Modelling shifts in social opinion through an application of classical physics

This paper explores the abstraction of classical physics and applies several metrics that explore the evolution of social opinion. These metrics include an abstraction of Newtonian kinematics: mass, position, speed, acceleration, and Newtonian dynamics, an abstraction of force. Poll data is fit to a 2nd-order polynomial and a logistic function. These fits are used to understand the acceleration of opinion shift, and we explore recent social, cultural, and environmental trends, such as views on global climate change. We compare our results with the evolution of communication technologies and time spent on devices over the past 120 years. We show that the model connects the evolution in opinion with an abstraction of a Galilean concept: acceleration is independent of mass. Finally, we discuss the model of social polarization and the non-linear effect of media such as echo chambers.

decisions in a large population (millions) over an extended period (at least 10 years). In this vein, the modeling of many interactions resembles physical systems where myriad small interactions produce a net effect.
The influence of mass media has dramatically increased, especially in recent years 15,28 . This recent influence begs the question, how much does it influence our opinions? What are the underlying forces that motivate opinion change? Are the drivers for opinion change individuals who decide on their own after learning about a topic, or is opinion change primarily media-driven and opinion shift occurs because most individuals are influenced by media personalities that convince the listener what they should believe? The answer is likely a combination of these two views. Several authors have done extensive research on this topic [16][17][18] . The question remains: how much do others influence us?
This paper poses a new approach to understanding social change using an abstraction of Newtonian laws. Though perhaps unorthodox approach, we show that abstraction of physical laws and principles produces dynamics and can explain long-term cultural trends and and opinion shifts. We parallel classical kinematics and dynamics, though we do not focus on individual dynamics, only the ensemble effects. Though quantum mechanics can treat the micro view of physical reality, this tool is generally inappropriate to explore phenomena at larger scales where classical mechanics is appropriate. We use this framework to explore both long-term trends in culture and opinions on environmental goals, namely views towards climate change.
One of the older polls in the USA includes the Gallup Polls, founded by George Gallup in 1935 and continues to provide numerous polls on various social opinions. This poll offers a long baseline of information for this study 19 .
We proceed by (1) formulating a generalization of Newton's Laws or an abstraction of Newtonian kinematics to study abstract motion. In "Methods", we describe the abstraction of classical physics and present poll data from the Gallup polls. Next, we present (2) a comparison of the model with information extracted from curves that fit the poll data ("Results"). Finally, in "Discussion", we discuss the results in the context of social models and explore other areas where the abstraction of classical mechanics may help study non-linear social behavior.

Social data and non-linear behavior.
To test the abstraction of Newtonian kinematics (ANK), we use Gallup polls of long-term trends in social opinion for several significant concerns in the USA 19,20 . Data is shown in Fig. 1. The left vertical axis displays the number of states for the first set (left legend) of data and the right hand axis shows percentage for the second (right legend). We also include PEW Research data on recent polling on environmental and climate data 21 .
The plot indicates the evolution of states on their opinion to either remove a ban or increase a prohibition. We divide the data into two sets of information: (1) data showing states that support (or prohibit) issues and (2) data of polling of popular topics. The former are indicated with thick lines in the figure, and circles at the end of these lines indicate a constitutional amendment or Supreme Court decision. The latter set of data include various trends: percentage of the population with no religious affiliation church membership, willingness to vote for a woman for president, desire to vote for a black president, ideal family size (Family Size), and woman's preference to work outside of the home (Family Life) 22,23 .
Two examples demonstrate the linear, polynomial, and logistic fits to the data, see Figs. 2 and 3. Additional figures showing the complete set of polls, are shown in Figs. 6 and 7. The full set of fits are shown in the Appendix C. We also show the non-linear behavior of each poll by comparing the R 2 for linear, 2nd-order polynomial, and logistic fits. A short description of the poll is given in Table 1. A linear fit might initially describe rising (or Newton's laws of motion. We define the abstraction of "distance" or "position" of people's opinion from a specific position, x −→ x , the abstraction of mass or the inertia to change of opinion from a norm m −→ m , the abstraction of the rate of change opinion or speed, v −→ v , and the abstraction of momentum, the product of the mass and speed, p −→ p . A natural extension of the abstraction of Newtonian kinematics is Newtonian dynamics (study of force). The abstraction of force, the push, pull or influence "acting" on a population with an opinion, F −→ F . The notion of time remains the same in the abstraction. Energy can also be considered in the abstraction. Potential energy can invoke the change of opinions V and kinetic energy T , the energy associated with the motion or change in 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 velocity. The abstraction represents kinematics created from a constant force, not different than the equations of motion generated from the Lagrangian of an object in a (nearly constant) gravitational potential such as the one 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 ascended to the top of the Tower of Pisa and subsequently dropped a massive and less massive ball. The contemporary thought was that the more massive ball would fall faster. Still, instead, both fell at the same rate, dispelling the centuries-old Aristotelian theory of gravity stating that objects fall at speed proportional to their mass. For completeness, the abstraction of the three classical laws are shown in Table 2. Appendix A discusses the abstraction of Newton's 3rd law.
The equations of motion for an object under constant force becomes,  www.nature.com/scientificreports/ with the natural abstraction, and therefore the acceleration and force becomes,   Extracting the data to produce acceleration and force. Acceleration and force are extracted from  Table 3 provides the values for t (year), x inf and v inf at the inflection point for each social and environmental poll. The acceleration a is determined using Eq. (4) at the inflection points for each poll and plotted verses year in Fig. 5. Likewise, the force term F = m a is determined using a (unitless) abstraction of mass, or value normalized to the USA population in 1910, m 1910 = 1.
The errors for each fit are determined using the 95% confidence interval determined using the standard MATLAB fit routine (MathWorks ) and constraining the curve to a maximum of either 50 or 100%. This error is subsequently used to determined the uncertainty in t inf . Uncertainties for x , and v o are determined from statistics in the Gallop polls, 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.

Results
The evolution of acceleration (circles) and normalized force (squares) are plotted in Fig. 5 along with their respective uncertainty for each poll data. One data outlier is seen in Fig. 5, the 2006 data point (Same-Sex Marriage). This point is characterized by a short base (short time between the first and last poll data) producing a large fit error. Two additional opinion polls were considered and plotted. These two points complement S1 and S6, polling for interracial marriage (inflection point at 1992) and the legalization of marijuana (inflection point at 2026). The 2078 point, the most projected point, represents polls for stricter environmental laws (E2) and results from data fitting to the slow rise of the logistic curve, producing an inflection point projected decades into the future. Similarly, the non-religious point (O1) is fitted to polling data that is slow rising, also producing an inflection point far off into the future.
Within statistical error, the evolution of the acceleration for the three data sets (Table 1) is flat, 3 × 10 −5 ± 0.0005 percentage points per year. Three environmental polls (data at the bottom of Table 1) have been included in the analysis. The normalized force increases by 0.002 ± 0.0025 percentage points per year (blue, solid line), or also flat within error. The increase, though larger than the acceleration, is nonetheless statistically Table 2. Newton's Laws and an abstraction.

Law Description Equation Abstraction
1st Law of Inertia: an object at rest will stay at rest, and an object in motion will continue at a constant speed and in a straight line unless acted on by a net external force The opinion of a population will remain as it is unless a net driving force (influence) acts upon it 2nd The rate of change of momentum of a body over time is directly proportional to the net driving force and moves in the same direction as the applied force The rate of change of the abstract momentum is directly proportional to the external influence (force) applied and occurs in the same direction as the force 3rd All forces between two objects exist in equal magnitude and opposite direction. If one object A exerts a force F A on a second object B, then B exerts a force F B on A, and the two forces are equal in magnitude and opposite in direction All influences acting on a population exists in equal magnitude and opposite direction: if one object A exerts a force F A on a second object B, then B exerts a force F B on A, and the two forces are equal in magnitude and opposite in direction  1959 1994 2002 1949 1952 1938 1960 1937 1974 2009 1994 2009

Discussion
The time spent on this new media format may provide users with a critical level of selective reinforcement. However, this change did not increase in acceleration. Though a slightly larger increase is seen in the force term, it is also statistically insignificant (Fig. 5). This abstraction of Newtonian mechanics is used to study the rate of opinion change of several social and environmental topics. The main finding is the rate of change of opinion has remained flat over the past 120 years. Returning to the physical analogy, a force provides acceleration on an object, the equivalent of a second-order derivative with respect to time, or rate of change of position with respect to time. This same dynamics is found in the rate of change of opinion with respect to time. The zero acceleration change might indicate that social media does not play a role in the changing opinion as a whole. On the other hand, micro-processes, including polarization of opinion, would be averaged out and therefore remain undetected in this analysis.
Areas to explore. The polarization of opinion can result from drivers, including media storms 29 , feedback loops (media affecting personalities and personalities affecting media attention), and information "echo chambers" [30][31][32] . Fits using a 2nd-order polynomial or the logistic function characterize the average acceleration of evolving opinion, but two polarized opinion subsets would remain hidden in our analysis. Understanding these non-linear effects is critical for better understanding the dynamics of polls and better discerning the malicious spread of misinformation 30 . Polarization might be studied in a similar vein as we approached this research using classical mechanics, but now using an approach for many-body interactions, the framework of thermodynamics. This approach might help elucidate underlying mechanisms and complement recent social simulations exploring this issue 33 . A further interest includes pursuing how media influences polarization; in theory, we could similarly address polarization, using the methods of this, yet exploring smaller subsets of the population that is influence to various degrees by echo chamber phenomena and other media effects.

Conclusions
Polls measuring the change in opinion have been analyzed using an abstraction of acceleration. We show that acceleration is flat over 120 years data. On the other hand, when the population is considered, the abstraction of force rises slightly over time, yet it too, is statistically insignificant. The effects of media, including social media, do not seem to alter the acceleration of opinion as demonstrated over one hundred years of poll taking. The study explores averages and the population may experience more rapid changes in opinion over time, but these are averaged out when considering the opposing opinions.

Data availability
All data used in this study are available from the corresponding author on request.