Chaos & Unsolvability: Solving the 3-Body Problem & the Role of Chaotic Motion

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In summary, the equations of motion for the 3-body problem are not exactly solvable due to their non-linear nature. This means that numerical methods are typically used to solve them. Chaos is characterized by sensitivity to initial conditions, making it difficult to find precise solutions for non-linear equations. While simple systems may exhibit chaotic motion, they can still be solved numerically to a certain degree of accuracy. However, they are not analytically solvable. The classical three body problem and electromagnetics are examples of non-linear problems that are not solvable for point particles, as they exhibit numerical instability. This is because they display critical sensitivity to initial conditions, similar to trying to predict the exact fall of a pencil when stood on its point.
  • #1
Ed Quanta
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Can something like the 3 body problem be shown to be unsolvable because it exhibits chaotic motion? In other words, must all solvable equations of motion be 100 percent free of chaos? Sorry if my question isn't clear.
 
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  • #2
The equations of motion that describe the 3-body problem are not exactly solvable concisely because they are non-linear. Usually the best method for solving them is numerical.

Chaos is best described as sensitivity to initial conditions. Thus in a system of non-linear equations if we alter the initial conditions by any amount we will see a different behavior in that system.

Here is the wikipedia write up on chaos theory:
http://en.wikipedia.org/wiki/Chaos_theory

And one from math world:
http://mathworld.wolfram.com/Chaos.html

These should help you clarify some of the points you will need to understand if you would like to study this subject. Good Luck!
 
  • #3
Ed, it is possible to have chaotic motion for which the dynamic equations have a solution (for example, a driven pendulum).
 
  • #4
Ed: I think the answer to both questions is Yes, provide you are using "solvable" to mean integrable. Of course many simple systems exhibit chaos, but are still numerically solvable to some precision. But they are not analytically solvable.
 
  • #5
The classical three body problem is linear because F=ma is linear. Electrodynamics
is also linear and the charge distribution on a piece of metal is a constrained
many-body linear problem. But it too is not solvable for the point particles- the
solutions are NUMERICALLY unstable.

It is chaotic because the equations display a critical sensitivity to initial conditions,
like solving for which way a pencil will fall when stood perfectly on its point.
 

1. What is the 3-body problem?

The 3-body problem refers to the mathematical problem of predicting the motion of three objects, such as planets, under the influence of their mutual gravitational attraction. It is an unsolved problem in physics and astronomy, as it is difficult to find a general solution that can accurately predict the motion of the three bodies over time.

2. Why is the 3-body problem important?

The 3-body problem is important because it has significant implications for our understanding of the universe. Its unsolvability highlights the limitations of our current understanding and mathematical tools. The study of the 3-body problem has also led to the development of chaos theory and the recognition of chaotic motion in systems.

3. How does chaos theory relate to the 3-body problem?

Chaos theory is the study of complex and unpredictable systems that are highly sensitive to initial conditions. The 3-body problem is an example of a chaotic system, as small changes in the initial conditions can lead to drastically different outcomes. The study of the 3-body problem has contributed to the development of chaos theory and our understanding of chaotic motion.

4. Can the 3-body problem be solved?

No, the 3-body problem cannot be solved in the general case. While there are specific cases where a solution can be found, such as when the three bodies are in a particular arrangement or have equal masses, there is no general analytical solution for all possible scenarios. This is due to the complex and non-linear nature of the equations involved.

5. What is the significance of chaotic motion in the 3-body problem?

The presence of chaotic motion in the 3-body problem has significant implications for our understanding of the universe. It shows that even seemingly simple systems can exhibit complex and unpredictable behavior. The study of chaotic motion in the 3-body problem has also led to advancements in fields such as weather forecasting, economics, and biology.

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