The History of the Planar Elastica: Insights into Mechanics and Scientific Method
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Euler’s formula for the buckling of an elastic column is widely used in engineering design. However, only a handful of engineers will be familiar with Euler’s classic paper De Curvis Elasticis in which the formula is derived. In addition to the Euler Buckling Formula, De Curvis Elasticis classifies all the bent configurations of elastic rod—a landmark in the development of a rational theory of continuum mechanics. As a historical case study, Euler’s work on elastic rods offers an insight into some important concepts which underlie mechanics. It sheds light on the search for unifying principles of mechanics and the role of analysis. The connection between results obtained from theory and those obtained from experiments on rods, highlights two different approaches to scientific discovery, which can be traced back to Bacon, Descartes and Galileo. The bent rod also has an analogy in dynamics, with a pendulum, which highlights the crucial distinctions between initial value and boundary value problems and between linear and nonlinear differential equations. In addition to benefiting from the overview which a historical study provides, the particular problem of the elastica offers students of science and engineering a clear elucidation of the connection between mathematics and real-world engineering, issues which still have relevance today.