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 socalled
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 ≤ x ≤ a, the one that maximizes or minimizes
∫_{0}^{a} 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 socalled 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 v^{1/2} a minimum, or,
for constant M, ∫ ds v^{1/2} 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.
a.s.
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, 12001800 (1969, Harvard University Press), pp. 399406.
Documents Available:
 A beautiful, fullcolor scan of an
original copy of E65
is available at the website of the
Posner Memorial Collection, located at the Carnegie
Mellon Library.
 For those with low bandwidth connections, The Euler Archive has made a monochromatic
verstion of the book from the Posner Memorial Collection avaialable, to be downloaded
chapter by chapter:
 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. 245310);
 Additamentum 2 De motu
projectorum in medio non resistente, per methodum maximorum ac minimorum determinando
(pp. 311320).
 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. 1334 (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. 197241 (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. 151191 (1985).
 Fraser CG., “The calculus as algebraic analysis  some observations on mathematicalanalysis in the 18thcentury.” Archive for History of Exact Sciences, 39 (4), pp. 317335 (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, 18001830.” Annals of Sciences, 41 (1), pp. 133 (1984).
 Roche J., “What is potential energy?.” European Journal of Physics, 24 (2), pp. 185196 (Mar 2003).
 Vagliente VN, Krawinkler H., “Euler's paper on statically indeterminate analysis.” Journal of Engineering MechanicsASCE, 113 (2), pp. 186195 (Feb 1987).
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