Example of integrable system

In summary, the problem is to find 3 conserved quantities in involution for a system of 3 unit masses moving around a unit circle with a repulsive force between them. The total mechanical energy and the total angular momentum are integrals of motion, and the third conserved quantity could be the total linear momentum, the total momentum along the radial direction, or the total angular momentum around a specific axis.
  • #1
luisgml_2000
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Homework Statement



3 unit masses move around a unit circle. Between them there is a repulsive force that decreases with the angle between the particles as

[tex]
F_{ij}=\exp(-(\theta_i-\theta_j))
[/tex]

The problem is to find 3 conserved quantities in involution.

Homework Equations





The Attempt at a Solution



From physical considerations we can see that both the total mechanical energy and the total angular momentum are integrals of motion, but I'm missing the 3rd one. Any idea?
 
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  • #2


Hi there,

Thank you for your post. It's great to see that you are working on a problem related to circular motion and forces. I have a few ideas for the third conserved quantity that you can explore.

One possibility is the total linear momentum of the system. Since the particles are moving in a circle, their motion can also be described as a combination of linear motion along the tangents of the circle. Therefore, the total linear momentum of the system should also be conserved.

Another possibility is the total momentum along the radial direction. Since the particles are moving around a circle, their distance from the center of the circle remains constant. This means that the total momentum along the radial direction (perpendicular to the tangents) should also be conserved.

You can also explore the total angular momentum around a specific axis, such as the center of the circle or one of the particles. This would depend on the specific setup of your problem and the axis of rotation.

I hope these ideas give you some direction to find the third conserved quantity. Keep exploring and don't hesitate to reach out if you have any further questions.

Best of luck with your research!
Scientist
 

1. What is an integrable system?

An integrable system is a mathematical model that describes a physical system in which the equations of motion can be solved exactly, meaning that the solution can be expressed in terms of simple functions such as polynomials or trigonometric functions.

2. Can you give an example of an integrable system?

One example of an integrable system is the harmonic oscillator, which describes the motion of a mass attached to a spring. The equations of motion for this system can be solved exactly using trigonometric functions.

3. How do integrable systems differ from non-integrable systems?

Integrable systems are characterized by the existence of a sufficient number of conserved quantities, which allow for the exact solution of the equations of motion. Non-integrable systems, on the other hand, do not have enough conserved quantities and therefore cannot be solved exactly.

4. What are the applications of integrable systems?

Integrable systems have a wide range of applications in physics, engineering, and mathematics. They are used to model various physical systems such as the motion of planets, the behavior of fluids, and the dynamics of particles. They also have practical applications in fields such as cryptography and data encryption.

5. How are integrable systems studied and analyzed?

Integrable systems are studied using a combination of analytical and numerical methods. Analytical methods involve deriving equations and solving them using mathematical techniques. Numerical methods involve using computers to simulate the behavior of the system and analyze its properties. Both approaches are important in understanding the behavior of integrable systems.

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