# The n-body problem revisited

#### Carl James Mesaros

Dear My Math Forum Community:
Could the n-body problem be a macroscopic manifestation of the Heisenberg and Born Uncertainty Principle? Thank you.

#### SDK

What does this even mean? The $n$ body problem asks how celestial bodies move in their mutual gravitation field. It isn't a manifestation of anything.

Moreover, almost any model for the $n$ body problem is deterministic so it can't possibly be related to say the Heisenberg uncertainty principle which is a statement about how well one can simultaneously understand a probability distribution and its Fourier transform.

It seems like you are just saying some words you heard in a class. If this isn't the case, try to ask a more pointed and coherent question.

3 people

#### topsquark

Math Team
What does this even mean? The $n$ body problem asks how celestial bodies move in their mutual gravitation field. It isn't a manifestation of anything.

Moreover, almost any model for the $n$ body problem is deterministic so it can't possibly be related to say the Heisenberg uncertainty principle which is a statement about how well one can simultaneously understand a probability distribution and its Fourier transform.

It seems like you are just saying some words you heard in a class. If this isn't the case, try to ask a more pointed and coherent question.
Actually we can talk about a Quantum n body problem. It suffers from the same Mathematic problems as the n body problem in Classical Mechanics.

For example we do have a simple case of the 2 body problem: The hydrogen atom (proton and electron) has a closed form solution. But trying to do this with deuterium (1 proton, 1 neutron, and an electron) is impossible. You would have to approximate a solution.

There is no macroscopic version of the Heisenberg Uncertainty Principle. That is a purely QM effect. Nor is there a macroscopic version of Born's Lemma. We'd have to assume that a macroscopic particle has a wave-like character and it doesn't.

I have to agree with SDK. I can't see where you would find an example of this statement in any respectable Physics paper or text. It sounds like it's made up. I would suggest you abandon your reading material and find a better source.

-Dan

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2 people

#### Carl James Mesaros

What led me to ask this question certainly from a rather insufficient knowledge of both the n - body problem and the Uncertainty Principle. The basic premise that I went on is that the position of a body in the n-body problem cannot be determined with certainty, just as it is with a body in the Heisenberg Uncertainty Principle. I must apologize for being so
presumptuous as to ask such a question without having my facts straight!

#### topsquark

Math Team
What led me to ask this question certainly from a rather insufficient knowledge of both the n - body problem and the Uncertainty Principle. The basic premise that I went on is that the position of a body in the n-body problem cannot be determined with certainty, just as it is with a body in the Heisenberg Uncertainty Principle. I must apologize for being so
presumptuous as to ask such a question without having my facts straight!
It happens. It's better to ask the question than for it to be a stopping point for you at some later time. Always ask questions!

-Dan

#### SDK

What led me to ask this question certainly from a rather insufficient knowledge of both the n - body problem and the Uncertainty Principle. The basic premise that I went on is that the position of a body in the n-body problem cannot be determined with certainty, just as it is with a body in the Heisenberg Uncertainty Principle. I must apologize for being so
presumptuous as to ask such a question without having my facts straight!
I think you are confusing a system which is chaotic with a stochastic system. The classical $n$-body problem is a deterministic, but chaotic system. Actually this isn't quite true. For $n = 2$ it is known to be integrable (hence not chaotic) and I think it has only been proved chaotic for $n =$ 3 and 4, but certainly it is widely believed to be true for all $n \geq 3$. Anyway, being chaotic does not mean that the position can not be determined. For example, we know exactly where the moon is despite its position being determined (primarily) by the gravitational field of other bodies.

The Heisenberg uncertainty principle on the other hand is not related to chaos at all. It says that if a probability distribution has a small variance, then its Fourier transform has a large variance. When applied to objects which have wave-like properties, their position/momentum obey distributions and one is the Fourier transform of the other. So this can be interpreted as saying that the position of a particle and its momentum can not both be precisely determined. This is a fundamental property of the fact that the system is stochastic.

tldr; To the best of my knowledge there is no relationship between the $n$-body problem and the Heisenberg uncertainty principle.

1 person

#### topsquark

Math Team
The Heisenberg uncertainty principle on the other hand is not related to chaos at all. It says that if a probability distribution has a small variance, then its Fourier transform has a large variance. When applied to objects which have wave-like properties, their position/momentum obey distributions and one is the Fourier transform of the other. So this can be interpreted as saying that the position of a particle and its momentum can not both be precisely determined. This is a fundamental property of the fact that the system is stochastic.

tldr; To the best of my knowledge there is no relationship between the $n$-body problem and the Heisenberg uncertainty principle.
Oooh! I like the wording on the variance thing. Nice job.

You are correct: There is no connection between the Heisenberg principle and the n-body problem.

-Dan