Why does math describe physics so perfectly?

[mpatryluk]

How is it that the numbers we get for our equations describe the laws of physics so cleanly? As a good example, take the equation for gravitational attraction.

The strength of the gravitational attraction is divided by the distance squared, AKA the distance x the distance itself.

But r^2 seems like too perfect of a coincidence. Why wouldn't it be to the power of some random non whole number that reflected the randomness of the universe?
i.e.

r^2.02934

This means there must be something special and fundamental about whole number exponentials, but I am not sure what it is or why it is.

So why exactly is it that the distance is multiplied by itself exactly once in that equation, and many others?

 

[Vanadium 50]

In the case you mention, it's because we live in 3.00000 dimensional space, and that power is d-1.

 

[Fredrik]

1/r² has the nice property that the orbit of a planet is an ellipse (if we neglect the influence of the other planets).

General relativity predicts that the orbits are not exactly ellipses. If I remember correctly, it predicts a result that corresponds to what we'd get from Newtonian mechanics with terms proportional to 1/r², 1/r³, 1/r⁴, and so on in the formula for the gravitational force.

 

[mpatryluk]

In the case you mention, it's because we live in 3.00000 dimensional space, and that power is d-1.

Ahh, i didnt expect an answer relating to the dimensionality of space, that's very interesting, and makes perfect sense.

Unfortunately, i have no idea why it makes perfect sense, and don't understand the logic of the relationship between number of dimensions and degree of exponents.

Dare i ask why? Or would that be too complicated?

Also, that d-1 rule applies as a blanket for any dimensional space? Does that mean that a 9 dimensional space would have r^8?

 

[Vanadium 50]

In this case, the strength of the field in N dimensions is inversely related to the surface of an N-sphere or radius r. In 3 dimensions, the area is 4pi r^2, and there is your 1/r^2. In 9 dimensions, the area of a 9-sphere is something like 32pi^4/105 r^8, so there you get a 1/r^8 field.