, I mean without moduli spaces and ergodic theory?
No.

In addition I greatly appreciate it if you please suggest good and accessible resources for a beginner in moduli spaces and ergodic theory assuming understanding of the essential topology used in general relativity.
...that's hard to judge. Your best starting point would be a lecture series or something, i.e.
http://www.math.psu.edu/sarig/506/ErgodicNotes.pdf
http://www-personal.umich.edu/~eclader/ModuliSpacesMiniCourse.pdf
... then work out what you need from there.
 
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1. What is the three-body problem?

The three-body problem is a famous mathematical and physical problem that involves predicting the motion of three objects, such as planets or stars, that are affected by each other's gravitational pull. It is one of the oldest unsolved problems in physics and has been studied for centuries.

2. Why is the three-body problem difficult to solve?

The three-body problem is difficult to solve because the motion of three interacting objects cannot be predicted using simple mathematical equations. Unlike the two-body problem, where the objects only interact with each other, the third body adds complexity and makes it almost impossible to find a closed-form solution. This is why the three-body problem is often referred to as a "chaotic" system.

3. How have scientists tried to solve the three-body problem?

Scientists have used various approaches to try and solve the three-body problem, including numerical simulations, perturbation theory, and symplectic integrators. These methods involve using computers to solve the equations of motion for the three bodies, or approximating the solutions through mathematical techniques.

4. Why is making new sense of the three-body problem important?

Understanding and solving the three-body problem has important implications for various fields, including astronomy, astrophysics, and celestial mechanics. It can help us better understand the behavior of planetary systems, the stability of orbits, and the long-term evolution of the universe.

5. What recent developments have been made in solving the three-body problem?

In recent years, there have been significant developments in solving the three-body problem, including the discovery of new families of periodic orbits and the development of new numerical methods that can accurately simulate the motion of three bodies over long periods of time. These advancements have allowed scientists to make new sense of the three-body problem and make progress towards finding a general solution.

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