**RESEARCH TRIANGLE PARK, N.C. --** When Isaac Newton formulated the laws of motion during the late 17th century, he had to use the language of geometry to communicate the ideas of differential calculus to his audience.

In his most distinguished work, Philosophiae Naturalis Principia Mathematica, Newton used geometrical arguments to introduce the unifying concept of motion, because geometry was what scientists understood back then.

In much the same way, processes that don’t follow Newtonian mechanics face a similar communication issue today.

“Currently, we find all kinds of arguments based on scaling,” said Dr. Bruce West, the Army’s senior research scientist for mathematics and statistical physics and a researcher at the U.S. Army Combat Capabilities Development Command's Army Research Laboratory's Army Research Office. “Just as Newton had to use geometry to describe…motions through differential equations, today we use scaling to describe the non-Newtonian behavior of complex systems.”

According to West, fractional calculus represents the underlying principles of non-Newtonian behavior, which scientists often explain through scaling.

In the 20th century, researchers found that Newton’s laws of motion did not apply to extremely large and extremely small objects as well as those that moved extremely fast. As a result, they had to completely change their understanding of their world.

“What this does is it constricts the applicability of where Newton’s laws are valid to a smaller and smaller domain,” West said. “This is not the demise of Newtonian physics, but it shows the increasingly restricted domains of Newtonian science.”

Ultimately, scientists uncovered a solution for each of these contractions in the physical sciences. They used general relativity to describe the motions of extremely large objects, quantum mechanics to describe the motions of extremely small objects and special relativity to describe the motions of extremely fast-moving objects.

These explanations remedied these particular discrepancies in Newton’s laws; however, they still don’t account for a variety of other domains, such as sociology and the life sciences, that don’t follow the Newtonian understanding of motion.

The contradictions in these cases lies with Newton’s incomplete understanding of time, he said.

“For over 300 years now, we’ve had this notion of time being this smooth-flowing stream that flows uninterrupted in one direction,” West said. “Yet, there are some experiments that say that that’s not what happens. Newton’s laws are great for describing the flow of water, but not the flow of honey or oil.”

West explained that Newton’s laws do not account for the concept of history, in which a moving object retains a memory of more than its initial state as it travels to its destination.

For researchers to predict the motion of viscous materials like honey and oil, they need to consider the history of the material’s movements that led up to its current position. Because ordinary calculus cannot describe this kind of motion, researchers defined these types of materials as non-Newtonian fluids.

“What makes a phenomenon non-Newtonian is the fact that, if it’s timed, the time has memory,” West said. “Where it goes in the future depends on its entire past history. That’s where fractional calculus comes in.”

Fractional calculus, West said, serves as another way to describe the mathematical concept of scaling, which seeks to calculate the properties and behaviors of systems at multiple levels of complexity.

Nonlocality, which describes the apparent ability of objects to instantly know about each other’s state almost as if in anticipation of future events, gives fractional derivatives a built-in ability to incorporate memory effects.

In other words, fractional calculus provides incredibly useful tools for modeling extremely complex phenomena.

West illustrated this example with the normal sinus rhythm, which physicians are taught models the rhythm of a healthy heart.

During the 2000s, researchers introduced the concept of scale-free networks to describe networks whose variability follow an inverse power law distribution. West argues that this model explains the heart behavior much better than the outdated beliefs surrounding the normal sinus rhythm.

“The normal sinus rhythm is a myth; it doesn’t exist,” West said. “The normal sinus rhythm [actually] has scale-free behavior. What happens at one point in your body depends on what happens everywhere in your body. It’s not just local.”

For West, the introduction of fractional calculus to supplement Newtonian physics represents just one of many ways in which humans have surpassed their previous understanding of the universe.

West, a guest on the laboratory’s What We Learned Today podcast, Oct. 8, 2020, explains how scientific progress manifests itself in a universal ritual where researchers continually construct models, gather more data and struggle against contradictions that stop them in their tracks until they develop a new, more comprehensive model.

“Every science evolves through empirical paradox,” West said. “And it’s in the resolution of the paradox that you learn a new way to think and a new way to view the world.”

This fundamental process of learning will become even more essential to the Army of the future once Multi-Domain Operations increase the complexity of interactions between different human and technological systems, he said.

In preparation for this future, West anticipates the rise of complexity science, which will help advance the flexibility and adaptability of cyber networks for warfare.

“Paradoxes are going to be at every turn because of the complexity of the interactions,” West said. “How the paradox is resolved is going to be fundamental in whether we win or not.”

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**CCDC ****Army Research Laboratory**** is an element of the ****U.S. Army Combat Capabilities Development Command****. As the Army’s corporate research laboratory, ARL is operationalizing science to achieve transformational overmatch. Through collaboration across the command’s core technical competencies, CCDC leads in the discovery, development and delivery of the technology-based capabilities required to make Soldiers more successful at winning the nation’s wars and come home safely. CCDC is a major subordinate command of the ****Army Futures Command****.**

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