Why are some equations not solvable?

[DecayProduct]

What I mean is, I have read how many equations that model physical phenomena often aren't directly solvable. Instead they are approached by successive approximations. And we get answers that are close enough.

My question is, if a physical phenomenon seems to follow a mathematical pattern, why is it we can only approximate the math, instead of solve an exact set of equations. Obviously the phenomenon is not doing successive approximations to approach the actual state it ends up in, it just does it. If a physical system can "do" something, why is there not an exact equation to describe it? I hope the question makes sense.

 

[Mentallic]

A very interesting thought

I believe that the mathematics required to describe all the natural processes is much more complicated than we can fathom. Imagine for e.g. a projectile: take into consideration all the possible forces at work. The fluctuations of drag, gravity, spin of the projectile creating pressure differences etc. All these variables ARE reality/physical phenomena. Our math to describe a projectile's motion attempts to consider only the most significant forces and thus simplifies much of this because most of the other forces would be negligible.

Now I know this doesn't really answer your question, but I'm just trying to make the point that even if the math doesn't give an exact solution, this is the least of our concerns as the math has already been simplified and is thus approximate in the end anyway.

Also, I believe that we haven't "discovered" all the math that could possibly exist and as such we can't give exact solutions just yet.

e.g. let's say some equation that describes a physical phenomena is given by x=2sin(x)
So we can't give an exact solution to this problem, but I believe that in the undetermined future, there will be a way to solve x in this equation as an exact form.

 

[Phrak]

What I mean is, I have read how many equations that model physical phenomena often aren't directly solvable. Instead they are approached by successive approximations. And we get answers that are close enough.

My question is, if a physical phenomenon seems to follow a mathematical pattern, why is it we can only approximate the math, instead of solve an exact set of equations. Obviously the phenomenon is not doing successive approximations to approach the actual state it ends up in, it just does it. If a physical system can "do" something, why is there not an exact equation to describe it? I hope the question makes sense.

I've been bashing my head over one for better than a month--or has it been 14 years?
Could you give some examples?

 

[DecayProduct]

I've been bashing my head over one for better than a month--or has it been 14 years? Could you give some examples?

A three-body problem in celestial mechanics is the one that got me thinking on this route. Apparently, a system involving two bodies is fairly simple to solve using the laws of Newton and Kepler, but add a third object to the mix, and all hell breaks loose. Well, all these bodies are doing what they are doing, and they are doing it directly. There is no approximation in the real system. So is it that either mathematics doesn't describe the world accurately, or we just don't know how to do the right math?

 

[maverick_starstrider]

In a nutshell: Because of Godel's incompleteness theorem we know that some of these insolubly things will never be solubly which often leads to a lot of people making the sometimes erroneous assumption that just because we haven't found a closed form solution yet then there must not be one. But occasionally one of these problems gets solved. As a physicist the best example that comes to mind is Onsager's solution for a 2D Ising system, if you read it it's a huge amount of math but he did find an analytical solution.

P.S. It is not correct to say that a series solution is an approximate solution. The infinite taylor series 1+x+\frac{x^2}{2}+... is EXACTLY e^x. No approximation has been made. It is only when you truncate terms that it is an approximation

 

[maverick_starstrider]

I should also comment that the current state of math and physics is that they are two entirely divorced entities. Math is an entirely self-consistent abstract construction, it makes no promises that it is an all encompassing model of our physical universe. Therefore, philosophically, there is really no requirement that all physical phenomena must be exactly mathematically modelable using our most common axiomatic system of math (calculus of the real and complex numbers).

For example, when applying math to physics we often toss aside "unphysical" solutions to a given equation. There is no MATHEMATICAL reason to do this. Now you can say that maybe there is a deeper physical theory which will be entirely mathematically consistent and provide a mathematical reason for discarding those solutions but as it is right now we simply remove them.

 

[Phrak]

maverick. Given what you've said, I think a good question here is "Can the three body problem be expressed as an infintite sum?

I think this is equivalent to asking, given all the initial conditions, can one define x₃ = f(x₁, x₂, x₃, y₁, y₂, y₃, z₁, z₂, z₃, t) as an infinite sum, for instance?

In which case, it becomes the same manner of function as the expontential function, doesn't it?