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Euler's Navigation Variational Problem

In a 1747 publication, De motu cymbarum remis propulsarum in fluviis (“On the motion of boats propelled by oars in rivers”), Leonhard Euler (1707-1783) works out various instances of a boat moving at constant speed across a stream flowing in straight streamlines at assigned speeds, in which one of these gives rise to a variational problem consisting of finding the quickest crossing path between two points on opposite side of the river banks, which is generally known as the navigation variational problem. This problem together with the well-known catenary and brachistochrone problems, are considered classical examples in the calculus of variations. Here, we shall present a brief account on Euler’s recurrent interests in calculus of variations, mainly laid out in three publications that span between 1738 and 1744. Particular focus will be given to Euler’s navigation variational problem. A brief account on Lagrange’s contributions to variational calculus is also presented.