The Falsifiability of Newton’s Laws

Consider the following thought experiment: we setup an experiment consisting initially of objects and known forces, and we run Newton’s laws to predict the motion of the objects. We then run the experiment, and the observed results do not match the predicted calculations. Can we just attribute the discrepancy to a new force that we haven’t yet studied, and still satisfy Newton’s laws with this just-defined unknown force? Can we explain all discrepancies with Newton’s laws in this way? Then, are the laws really saying anything materially? Can we think of forces as convenient mathematical devices to represent “contributions of mass times acceleration”?

In other words, taken away from the specific laws of gravity and the other forces, are Newton’s laws themselves falsifiable?

In this article, we construct a specific experiment that actually can falsify Newton’s laws. This setup is based on one of the main consequences of Newton’s laws that is often touted early on in physics classrooms: superposition.

The Experiment

Consider a setup as follows: identical objects A and B are initially placed a distance $r$ apart. We may WLOG define a coordinate system so that their midpoint is at the origin. They are released and their motions are tracked. From symmetry, we need to track one of the objects only, say the motion of B, where a function is recorded, $x_{B,d}(t)$ , transcribing the position when the objects are initially $d$ apart. This may not be the same thing as the motion once the position of the object reaches $\frac{r}{2}$ , since the values of velocity and acceleration (which are zero at the start of the experiment) may differ — in fact, due to higher-order derivative dependence in Newton’s laws, we expect that these will be different. In fact, let’s do the following: we take data on object B of the form $\left( x_{0B},t,x_{B},v_{B},a_{B} \right)$ , noting the initial distance apart, the time, the position at that time, the velocity at that time, and the acceleration at that time.

Now, place an object C identical to both, so that all three objects initially form an equilateral triangle, with side length $r$ . We may WLOG define a coordinate system so that A is at the origin, all initial positions are in the quadrant $x,y \geq 0$ , and C is on the $x$ -axis (so C is at $(r,0)$ ). From Newton’s laws and the data above, can we derive what the motions of A, B, and C must look like? This experiment can then reliably falsify the laws.

Let the masses of the objects be $m$ . In the first experiment, the forces on A and B were symmetric by design. We have, for B, that

$\displaystyle ma_{B} = F_{net,B} = F_{A \rightarrow B}.$

We only know that the force must be a function of distance from A (not position on an arbitrary coordinate system), velocity of B, and acceleration of B — we don’t know what the form of the function is. The distance from A in our coordinate system is $2x_{B}$ , so $F\left( 2x_{B},v_{B},a_{B} \right) = ma_{B}$ . We will let the data values from the first experiment be subscripted with “1”, so that $F\left( 2x_{B},v_{B},a_{B} \right) = ma_{B1}$ . This is the form of the force of any of the objects A, B, or C on any other. We then have $\overline{F}\left( 2x_{B},v_{B},a_{B} \right) = ma_{B}\widehat{x}$ , with the bar indicating the vector quantity.

For the setup with A, B, and C: by design, this is symmetric. Thus, the motions of A and C should be symmetrical to that of B. If they are not, then there is an error in the setup of the experiment, so we now focus only on the motion of B. Take our coordinate system so that A is always at $(0,0)$ . By symmetry, B must be at $\left( x_{B},y_{B} \right)$ with $\frac{y_{B}}{x_{B}} = \tan 60^{\circ} = \sqrt{3}$ , so B is at $\left( x_{B},x_{B}\sqrt{3} \right)$ . Then by the symmetry C is at $\left( 0,2x_{B} \right)$ . The forces on B are $F_{A \rightarrow B}$ and $F_{C \rightarrow B}$ , and by symmetry their magnitudes are identical, so

$\displaystyle \overline{F_{net}} = \overline{F_{A \rightarrow B}} + \overline{F_{C \rightarrow B}} = F\left( \widehat{x}\cos 60^{\circ} + \widehat{y}\sin 60^{\circ} \right) + F\left( \widehat{x}\cos\left( 180^{\circ} - 60^{\circ} \right) + \widehat{y}\sin\left( 180^{\circ} - 60^{\circ} \right) \right)$

$\displaystyle = F\sqrt{3}\ \widehat{y}.$

Thus,

$\displaystyle ma_{Bx} = F_{net,x} = 0,$

but this is clear from symmetry (to be clear, since this can seem contradictory: the second derivative with respect to time of the position of B *relative to the position of A* is 0.)

Thus, we use the $y$ coordinate:

$\displaystyle ma_{By} = F_{net,y} = F\sqrt{3} = ma_{B1}\sqrt{3}$

$\displaystyle \rightarrow a_{By} = a_{B1}\sqrt{3}.$

Let us take a moment to clear this up: the $F$ comes from the previous experiment and is the force when two of the objects are some distance say $r^{'}$ apart and at some velocity $v$ . (We know force cannot depend on time or position alone; the dependence on position comes from our imposition of a coordinate system and is thus valid only in our specific coordinate system.) In that experiment, the distance between the objects is just $2x_{B}$ . Thus, we have the following concrete prediction:

1. Perform the first experiment and take the data above: $\left( x_{0B},t,x_{B},v_{B},a_{B} \right)$ . Note down the acceleration of B as a function of $x_{B}$ and $v_{B}$ ; this will be a function $a_{B1}\left( x_{B},v_{B} \right)$ , not a relation (there will not be separate dependence on $x_{0B}$ or $t$ .)

2. Prediction: the acceleration of B with respect to A, or the second derivative of the position of B with respect to A, in the second experiment in the y direction, call it $a_{By}$ , will be equal to $\sqrt{3} a_{B1}\left( x_{B},v_{B} \right)$ , where $x_{B}$ is the position of B with respect to A in the x direction.

This settles the question of falsifiability, and it could be a useful example in a pedagogical setting.

Written July and August 2020.

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