What is the Three-Body Problem in Physics and Why is it Considered Unsolvable?

In summary, there are a few examples of ODEs in physics that are unsolvable analytically, such as a pendulum and coupled springs. However, the Three-Body Problem is considered the most challenging and has yet to be solved for all situations.
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Hi, I'm trying to come up with some examples of ODEs (not PDES) in physics that are unsolvable analytically, and for some reason I'm drawing a blank.

There are a few obvious examples from classical mechanics such as a pendulum and coupled springs etc... but beyond that I can't seem to recall coming across any.

Can anyone think of some ODEs that describe interesting physics and whose derivation can be understood without too much background knowledge?
 
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1. What are Unsolvable ODEs in physics?

Unsolvable ODEs (ordinary differential equations) in physics refer to mathematical equations that describe the behavior of physical systems and cannot be solved using known mathematical methods. These equations often involve complex variables and their derivatives, making it challenging to obtain a closed-form solution.

2. What causes an ODE to be unsolvable?

There are several factors that can make an ODE unsolvable in physics. One common reason is the nonlinear nature of the equation, which means that the variables and their derivatives are raised to powers other than 1. Another factor could be the presence of boundary conditions or initial values that do not have a closed-form solution.

3. Are there any techniques for solving unsolvable ODEs?

While there is no general method for solving unsolvable ODEs, there are some techniques that can be used for specific types of equations. These include numerical methods, series solutions, and perturbation methods. However, these techniques may not always provide an exact solution and may require some approximations.

4. How do unsolvable ODEs impact physics research?

Unsolvable ODEs can have a significant impact on physics research as they limit our ability to fully understand and model physical systems. They can also hinder the development of new theories and technologies. As a result, scientists often need to find alternative approaches or make simplifying assumptions to work with these equations.

5. Can unsolvable ODEs be solved in the future?

It is difficult to predict if unsolvable ODEs in physics will ever be solved completely. While advancements in mathematics and computing have allowed us to solve more complex equations, there will always be some equations that are beyond our current capabilities. However, ongoing research and development in these fields may lead to new techniques for solving these equations in the future.

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