## E65 -- Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti

(A method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense)

Summary:

This work is concerned with the calculus of variations. Euler's main contribution to this subject is that he changed it from a discussion of essentially special cases to a discussion of very general classes of problems. This work includes a listing of 100 special problems that Euler considers to illustrate his methods. Euler also demonstrates a general procedure for writing down the so-called Euler differential equation or first necessary condition. This is also the first work in which the principle of least action (which Euler states and discusses) is presented; the principle is the first deep insight (apart from Fermat's principle of least times) of how the calculus of variations comes into play in physics.

Among the problems that Euler looks at in order to demonstrated his methods are:
• Find, among all plane curves y = y(x), 0 ≤ xa, the one that maximizes or minimizes ∫0a Z dx, where Z is a "determinate" function of x, y, p = dy/dx, q = dp/dx, r = dz/dx, etc.
• Find the shape of the brachystochrone curve when the medium through which the heavy particle falls restricts the motion that depends only on the particle's velocity.
• Find the plane curve that a heavy particle will follow so that it falls in the shortest possible line through a resisting medium.
• Find the geodesic joining two fixed points on a given concave or convex surface when:
• the geodesic can be any curve on the surface.
• the geodesic must satisfy an accessory condition such as an isoperimetric condition.
• the geodesic must satisfy an arbitrary number of accessory conditions.
Euler also finds a simple (but the first) instance of the Lagrange multiplier method. He takes the Brachystochrone problem and modifies it as follows: determine the curve, joining two points in a vertical plane, that a heavy particle will trace as it falls in a resisting medium so that the particle falls in the least amount of time. By considering this problem, Euler arrives at the first necessary condition for the so-called Lagrange problem. In addition Euler derives a fundamental condition that is invariant under "general" transformations of the coordinate axes.

As mentioned above, this work contains the first publication of the principle of least action, which Euler formulates as follows: Let the mass of the projected body be M, let v be half the square of the velocity of the projected body, and let the element of arclength along the prescribed path be ds. Among all curves passing through the same end points, the desired one makes the integral ∫ M ds v1/2 a minimum, or, for constant M, ∫ ds v1/2 a minimum. This principle applies to any number of bodies or particles, but it seems to run into a difficulty when one considers the motion in a resisting medium. Note that the principle of least action is now usually attributed to Maupertuis.

For a more detailed explanation, see Herman H. Goldstine's A History of the Calculus of Variations from the 17th through the 19th Century.
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Publication:
• Originally published as a book in 1744
• Opera Omnia: Series 1, Volume 24
• An English translation of the first of selections of E252, along with some brief commenatary, is published in D. J. Struik's A Source Book in Mathematics, 1200-1800 (1969, Harvard University Press), pp. 399-406.
Documents Available:
• Original publication: A beautiful, full-color scan of an original copy of E65 is available at the website of the Posner Memorial Collection, located at the Carnegie Mellon Library.
• Original publication: A low-bandwidth version of the Posner copy is available directly from the Euler Archive:
• Caput 1 De methodo maximorum et minimorum ad lineas curvas inveniendas applicata in genere.
• Caput 2 De methodo maximorum et minimorum ad lineas curvas inveniendas absoluta.
• Caput 3 De inventione curvarum maximi minimive proprietate praeditarum.
• Caput 4 De usu methodi hactenus traditae in resolutione varii generis quaestionum.
• Caput 5 Methodus, inter omnes curvas eadem proprietate praeditas, inveniendi eam quae maximi minimive proprietate gaudeat.
• Caput 6 Methodus, inter omnes curvas pluribus proprietatibus communibus gaudentes.
In addition to the 6 chapters (pp. 1- 244), two “Additamenta” can be found, namely:
• Additamentum 1 De curvis elasticis (pp. 245-310);
• Additamentum 2 De motu projectorum in medio non resistente, per methodum maximorum ac minimorum determinando (pp. 311-320).
• A German translation has been done by Alexander Aycock and Artur Diener: E65
• An English translation of the first Additamentum appears in

• The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E65 include:
• Fellmann EA., “The 'Principia' and continental mathematician + Newton.” Notes and Records of the Royal Society of London, 42 (1), pp. 13-34 (Jan 1988).
• Fraser C., “Lagrange, J. L. early contributions to the principles and methods of mechanics.” Archive for History of Exact Sciences, 28 (3), pp. 197-241 (1983).
• Fraser C., “Lagrange, J. L. changing approach to the foundations of the calculus of variations.” Archive for History of Exact Sciences, 32 (2), pp. 151-191 (1985).
• Fraser CG., “The calculus as algebraic analysis - some observations on mathematical-analysis in the 18th-century.” Archive for History of Exact Sciences, 39 (4), pp. 317-335 (1989).
• Goldstine HH., A History of the Calculus of Variations from the 17th through the 19th Century
• Grattanguinness I., “Work for the workers - advances in engineering mechanics and instruction in France, 1800-1830.” Annals of Sciences, 41 (1), pp. 1-33 (1984).
• Oldfather, W., Ellis, C., & Brown, D. (1933). "Leonhard Euler's Elastic Curves." Isis, 20 (1), 72-160. This work includes an English translation of the first Additamentum.
• Roche J., “What is potential energy?.” European Journal of Physics, 24 (2), pp. 185-196 (Mar 2003).
• Vagliente VN, Krawinkler H., “Euler's paper on statically indeterminate analysis.” Journal of Engineering Mechanics-ASCE, 113 (2), pp. 186-195 (Feb 1987).