Three-Body Problem with Symmetry: Finite Masses and Spring Constants Considered

  • Thread starter Hans Stricker
  • Start date
In summary, the problem of three bodies with two having mass M and one having mass m can be easily solved when one of the masses is considered infinite or when the spring constant is 0. However, even when the masses are not infinite or equal, there is still hope for a closed or approximate solution using the lagrangian or hamiltonian formalism, or by considering conservation of momentum.
  • #1
Hans Stricker
5
0
Consider the problem of three bodies two of which having mass M, one of them having mass m. Body m is in the middle between the other two, coupled to them by two equal linear springs in rest. Now fix the two bodies M and move body m for a small amount perpendicular to the connection line. Now let loose the three of them.

The solution is trivial when we assume m to be finite and M to be infinite (or vice versa), or when the spring constant is 0.

But the problem is so symmetric, that there might be hope to get a closed or approximate solution even for finite m >> M, M >> m, or even M = m?
 
Physics news on Phys.org
  • #2
Definitely. Its a trivial problem in the lagrangian or hamiltonian formalism. If you are not familiar with those, think about conservation of momentum---and I think you can still get the right answer (and definitely the right idea of the answer).
 

1. What is the simple three-body problem?

The simple three-body problem is a classical mechanics problem that involves studying the motion of three point masses in a gravitational field, without taking into account the effects of other external forces. The three bodies interact with each other through the force of gravity, and the problem is to determine their positions and velocities at any given time.

2. How is the simple three-body problem solved?

The simple three-body problem is solved using mathematical equations and numerical methods. The most common method used is the numerical integration of the equations of motion, which involves calculating the positions and velocities of the three bodies at discrete time steps. Other methods, such as analytical and perturbative solutions, are also used depending on the specific scenario.

3. What are the main challenges in solving the simple three-body problem?

One of the main challenges in solving the simple three-body problem is that there is no general analytical solution for all possible scenarios. This means that each situation must be studied separately and different methods must be used to find a solution. Another challenge is that the motion of the three bodies can be chaotic and difficult to predict, making it challenging to accurately model and solve the problem.

4. What are some real-life applications of the simple three-body problem?

The simple three-body problem has many applications in the fields of astrophysics, celestial mechanics, and spacecraft trajectory design. It is used to study the motion of celestial bodies, such as planets, moons, and asteroids, in the solar system. It is also used to design spacecraft trajectories and understand the dynamics of satellite systems.

5. Are there any limitations to the simple three-body problem?

Yes, there are some limitations to the simple three-body problem. It assumes that the three bodies are point masses and do not have any physical dimensions. It also does not take into account other external forces, such as atmospheric drag or tidal forces, which can have significant effects on the motion of the bodies in real-life scenarios. Additionally, the problem becomes more complex when there are more than three bodies involved, making it difficult to generalize the results.

Similar threads

  • Classical Physics
Replies
1
Views
676
  • Classical Physics
Replies
2
Views
775
Replies
30
Views
2K
  • Classical Physics
Replies
3
Views
651
  • Classical Physics
4
Replies
131
Views
4K
  • Classical Physics
3
Replies
84
Views
4K
Replies
3
Views
960
  • Classical Physics
Replies
3
Views
645
  • Introductory Physics Homework Help
Replies
29
Views
823
Replies
86
Views
4K
Back
Top