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Volume 6, Issue 1 (2026)Read More

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Translation & Commentary
19 August 2021

Euler's three-body problem

Sylvio Bistafa
In physics and astronomy, Euler's three-body problem is to solve for the motion of a body that is acted upon by the gravitational field of two other bodies. This problem is named after Leonhard Euler (1707-1783), who discussed it in memoirs published in the 1760s. In these publications, Euler found that the parameter that controls the relative distances among three collinear bodies is given by a quintic equation. Later on, in 1772, Lagrange dealt with the same problem, and demonstrated that for any three masses with circular orbits, there are two special constant-pattern solutions, one where the three bodies remain collinear, and the other where the bodies occupy the vertices of two equilateral triangles. Because of their importance, these five points became known as Lagrange points. The quintic equation found by Euler for the relative distances among the collinear bodies was also found later by Lagrange, and because of that, Euler has also been given credit for the discovery of the three collinear Lagrange points. A practical application of the collinear points for satellite location is also presented.
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Articles & Notes
19 September 2022

Basel Problem: Historical perspective and further proofs from stochastic processes

Uwe Hassler, Mehdi Hosseinkouchack
In this note, we offer a historical perspective on solutions of the Basel problem. In particular, we have a closer look at some of the less famous results by Euler E41 and provide a review of a selection of the assemblage of earlier proofs. Moreover, we show how to generate further proofs using Karhunen-Lo\`{e}ve expansions of stochastic processes.
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Articles & Notes
22 August 2023

Euler’s Variational Approach to the Elastica

Sylvio Bistafa
The history of the elastica is examined through the works of various contributors, including those of Jacob and Daniel Bernoulli, since its first appearance in a 1690 contest on finding the profile of a hanging flexible cord. Emphasis will be given to Leonhard Euler’s variational approach to the elastica, laid out in his landmark 1744 book on variational techniques. Euler’s variational approach based on the concept of differential value is highlighted, including the derivation of the general equation for the elastica from the differential value of the first kind, from which nine shapes adopted by a flexed lamina under different end conditions are obtained. To show the potential of Euler’s variational method, the development of the unequal curvature of elastic bands based on the differential value of the second kind is also examined. We also revisited some of Euler’s examples of application, including the derivation of the Euler-Bernoulli equation for the bending of a beam from the Euler-Poisson equation, the pillar critical load before buckling, and the vibration of elastic laminas, including the derivation of the equations for the mode shapes and the corresponding natural frequencies. Finally, the pervasiveness of Euler’s elastica solution found in various studies over the years as given on recent reviews by third parties is highlighted, which also includes its major role in the development of the theory of elliptic functions.
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Articles & Notes
4 September 2025

Euler’s Original Derivation of Elastica Equation

Shigeki Matsutani
Euler derived the differential equations of elastica by the variational method in 1744, but his original derivation has never been properly interpreted or explained in terms of modern mathematics. We elaborate Euler's original derivation of elastica and show that Euler used Noether's theorem concerning the translational symmetry of elastica, although Noether published her theorem in 1918. It is also shown that his equation is essentially the static modified KdV equation which is obtained by the isometric and isoenergy conditions, known as the Goldstein-Petrich scheme.
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