To explain the origins of sociomechanics, I need to give a bit of history.

I first encountered system dynamics in the mid-1990s and found it useful in my research. It constructed better models than the more orthodox differential equation approach. Also, the stock flow diagrams made it much easier to share models with people for whom maths was not their language1.


I started teaching it to final-year mathematics students in 1999, and it became a popular course. The concept of feedback proved powerful in building models and explaining behaviour. For example, think of the spread of a disease where infected people remain infected. The SI model, which describes the spread of the infection, has two loops: reinforcing, R1, and balancing, B1, Figure 1.

Figure 1: SI Model

The stocks have S-shaped behaviour, as shown in Figure 2, which is easily explained by the loops. The reinforcing loop B1 dominates the growing exponential phase of the infected, and B1 explains the slowing down phase. A similar explanation works for the decline in susceptibles. This is the explanation given in the books on system dynamics and the one I gave each year to my students.

Figure 2: Behaviour of Infected and Susceptible

However, my students were maths students, and they would ask, “Why?” “Prove it!” I would show them that if you remove the R1 loop in Figure 1 then it only slows down. So the accelerating phase of Figure 2 must have been caused by R1. You can demonstrate the dominance of B2 in the second phase by removing this loop from Figure 12.

Loop Impact

But the students went further, and asked, “Can you measure how much a loop accelerates a stock?” Searching for answers to this question led me to a paper by Mohammad Mojtahedzadeh et al. (2004) on the Pathway Participation Metric (PPM)3. I distilled his work down to a process the students could follow and try out for themselves. Questions on the calculus of pathways became a feature of my examination questions until the course ended on my retirement. I presented the work at an International System Dynamics Conference in 20124 and it was published as the Loop Impact Method (Hayward and Boswell, 2014)5.

Although loop impact is the foundational method of sociophysics, I had not used that word at this stage.

Newton’s Laws and System Dynamics

I realised very early on that this was not just a mathematical technique for measuring the effects of feedback loops. The fundamental equation that described the acceleration in stock behaviour looked very much like Newton’s Second law of Motion. Let I stand for the Infected stock. The equations can be written:

Equation 1: Differential Equations of the SI Model

Differentiate the I equation by time to obtain the loop impacts:

Equation 2: Acceleration of I Determined by Loop Impacts

The impacts of the two loops, R1 and B1, are contained in the brackets of the right hand side of the equation. The left hand side is the acceleration of the infected stock. Feedback loops cause the trajectory to curve in the same way forces in mechanics cause particles to deviate from uniform motion. The actual “forces” of the loops are the impacts multiplied by the rate of change of I — its net flow.

It became clear that there were three laws of system dynamics behaviour analogous to Newton’s three laws. Also, the concepts of mass and momentum were useful descriptors of model behaviour. I presented this interpretation of the Loop Impact method at the 2015 International System Dynamics Conference6.

Newtonian Framework

Publishing this application of mechanics to system dynamics proved problematic. To achieve publication, my colleague and I presented the work as the “Newtonian Interpretive Framework”. This framework was a set of rules for interpreting System Dynamics models in Newtonian mechanical terms. Indeed the framework can be applied directly to the social situations captured by the model. In this sense we had a methodology for using mechanics to understand dynamic social behaviour7. We still had not called it sociomechanics!

We pursued the work further publishing the ideas in the mathematical modelling literature. In particular, we published in a physics journal known for its promotion of sociophysics — the application of physics to sociological problems8. We invented the name sociomechanics as an appropriate description of our methodology — a subset of sociophysics

Energy and Power

With further work, we have been to show that feedback loops inject, remove and transfer energy. That is a form of dynamic energy that modifies the rate of change of stocks9. Change in stocks may be described as a type of kinetic energy. The rate of change of energy, called power, is also defined. Power is proportional to loop impact and an easier concept to grasp.

Figure 3: Energy flows in the SI model

The work is ongoing.


  1. I used an early version of Stella. Stella is now owned and distributed by isee systems. ↩︎
  2. This method was developed by David Ford: Ford D. 1999. A behavioral approach to feedback loop dominance analysis. System Dynamics Review, 15(1), 3–36. ↩︎
  3. Mojtahedzadeh M, Andersen D, Richardson GP. 2004. Using Digest to implement the pathway participation method for detecting influential system structure. System Dynamics Review, 20(1), 1–20. ↩︎
  4. Hayward J. 2012. Model Behavior and the Strengths of Causal Loops: Mathematical Insights and a Practical Method. Presented at the 30th International Conference of the System Dynamics Society, St. Gallen, Switzerland, July 2012 ↩︎
  5. Hayward J, Boswell GP. 2014. Model behaviour and the concept of loop impact: A practical method. System Dynamics Review, 30(1–2), 29–57. ↩︎
  6. Hayward J. 2015. Newton’s Laws of System Dynamics. Presented at the 33rd International Conference of the System Dynamics Society, Cambridge, MA, July 2015. ↩︎
  7. Hayward J, Roach PA. 2017. Newton’s Laws as an Interpretive Framework in System Dynamics. System Dynamics Review, 33(3-4), 183-218. DOI: 10.1002/sdr.1586. ↩︎
  8. Hayward J, Roach PA. 2019. The concept of force in population dynamics, Physica A: Statistical Mechanics and its Applications, 531, 121736, DOI: 10.1016/j.physa.2019.121736. ↩︎
  9. Hayward J, Roach PA. 2022. The Concept of Energy in the Analysis of System Dynamics ModelsSystem Dynamics Review, 38(1), 5-40, ↩︎