### Social Data and Nonlinear Behavior

To test the *abstraction of newtonian kinematics* (ANK), we use Gallup polls on long-term trends in social opinion for several important concerns in the United States^{19.20}. The data is presented in Fig. 1. The left vertical axis shows the number of states for the first set (left legend) of data and the right axis shows the percentage for the second (right legend). We also include data from PEW Research on recent environmental and climate data surveys^{21}.

The plot indicates the evolution of states on their opinion to lift a ban or increase a ban. We divide the data into two sets of information: (1) data showing which states support (or prohibit) the issues and (2) poll data on popular topics. The former are indicated by thick lines in the figure, and the circles at the end of these lines indicate a constitutional amendment or Supreme Court decision. This latest dataset includes various trends: percentage of population with no religious affiliation, willingness to vote for a female president, desire to vote for a black president, ideal family size (family size), and family preference. woman to work. outside the home (family life)^{22.23}.

Two examples illustrate linear, polynomial and logistic fits to the data, see Figs. 2 and 3. Additional figures showing the complete set of soundings, are shown in Figs. 6 and 7. The full set of adjustments is presented in Appendix C. We also show the nonlinear behavior of each survey by comparing the (R^2) for linear, 2nd order polynomial and logistic fits. A brief description of the poll is given in Table 1. A linear fit might initially describe increasing (or decreasing) data from Gallup polls. Still, the trend is usually better fitted by a 2nd order polynomial or logistic curve, because opinion starts to change faster. The logistic curve,

$$begin{aligned} P

(1)

describes an evolving phenomenon and the value *a* characterizes data leveling. The maximum values are (a=100) for percentage data, or (a = 50) for state data. Values *k* and (for) fit the data. Appendix B provides additional information on the logistic adjustment of these data.

### Newton’s laws of motion

We define the abstraction of “distance” or “position” of people’s opinion from a specific position, (x longrightarrow widetilde{x})abstraction from mass or inertia to change of opinion from a norm (m longrightarrow widetilde{m})the abstraction of the rate of change of opinion or speed, (v longrightarrow widetilde{v})and the abstraction of momentum, the product of mass and velocity, (p longrightarrow widetilde{p}). A natural extension of the Newtonian kinematics abstraction is Newtonian dynamics (study of force). The abstraction of force, push, attraction or influence “acting” on a population having an opinion, (F longrightarrow widetilde{F}). The notion of time remains the same in the abstraction.

Energy can also be considered in the abstract. Potential energy can invoke the change of opinions (widetilde{V}) and kinetic energy (widetilde{T}), the energy associated with movement or change of opinion, towards or away from a norm. The lack of physical dimensions means that the abstraction is meaningful for speed but not a vector quantity such as speed. The abstraction represents kinematics created from a constant force, no different from the equations of motion generated from the Lagrangian of an object in a (nearly constant) gravitational potential such as that found near the Earth’s surface. A sketch of the abstraction is given in Fig. 4. Here we see Galileo’s famous experiment reproduced. He climbed to the top of the Tower of Pisa and then dropped a massive, less massive ball. The contemporary thought was that the more massive ball would fall faster. Yet instead the two fell at the same rate, dispelling the age-old Aristotelian theory of gravity that objects fall at a rate proportional to their mass. For completeness, the abstraction of the three classical laws is presented in Table 2. Appendix A deals with the abstraction of Newton’s 3rd law.

The equations of motion of an object subjected to a constant force become,

$$begin{aligned} x

(2)

with natural abstraction,

$$begin{aligned} {widetilde{x}}

(3)

and so the acceleration and the force become,

$$begin{aligned} widetilde{a}= frac{2 cdot (widetilde{x}

(4)

showing that force is proportional to mass.

### Extracting data to produce acceleration and force

Acceleration and force are taken from Fig. 1 using logistical adjustments for each survey. The weather *you*, or year, is marked at the inflection point of the logistic curve. This is shown in Figs. 2 and 3 for the years 1915 and 2005 respectively. The acceleration is determined at the point of inflection and at the moment *you* is measured from the first query data point (t_1) at the inflection point (t_{inf})Where (t ~=~ t_{inf} – t_1). Likewise, the position (widetilde{x}

(5)

Table 3 provides the values for *you* (year), (widetilde{x}_{inf}) and (widetilde{v}_{inf}) at the inflection point of every social and environmental poll. Acceleration (widetilde{a}) is determined using the equation. (4) at the inflection points for each poll and year verses plotted in Fig. 5. Similarly, the term force (widetilde{F}= mwidetilde{a}) is determined using a (unitless) abstraction of mass, or a value normalized to the population of the United States in 1910, (widetilde{m}_{1910} = 1).

Errors for each fit are determined using the 95% confidence interval determined using the standard MATLAB fitting routine (MathWorks(circledR)) and constraining the curve to a maximum of 50 or 100%. This error is then used to determine the uncertainty of (t_{inf}). Uncertainties for (widetilde{x})and (widetilde{v}_o) are determined from Gallop poll statistics, ranging from 1000 to 4000 for each data point. Finally, standard error propagation rules are applied to Eq. (4) to determine the error for the acceleration and the normalized force.