Unified Physics Theory Explains Animals' Running, Flying, Swimming

A single unifying theory can describe the essential physics of how all sorts of animals get around, from flying insects, to fish, to humans, according to researchers at Penn State and Duke University. The research team found that all animals bear the same stamp of physics in their design. The findings, published in the January 2006 issue of the Journal of Experimental Biology, challenge the notion that fundamental differences exist between apparently unrelated forms of locomotion.

"From simple physics, based only on gravity, density, and mass, you can explain within an order of magnitude many features of flying, swimming, and running," said James Marden, professor of biology at Penn State. "It doesn't matter whether the animal has eight legs, four legs, two legs, or even if it swims with no legs." The findings have important implications for understanding factors that guide evolution by suggesting that many important functional characteristics of animal shape and locomotion can be predicted by the laws of physics. The findings also offer an explanation for remarkable universal similarities in animal design that had long puzzled scientists, the researchers said.

"The similarities among animals that are on the surface very different are no coincidence," said Adrian Bejan, J. A. Jones Professor of Mechanical Engineering at Duke's Pratt School. "In fact, animal locomotion is no different from other flows, animate and inanimate: they all develop in space and in time such that they optimize the flow of material." In the case of animal locomotion, this means that animals move such that they travel the greatest distance while expending the least amount of energy, he said.

The researchers show that the so-called "constructal theory," a powerful analytical approach to describing virtually everything that moves, or flows, in nature, can explain basic characteristics of locomotion for every creature--how fast they get from one place to another and how rapidly and forcefully they step, flap, or paddle in relation to their mass. First conceived by Bejan and published in 1996, the constructal law arises from the basic principle that flow systems evolve so as to minimize energy wasted to friction or other forms of resistance so that the least amount of useful energy is lost.

The researchers report that the constructal law predicts universal relationships between animals' body mass and speed, as well as the frequency and force of the strides, beats, or undulations that propel their bodies forward. "Running, swimming and flying occur in vastly different physical environments and, likewise, involve quite different body mechanics," Bejan said of the new application of constructal law. "Nonetheless, there are strong convergences in certain functional characteristics of runners, swimmers and flyers." For example, the stride frequency of running vertebrates bears the same relationship to the animals' mass as does the rate at which fish swim. Similarly, the velocity of runners conforms to the same principles as the speed of birds in flight.

"The force generated by the muscular 'motors' of runners, swimmers, and flyers conforms with surprisingly little variation to a universal value dependent only on muscle mass," Marden said. Why this relationship should be so had remained a mystery, he said.

Marden said he first stumbled across the problem in the 1980s when studying the variability in flight performance of insects and other flying animals. He attached weights to them and got a "strange universal result." All the organisms he tested- birds, bats, insects--could all lift approximately the same amount of weight in relation to the size of their flight muscles regardless of their many other biological differences. "The size of the wings didn't matter; nothing else seemed to matter." Marden said. "It was fascinating, but there was no explanation for this commonality when so much about the animals seemed to be so different."

Years later, a student of Marden's suggested they analyze the function of jet engines to determine whether they, too, followed the same principle. Although Marden said he at first dismissed the idea as ridiculous, a 2002 report by the two in the Proceedings of the National Academy of Sciences showed that flying insects, birds and bats, running and swimming animals, piston engines, electric motors, and jets all showed the very same pattern. "We found that all of the motors used by humans and animals for transportation have a common upper limit of mass-specific net force output that is independent of materials and mechanisms," Marden said.

Unbeknownst to Marden, Bejan already had applied his constructal theory to a similar flight principle, the relationship of mass to flight speed in insects, birds, and airplanes, ranging from the extremes of a house fly to a Boeing 747. The result was first published in Bejan's book, Shape and Structure, From Engineering to Nature, in 2000 by Cambridge University Press.

A fortuitous meeting of Bejan and Marden at a conference in 2004 led them to extend Bejan's constructal theory from flying to running. The theory shows that, to maintain a constant speed, runners and flyers alike must expend energy to account for two mechanisms of work destruction: that which is destroyed at each jump and landing, or with each rise and fall in the air; and that which is lost to friction against the ground or air.

"To run or fly at optimal speed is to strike a balance between the vertical and horizontal loss of energy," Bejan said. Simple equations based on this idea closely predicted the actual velocities of animals running over a variety of terrains and the observed wingbeat frequencies of flying birds, bats, and insects, the current study reveals. "It was swimming that stumped us," Bejan said. "Everyone knows that, in water, fish are weightless." In other words, they explained, fish are neutrally buoyant, or nearly so, meaning that their tendency to float counteracts the force of gravity and they do not sink or rise. In essence, then, scientists have considered fish to move as though unaffected by gravity. Based on the data, swimmers exhibit the same body-mass scaling as runners and flyers. "The question was: How could a theory including gravity apply to swimming fish?" Marden said.

Bejan finally realized the answer. Although fish are neutrally buoyant, they still have to push water out of the way to move forward, he said. That water raises the surface--a phenomenon that often is imperceptible as it may be spread across an entire lake, stream, or ocean. "The water can only go up because the bottom and sides of the channel are rigid," Bejan said. "That bulge, however undetectable, is the fish's footprint." Fish must, therefore, work against gravity to lift an amount of water equal to their own mass for each body length they move forward. "It puts fish in the same physical realm as runners and flyers," Marden said.

The findings may have implications for understanding animal evolution, Marden said. One view of evolution holds that it is not a purely deterministic process; that history is full of chance contingencies. It is the idea purported by Steven Jay Gould and others that if you were to "rewind the tape" and run it again, evolution would proceed down a different path, Marden said. "Our finding that animal locomotion adheres to constructal theory tells us that--even though you couldn't predict exactly what animals would look like if you started evolution over on Earth, or it happened on another planet--with a given gravity and density of their tissues, the same basic patterns of their design would evolve again," Marden said.

The research was funded by the National Science Foundation.

CONTACTS

Jim Marden at Penn State: (+1) 814-863-1384, jhm10@psu.edu

Barbara K. Kennedy (PIO at Penn State): (+1) 814-863-4682, science@psu.edu

Adrian Bejan at Duke via Kendall Morgan (PIO at Duke): kendall.morgan@duke.edu