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I'm reading Mathematical Aspects of Classical and Celestial Mechanics, Second Edition by Arnold, Kozlov, and Neishtadt. It occurred to me that many people like to use third power when mention the law of universal gravitation.

For example,

$$F_ij=fracgamma m_i m_jr_ij$$ where $r_ij=r_i-r_j, gamma =textconst>0$

Is this just some conversion/tradition/habit, or does it carry any significance? i.e. the absolute value changed sign and derivative?

This is not about gravity but about making maths easier; the same idea comes up whenever you have a force in the radial direction. The reason that $bf r/r^3$ is a little easier to work with than $hatbf r/r^2$ is the following.

First, one can write it out in rectangular coordinates easily:
$$
fracbf rr^3 = frac1r^3 left( beginarrayc x\y\z endarray right)
$$

Secondly, because $hatbf r$ has a square root hidden in it which makes some manipulations trickier:
$$
hatbf r = (x^2 + y^2 + z^2)^-1/2(x bf i + y bf j + z bf k)
$$
Indeed, when doing things like differentiation, the first step is often to write $hatbf r$ as $bf r/r$.

Overall, the move to replace $hatbf r$ by $bf r/r$ reduces the number of different symbols in the mathematical expression and that is the main reason why it is regularly done. Having completed some algebra, one may then choose to present the result in terms of $hatbf r$ so as to draw attention to the overall scaling of the result, for example in order to make it clear that the gravitational law is an inverse square not an inverse cube law.

beginequation
mathbfF_ij boldsymbol=mathrm kdfracm_i m_jVert mathbfr_iboldsymbol- mathbfr_jVert^2 mathbfn_ij
tag1label1
endequation
where $:mathbfn_ij:$ the unit vector along the vector $:left(mathbfr_iboldsymbol- mathbfr_jright)$. But
beginequation
mathbfn_ij boldsymbol=dfracleft(mathbfr_iboldsymbol- mathbfr_jright)Vert mathbfr_iboldsymbol- mathbfr_jVert
tag2label2
endequation
so the third power
beginequation
mathbfF_ij boldsymbol=mathrm kdfracm_i m_jVert mathbfr_iboldsymbol- mathbfr_jVert^3 left(mathbfr_iboldsymbol- mathbfr_jright)
tag3label3
endequation

It is a matter of convenience:

Let $vecr_ij$ be a distance vector with magnitude $r_ij$ along the line connecting the masses $m_i$ and $m_j$. Then:

(I): $vecr_ij$ squared is a scalar whose value equals its magnitude squared. Proof: $vecr_ij^2=vecr_ij cdot vecr_ij=r_ij r_ij cos0=r_ij^2$.

(II): $vecr_ij$ can be written as $vecr_ij=r_ij hatr_ij$, with $hatr_ij$ being a unit vector having the same direction as $vecr_ij$. Proof: This goes as an axiom ;-)

If ones writes Newton’s law of gravity as
$$F_ij=fracγm_im_jvecr_ij^2=fracγm_im_jr_ij^2 tag1$$

Then (1) is an incomplete description of the gravitational force. Equation (1) only represents the magnitude of the gravitational force, as can be noticed by the fact that the right side of it is just a scalar.

It is a well-established observation that gravity is a force whose direction is along the line connecting the masses $m_i$ and $m_j$. As a force, gravity must be written as a vector, and therefore the right form would be

$$vecF_ij=fracγm_im_jr_ij^2 hatr_ij tag2$$

From (II), equation (2) can be written as

$$vecF_ij=fracγm_im_jr_ij^2 fracvecr_ijr_ij=fracγm_im_jr_ij^3 vecr_ij tag3$$

Typing vectors and unit vectors with arrows and “hats” atop (as in current case) is a little cumbersome. To solve for typing vectors with arrows, one can select other options as boldfacing them for example. To avoid unit vectors one uses the convention $fracvecrr$, i.e., the choice of (3) is a convenient form of writing these sort of equations, especially for works containing a huge amount of text.

It is not only easier (if one uses a WYSIWYG text editor of typing machine) to type equation (3) using these conventions, but visually cleaner as well. Just check:

beginequation
mathbfF_ij boldsymbol=mathrm γdfracm_i m_jmathbfr_ij^3 mathbfr_ij
tag4
endequation

As was pointed out by @AndrewSteane, there are even mathematical advantages by using this notation, hence, it is just more convenient to use the $fracbf rbf r^3$ notation.

The problem with this, however, is that people (almost) always is introduced to Newton’s law of gravity in its non-vector form (equation 1) and is expecting to find the inverse square law in a first glance, therefore, it may look odd for beginners until one gets used to it some point in future ;-)