Gsyfuy

In contrast to the other answers, I am going to go a different direction. This is not a question of mathematics, where we can just say "oh yeah just redefine your variables." This is a physics question.

Going further, The OP is assuming $m$ stands for the mass of an object, not some arbitrary proportionality constant that ends up being the mass. Therefore, having a law like $F=km^2a$ ($k$ as just some constant for unit's sake) would not leave the universe unchanged. This is saying, for example, that if we double the "amount of stuff" associated with an object, that we would need $4$ times as much force to achieve the acceleration. This is certainly not true in our universe today, so we cannot say this law would be found in a similar universe.

There is also the question of the relation between inertial mass and gravitational mass, which we take to be equal. i.e. we know the force of gravity to be proportional to the mass of the object that gravity is acting on. Therefore, if forces behaved like $F=km^2a$, then we would find that, near the surface of the earth, that the acceleration due to gravity would actually depend on the mass of the object! Certainly this is not the same as our universe.

I understand the other answers' attempts to try and teach a lesson about definitions in physics, and how there are some things that are around purely by convention. But this is not one of those cases I believe. The OP is starting from $m$ means mass. So we should start there too and then see what that means for the proposed new "force laws".

This would be like me saying that it would be fine to take what we understand kinetic energy to be and rewrite it as $mv$. Sure, I could say that I have "redefined" the velocity to be $v'=sqrt2 v$. And this is valid mathematically if we stay consistent with this in the rest of our equations. But if I start off by saying "let $v$ be the velocity of the object, i.e. the rate of change of displacement", then I cannot "redefine" my equations so that kinetic energy is $mv$.

"Force" is just a word. You don't need it at all to do Newtonian mechanics, if you start from conservation of energy and momentum and the idea of gravitational potential energy. "Force" is then just a mathematical invention to keep track of how the momentum is transferred between different bits of stuff, when things move around.

The world doesn't change just because humans invent random new words.

Then we would recall mass as $m' = m^2$ or $m' = 2m $. The whole idea of Newton's second law is that force and acceleration are proportional. We can then define inertial mass as the constant factor between them. So the world would look exactly the same.

Writing $F=kma$ and putting $k=1$ or $k=2,3...$ will not change the consequences of how things would actually behave.

All it does is put up a scaling factor to the value. But due to convenience, we agree on $k=1$ implying $ F=ma$.

Writing $a to 2a$ instead does not change anything but the convention you have to abide by, for it is the same argument as writing $kma=F$.

If $k=m$ as you have included ($F=m^2a$), things would be the same if you assume $m^{'}=m^2$ and assume that $m^{'}$ is the actual mass.

Besides, $m^{'}$ is meant to be a constant of proportionality, and so it makes no difference to the consequences of physics. $F=ma$ is based on the understanding that $F$ is proportional to $a$.

That is rather interesting, I've often thought of what would happen if you change a known relationship. If you doubled the mass, in your mass squared relationship, it would require four times the force to accelerate it the same. Another interesting change of relationship that happens in the real world is a rate of acceleration that increases or decreases over time. Try distance equals time cubed.

No matter the transformations you make into Newton's second law, as long as it's reverse transformation admits $rm a$ to be isolated.

$F(r,v,t)$ in Newton's second law as expressed by equation
$$ F(r,v,t)=ma tag{1}$$

Is a very abstract concept that must be adapted to solve the observed motions in the universe. More importantly, $F(r,v,t)$ is an abstraction of some kind of position, time, and velocity function that satisfy the experimental observations. Since equation (1) can be written

$$F'(r,v,t)=a tag{2}$$

and $F'(r,v,t)$ has no form a priori, any transformation in (1) that can be reversed into (2), will give the same results as Newton's second law. However, What one would call force is not necessarily what we currently call it but they should be easily converted into one another.

For example, lets analyze the definition of gravitational force in the advent that Newton's second law were

$$F(x,v,t)=2ma^2 tag{3}$$

In this particular case, the definition of gravitational force $F_G$ here must satisfy (at least) the following observed facts:

1. (By Galilee): “The acceleration of a falling body doesn’t depend on its mass”;


2. (By Kepler): “The motion of planets around the sun are ellipses…”

The simplest definition of $F_G$ that satisfies both conditions is

$$F_G=frac{4m^2}{r^4} tag{4}$$

However, with time, people would realize that $sqrt{F_G}$ was actually a better physical quantity, since more intuitive generalization of it, based on geometrical insights like Gauss law would arise. The same goes for the use of $m$ instead of $2m$.

One of the things that allowed us to invent and understand Newton's laws, which govern the motion of classical objects, is that they have the mathematical property of linearity. Linearity is a class of mathematical behaviors that people tend to be good at. If there are two or more forces acting on an object, the net force is the simple vector sum of the forces. If there is a system made of several masses, the total mass is the simple arithmetic sum of the individual masses.

For example, an Aristotelian misconception that was properly refuted until the Renaissance is that "heavier objects fall faster." An argument against this, which I believe was due to Galileo, is a thought experiment, as follows. Imagine three identical cannonballs dropped from the same height: they should fall at the same rate because they all have the same mass. But tie two of them together with a low-mass connector, like glue or silk, and you have two objects whose masses are different by a factor of two. Can connecting two heavy masses with a silk thread change their fall time by a factor of two? It's a clever thought experiment.

In your formulation, two identical masses $m$ accelerating in tandem with the same acceleration $a$ would have the same amount of force,

$$
F = m^2 a,
$$

acting on them. But if you tied them together so that you had a single object with mass $2m$ being acted upon by a force $2F$, then you would have

$$
2F = (2m)^2 a
$$

which predicts a different, smaller acceleration. Either you'd have to come up with an additional non-linear rule for adding forces, or you'd be describing a world that is different from the one where we live.