The Origins of Mechanical Conservation Principles and Variational Calculus in the 17th Century

The 17

th

century is considered as the cradle of modern natural sciences and technology as well as the begin of the age of enlightenment with the invention of analytical geometry by Descartes (1637), infinitesimal calculus by Newton (1668) and Leibniz (1674), and based on the rational mechanics by Newton (1687), initiated by Galilei (1638). In 1696, Johann Bernoulli posed the so-called brachistochrone problem in Acta Eruditorum, asking for solutions within a year’s time. Seven solutions were submitted and published in 1697, the most famous one by his brother Jacob Bernoulli, anticipating Euler’s idea of discrete equidistant support points and triangular test functions between three neighboured points, followed by the infinitesimal limit. Johann Bernoulli himself presented two intelligent solutions by joining geometrical and mechanical observations. Leibniz submitted a geometrical integration method for the differential equation of the cycloid and, what is important for this article, a short draft of a discrete or “direct variational” numerical approximation method, also using triangular test functions between neighboured support points with finite distances. This can be considered as a precursor of the finite element method. In connection with the brachistochrone, more general tautochrony problems were investigated, e.g. by Huygens and Newton. In conclusion many important developments of energy methods in mechanics using variational methods were already invented in the 17

th

century.