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: 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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