The method of finding periodic three-body orbits

In summary, the conversation discusses determining periodic orbits in the equal-mass three-body Newtonian gravitational problems. The suggested method is to confine the three bodies into a zero angular momentum space and calculate the gravitational force between the bodies to find the velocity/orbit. The concept of "algebraic symmetries" and "free group elements" are mentioned as tools to classify the orbits into distinct classes.
  • #1
Pefgjk
4
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Hello everyone!

I'm trying to get a deeper understanding on how to determine those periodic orbits in the equal-mass three-body Newtonian gravitational problems. The most general idea I know is to confine the three bodies into a zero angular momentum space, which sounds vague to me. What is the more detailed computational job here, to derive the numerical initial conditions?

Also, what is the general idea of "algebraic symmetries", and "free group elements", that are used to sort the "families" of orbits into distinct "classes".

Thanks in advance!
 
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  • #2
Zero angular momentum space seems to mean that the sum of the angular momentums ∑Iw=0. That means you have to know/figure out the mass, orbit speed, and radius.

You could also calculate the gravitational force between the bodies knowing their masses and distance between them and set that equal to the centripetal force to find the velocity/orbit.
 

1. How does the method of finding periodic three-body orbits work?

The method of finding periodic three-body orbits involves solving the equations of motion for three bodies under the influence of their mutual gravitational forces. This is typically done numerically, using computer simulations or mathematical techniques such as perturbation theory.

2. Why is it important to study periodic three-body orbits?

Studying periodic three-body orbits allows us to better understand the complex dynamics of multiple body systems in space, which is crucial for predicting the motions of celestial objects and spacecraft. It also has practical applications in fields such as astrodynamics and celestial mechanics.

3. What are some challenges of using the method of finding periodic three-body orbits?

One of the main challenges of using this method is the complexity of the equations involved, which can be difficult to solve analytically. This requires the use of computational methods and can be computationally intensive. Additionally, the method may not accurately predict long-term behavior due to the chaotic nature of three-body systems.

4. Can the method of finding periodic three-body orbits be applied to any three-body system?

The method can be applied to any three-body system, but it may not always yield a periodic orbit. In some cases, the three bodies may be too close or have certain initial conditions that result in chaotic behavior rather than a periodic orbit.

5. How does the method of finding periodic three-body orbits contribute to our understanding of the universe?

Studying periodic three-body orbits allows us to better understand the fundamental laws of physics, such as the law of gravitation, and how they govern the movements of celestial bodies. It also helps us to predict the motions of objects in space, which is important for space exploration and for understanding the origins and evolution of the universe.

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