1 1 The original Euler’s calculus-of-variations method: 2 Key to Lagrangian mechanics for beginners 3 Jozef Hanca) 4 Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia 5 ABSTRACT 6 Leonhard Euler's original version of the calculus of variations was geometric and easily visualized. Euler (1707- 7 1783) abandoned his version because of the absence in his day of the exact definition of a limit, adopting instead 8 the algebraic method of Lagrange. Lagrange’s elegant technique not only bypassed the need for intuition about 9 passage to the limit in calculus but also eliminated Euler’s geometrical insight. More recently, Euler's method 10 has been resurrected, shown to be rigorous and more general than Lagrange's, and employed in computer 11 solutions of physical processes. In our classrooms the study of advanced mechanics is still dominated by 12 Lagrange's method, which students often apply uncritically using "variational recipes". We describe an 13 adaptation of Euler's method that restores intuition and geometric visualization and has applications in almost all 14 of undergraduate physics, especially modern physics. Finally, we present Euler's method as a natural 15 introduction to computer-executed numerical analysis of boundary value problems and the finite element 16 method. 17 18 I. INTRODUCTION 19 In his pioneering 1744 work The method of finding plane curves that show some property of 20 maximum and minimum,1 Leonhard Euler introduced a general mathematical procedure or 21 method for the systematic investigation of variational problems. Along the way he formulated 22 the variational principle for mechanics, his version of the principle of least action.2 23 Mathematicians consider this event to be the beginning of one of the most important branches 24 of mathematics, the calculus of variations. Physicists also regard it as the first variational 25 treatment of mechanics, which later contributed significantly to analytic mechanics and 26 ultimately to the fundamental underpinnings of twentieth-century physics: general relativity 27 and quantum mechanics. 28 It is not certain3 when Euler first became seriously interested in variational problems and 29 properties. We know that he was influenced by Newton and Leibniz, but primarily by James 30 and Johann Bernoulli who were also attracted to the subject. The best known examples of 31 variational calculus include Fermat’s principle of least time ("between fixed endpoints, light 32 takes the path for which the travel time is shortest"), Bernoulli’s brachistochrone problem4 33 ("find a plane curve between two points along which a particle descends in the shortest time 34 under the influence of gravity"), and the so-called isoperimetric problem ("find the curve 35 which encloses the greatest area for a given perimeter"). 36 While each of Euler's contemporaries devised a special method of solution depending on the 37 character of the particular variational problem, Euler's own approach was purely mathematical 38 and therefore much more general. Employing geometrical considerations and his phenomenal 39 intuition for the limit-taking process of calculus, Euler established a method that allows us to 40 solve problems using only elementary calculus. 41 In 1755, the 19-year-old Joseph-Louis Lagrange wrote Euler a brief letter to which he 42 attached a mathematical appendix with a revolutionary technique of variations. Euler 43 immediately dropped his method, espoused that of Lagrange, and renamed the subject the 44 calculus of variations.5 Lagrange’s elegant techniques eliminated from Euler’s methods not 45 only the need for intuition about passage to the limit in calculus but also Euler’s geometrical 46 insight. It reduced the entire process to a quite general analytical manipulation which to this 2 1 day characterizes the calculus of variations. Since in Euler’s time the exact proof of the 2 passage to the limit could not be carried out,6 his method fell into disuse. 3 However, at the beginning of the twentieth century there was an interest in approximate 4 solutions of variational problems and partial differential equations. Euler’s method again 5 attracted the attention of mathematicians, and in the end the modern numerical analysis of 6 variational problems and differential equations7,8 fully vindicated Euler’s intuition. Euler’s 7 method rose like a phoenix and became one of the first direct variational methods.7 At 8 approximately the same time another direct methods were devised, the well-known Rayleigh- 9 Ritz method (1908) and its extension called Galerkin’s method (1915). Finally, all these 10 methods for solving differential equations led to the formulation of the very powerful Finite 11 Element Method (FEM) (1943, 1956)9,10 for carrying out accurate computer predictions of 12 physical processes. 13 For more than a century students in upper undergraduate classes in mechanics have been 14 taught to use Lagrange’s calculus of variations to derive the Lagrange equations of motion 15 from Hamilton’s principle, which is also known as the principle of least action (so renamed by 16 Landau and Feynman). However, there appears to be a conflict between the larger view of 17 theoretical physics and the pedagogy of upper level physics classes: (1) From the viewpoint of 18 modern theoretical physics, variational treatments such as the principle of least action with 19 Lagrangian formalism are "core technology” that can be extended to describe systems beyond 20 mechanics.11 (2) Because the calculus of variations is not widely taught in upper level physics 21 classes, students often meet it first in an advanced mechanics class, find the manipulations 22 daunting, do not develop a deep conceptual understanding of either the new (to them) 23 mathematics or the new physics, and end up memorizing "variational recipes." 24 Is there a strategy or method that allows us to introduce important variational treatments of 25 mechanics and physical laws as "core technology" while at the same time teaching students to 26 apply it with understanding and insight? 27 In the present paper we propose that Euler’s method be used as an introductory procedure 28 instead of the Lagrangian technique of variations. Section II is devoted to a detailed 29 description of Euler’s method from his 1744 work, together with notes for its pedagogic use. 30 We will see that Euler's method permits visualization useful for learning physics and requires 31 only elementary calculus. 32 In Sec. III we illustrate the computer application of Euler's method using interactive software 33 which allows students to carry out their own investigations of the principle of least action and 34 Lagrangian mechanics. Contemporary education research demonstrates that student-directed 35 exploration of a subject helps to develop genuine understanding of concepts and their 36 applications. 37 The final Sec. IV gives a list of other applications of the method which apply it to almost all 38 of undergraduate physics, especially modern physics. 39 40 II. BASIC IDEAS OF EULER’S METHOD 41 A. Euler’s original considerations 42 The exact transcription of original Euler’s derivations from his 1744 work is reproduced in 43 Goldstine’s book3. Several other mathematics and physics books offer somewhat modified 44 versions.12 Here we present only the essential considerations of the Euler approach.13 3 1 Euler’s starting point was his ingenious reduction of the variety of variational problems to a 2 single abstract mathematical form. He recognized that solving the variational problem means 3 finding an extremal (or more strictly stationary) value of a definite integral. 4 As a first example Euler presents a solution of the simplest variational problem: A function 5 F = F(x, y, y′) has three variables: the independent coordinate x, the dependent coordinate y, 6 and its derivative y′ with respect to the independent coordinate. Our problem is to determine a 7 the curve y = y(x), with 0 ≤ x ≤ a , which will make the definite integral ∫ F(x, y, y′)dx 0 8 extremal. Such an integral occurs in the brachistrochrone problem or in the description of 9 motion using the principle of least action. 10 Euler presents three crucial ideas which allowed him to solve the problem using only 11 elementary calculus: 12 (1) divide the interval between x = 0 and x = a into many small subintervals, each of width 13 ∆x ; 14 (2) replace the given integral by a sum ∑ F(x, y, y′)∆x . In each term of this sum evaluate 15 the function F at the initial point x, y of the corresponding subinterval and approximate the 16 derivative dy/dx by the slope of the straight line between initial and final points of the 17 subinterval; 18 (3) employ a visualized “geometrical” way of thinking. 19 Goldstine’s book3 displays (p.69) the original Euler’s figure (Fig. 1) in which the curve anz 20 represents the unknown extremal curve y = y(x): 21 22 23 Fig. 1. Original Euler’s figure used in his derivation 24 25 Figure 1 illustrates the fact that if we shift an arbitrarily chosen point on the curve, for 26 example point n, up or down by an increment nv, then in addition to the obvious change in the 27 ordinate yn of the point n, there are also changes in neighboring segments mn and no and their 28 slopes. All other points and slopes remain unchanged. These changes mean that only two 29 terms in the integral sum ∑ F(x, y, y′)∆x are affected, the first corresponding to segment mn 4 1 and the second to segment no. We can also say the variation of yn causes the integral sum to 2 be a function of single variable yn .