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CALCULUS OF VARIATIONS

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Originally appearing in Volume V27, Page 918 of the 1911 Encyclopedia Britannica.
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CALCULUS OF VARIATIONS, in mathematics. The calculus of variations arose from the attempts that were made by origin mathematicians in the 17th century to solve problems olthe of which the following are typical examples. (i) It tutus• is required to determine the form of a chain of given length, hanging from two fixed points, by the condition that its centre of gravity must be as low as possible. This problem of the catenary was attempted without success by Galileo Galilei (1638). (ii) The resistance of a medium to the motion of a body being assumed to be a normal pressure, proportional to the square of the cosine of the angle between the normal to the surface and the direction of motion, it is required to deter-mine the meridian curve of a surface of revolution, about an axis in the direction of motion, so that the resistance shall be the least possible. This problem of the solid of least resistance was solved by Sir Isaac Newton (1687). (iii) It is required to find a curve joining two fixed points, so that the time of descent along this curve from the higher point to the lower may be less than the time along any other curve. This problem of the brachistochrone was proposed by John (Johann) Bernoulli (1696). The contributions of the Greek geometry to the subject consist of a few theorems discovered by one Zenodorus, of whom little Early is known. Extracts from his writings have been pre- history. served in the writings of Pappus of Alexandria and Theon of Smyrna. He proved that of all curves of given perimeter the circle is that which encloses the largest area. The problems from which the subject grew up have in common the character of being concerned with the maxima and minima of quantities which can be expressed by integrals of the form f x'F(x, y, y')dx, xo in which y is an unknown function of x, and F is an assigned function of three variables, viz. x, y, and the differential coefficient of y with respect to x, here denoted by y'; in special cases x or y may not be explicitly present in F, but y' must be. In any such problem it is required to determine y as a function of x, so that the integral may be a maximum or a minimum, either absolutely or subject to the condition that another integral or like form may have a prescribed value. For example, in the problem of the catenary, the integral fx'oy (, +y") Idx must be a minimum, while the integral f (x' (1 +y'2) ldx J xo has a given value. When, as in this example, the length of the sought curve is given, the problem is described as isoperimetric. At the end of the first memoir by James (Jakob) Bernoulli on the infinitesimal calculus (169o), the problem of determining the form of a flexible chain was proposed. Gottfried Wilhelm Leibnitz gave the solution in 1691, and stated that the centre of gravity is lower for this curve than for any other of the same length joining the same two points. The first step towards a theory of such problems was taken by James Bernoulli (1697) in his solution of the problem of the brachistochrone. He pointed out that if a curve, as a whole, possesses the maximal or minimal property, every part of the curve must itself possess the same property. Beyond the discussion of special problems, nothing was attempted for many years. The first general theory of such problems was sketched by Leon-hard Euler in 1736, and was more fully developed by him in his Eisler. treatise Methodus inveniendi . . . published in 1744. He generalized the problems proposed by his predecessors by admitting under the sign of integration differential coefficients of order higher than the first. To express the condition that an integral of the form f xIF(x, y, y', y",... y"))dx xo may be a maximum or minimum, he required that, when y is changed into y+u, where u is a function of x, but is everywhere " infinitely " small, the integral should be unchanged. Resolving the integral into a sum of elements, he transformed this condition into an equation of the form aF d aF d2 aF d" aF EuAx ay ax ay 9y —.. +(—i)" . ax" -9y")] =o, and he concluded that the differential equation obtained by equating to zero the expression in the square brackets must be satisfied. This equation is in general of the 2nth order, and the 2n arbitrary constants which are contained in the complete primitive must be adjusted to satisfy the conditions that y, y', y', . y("'1) have given values at the limits of integration. If the function y is required also to satisfy the condition that another integral of the same form as the above, but containing a function di instead of F, may have a prescribed value, Euler achieved his purpose by replacing F in the differential equation by F-1 TO, and adjusting the constant a so that the condition may be satisfied. This artifice is known as the isoperimetric rule or rule of the undetermined multiplier. Euler illustrated his methods by a large number of examples. The newt theory was provided with a special symbolism by Joseph Louis de la Grange (commonly called Lagrange) in a series of memoirs published In 176o-62. This symbolism was afterwards adopted by Euler (1764), and Lagrange Lagrange. is generally regarded as the founder of the calculus of variations. Euler had been under the necessity of resolving an integral into a sum of elements, recording the magnitude of the change produced in each element by a slight change in the unknown function, and thence forming an expression for the total change in the sum under consideration. Lagrange proposed to free the theory from this necessity. Euler had allowed such changes in the position of the curve, along which the integral, to be made a maximum or minimum, is taken, as can be produced by displacement parallel to the axis of ordinates. Lagrange admitted a more general change of position, which was called variation. The points of the curve being specified by their co-ordini tee, x, y, z, and differentiation along the curve being denoted, as usual, by the symbol d, Lagrange considered the change produced in any quantity Z, which is ex-pressed in terms of x, y, z, dx, dy, dz, d2x, ... when the co-ordinates x, y, z are changed by " infinitely " small increments. This change he denoted by bZ, and regarded as the variation of Z. He ex-pressed the rules of operation with b by the equations bdZ =dbZ, bfZ =fbZ. By means of these equations fbZ can be transformed by the process of integration by parts into such a form that differentials of variations occur at the limits of integration only, and the transformed integral contains no differentials of variations. The terms at the limits and the integrand of the transformed integral must vanish separately, if the variation of the original integral vanishes. The process of freeing the original integral from the differentials of variations results in a differential equation, or a system of differential equations, for the determination of the form of the required curve, and in special terminal conditions, which serve to determine the constants that enter into the solution of the differential equations. Lagrange's method lent itself readily to applications of the generalized principle of virtual velocities to problems: of mechanics, and he used it in this way in the Mecanique analytique (1788). The terminology and notation of mechanics are still largely dominated by these ideas of J..agrange, for his methods were powerful and effective, but they are rendered obscure by the use of " infinitely " small quantities, of which, in other departments of mathematics, he subsequently became an uncompromising opponent. The same ideas were: applied by Lagrange himself, by Euler, and by xten- other mathematicians to various extensions of the cal- ;ions culus of variations. These include problems concerning of La. integrals of which the limits are variable in accordance granges with assigned conditions, the extension of Euler's rule of method. the multiplier to problems in which the variations are restricted by conditions of various types, the maxima and minima of integrals involving any number of dependent variables, such as are met with in the formulation of the dynamical Principle of Least Action, the maxima and minima of double and multiple integrals. In all these cases Lagrange's methods have been applied successfully to obtain the differential equation, or system of differential equations, which must be satisfied if the integral in question is a maximum or a minimum. This equation, or equations, will be referred to as the principal equation, or principal equations, of the problem. The problems and method of the calculus admit of more exact formulation as follows: We confine our attention to the case where the sought curve is plane, and the function F contains no differential coefficients of order higher than the first. Then the problem is to determine a curve joining two fixed points (xo, yo) and (xi, yi) so that the line integral ff x1F(x, y, y')dx xo taken along the curve may be a maximum or a minimum. When it is said that the integral is a minimum for some curve, it is meant that it must be possible to mark a finite area in the plane of (x, y), so that the curve in question lies entirely within this area, and the integral taken along this curve is less than the integral taken along any other curve, which joins the same two points and lies entirely within the delimited area. There is a similar definition for a maxi-mum. The word extremum is often used to connote both maximum and minimum. The problem thus posed is known as the First Problem of the Calculus of Variations. If we begin with any curve The symbol S. Formulation of the First Problem. joining the fixed end points, and surround it by an area of finite breadth, any other curve drawn within the area, and joining the same end points, is called a variation of the original curve, or a varied curve. The original curve is defined by specifying y as a function of x. Necessary conditions for the existence of an extremum can be found by choosing special methods of variation. One method of variation is to replace y by y+eu, where u is a function of x, and a is a constant which may be taken as small as we please. The function u is independent of e. It is differentiable, and its differential coefficient is continuous within the interval of Weak integration. It must vanish at x=xo and at x=xt. This wile- method of variation has the property that, when the dons. ordinate of the curve is but slightly changed, the direction of the tangent is but slightly changed. Such variations are called weak variations. By such a variation the integral is changed into f x1F(x, y+eu, y'+eu')dx. 0 and the increment, or variation of the integral, is f x''1F(x,y+eu, y'+eu')—F(x, y, y') ldx. In order that there may be an extremum it is necessary that the variation should be one-signed. We expand the expression under the sign of integration in powers of E. The first term of the expansion contributes to the variation the term e fxi(—u+aFu')dx. xo ay ay This term is called-the first variation. The variation of the integral cannot be one-signed unless the first variation vanishes. On trans-forming the first variation by integration by parts, and observing that u vanishes at x =xo and at x =x1, we find a necessary condition for an extremum in the form f xi OF d aF xo \--ay ax ay) ndx=o. It is a fundamental theorem that this equation cannot hold for all admissible f unctions u, unless the differential equation d aF OF dx a y' ay -0 is satisfied at every point of the curve along which the integral is taken. This is the principal equation for this problem. The Station- curves that are determined by it are called the stationary a curves, or the extremals, of the integral. We learn that curves. the integral cannot be an extremum unless it is taken along a stationary curve. A difficulty might arise from the fact that, in the foregoing argument, it is tacitly assumed that y, as a function of x, is one-valued; and we can have no a priori ground for assuming that this is the case for the sought curve. This difficulty might be met by an appeal to James Bernoulli's principle, according to which every arc of a stationary curve is a stationary curve between the end points of the arc—a principle which can be proved readily by adopting such a method of variation that the arc of the curve between two points is displaced, and the rest of the curve is not. But another method of meeting it leads to important developments. This is the method of parametric representation, introduced by K. Weierstrass. According to this method the curve is defined by specifying x and y as one-valued functions of a parameter O. The integral is then of the form f fof(x, y, x, y)do, where the dots denote differentiation with respect to 0, and f is a homogeneous function of x, y of the first degree. The mode of dependence of x and y upon 0 is immaterial to the problem, provided that they are one-valued functions of 0. A weak variation is obtained by changing x and y into x+eu, y+ev, where u and v are functions of 0 which have continuous differential coefficients and are independent of e. It is then found that the principal equations of the problem are d ofd af_ of=o To ax ax= de ay ay These equations are equivalent to a single equation, for it can be proved without difficulty that, when f is homogeneous of the first degree in x, d at at I d at a2f 02f +f(xy5,x) Z de ax— ax l =—x ? a ay —ay S yax—azay where /a t a2f a2f I a2f 162 ax2a)axay =x j' 2 a ' The stationary curves obtained by this method are identical with those obtained by the previous method. The formulation of the problem by the parametric method often enables us to simplify the formation and integration of the principal equation. A very simple example is furnished by the problem problem: Given two points in the plane of (x, y) on the same side of the axis of x, it is required to find a curve oftbe joining them, so that this curve may generate, by revolu- catenold. tion, about the axis of x, a surface of minimum area. The integral to be made a minimum is ry(i2+y2)Ido, and the principal equation is d (]) yx a (Z2 — o +.y2)i — of which the first integral is Yx (x2+5,2)1= c, 1+(dx)2': c and the stationary curves are the catenaries y=c cosh((x—a)fc}. The required minimal surface is the catenoid generated by the revolution of one of these catenaries about its directrix. The parametric method can be extended without difficulty so as to become applicable to more general classes of problems. A simple example is furnished by the problem of forming the equa- tions path of of the path of a ray of light in a variable medium. According to Fermat's principle, the integral fads. is a aray• minimum, ds representing the element of arc of a ray, and µ the refractive index. Thus the integral to be made a minimum is f eo (z2+y2+z2)ido. The equations are found at once in forms of the type or de 12(x2+5,2+9)l ax(x2+5,2+22)1=0; and, since (x2+5,2+i2)%do=ds, these equations can be written in the usual forms of the type d dx agds (1` as) - ax—°' The formation of the first variation of an integral by means of a weak variation can be carried out without difficulty in the case of a simple integral involving any number of dependent variables and differential coefficients of arbitrarily high orders, and also in the cases of double and multiple integrals; and the quantities of the type eu, which are used in the process, may be regarded as equivalent to Lagrange's Sx, by, . . . The same process may not, however, be applied to isoperimetric problems. If the first varia- tion of the integral which is to be made an extremum, Rule of subject to the condition that another integral has a pre- the mufscribed value, is formed in this way, and if it vanishes, the Heller. curve is a stationary curve for this integral. If the prescribed value of the other integral is unaltered, its first variation must vanish; and, if the first variation is formed in this way, the curve is a stationary curve for this integral also. The two integrals do not, however, in general possess the same stationary curves. We can avoid this difficulty by taking the variations to be of the form eiui+e2142, where el and e2 are independent constants; and we can thus obtain a completely satisfactory proof of the rule of the undetermined multiplier. A proof on these lines was first published by P. Du Bois-Reymond (1879). The rule had long been regarded as axiomatic. The parametric method enables us to deal easily with the problem of variable limits. If, in the First Problem, the terminal point (xi, y,) is movable on a given guiding curve 0(x1, yi) =o, the first variation of the integral can be written of ax+ of f el d a1 u+ d of do, ay~_xl 95,l e L aoax ax } do ay ay] where (xi+eut, y1+evl) is on the curve ¢(xi, 5,i) =o and ut, vi denote the values of u; v at (xi, y1). It follows that the required curve must be a stationary curve, and that the condition of am at am=o axayi ayaxi The corresponding condition in the case of The first variation. Para-metric method. must hold at (xi, 5,l)• the integral fxo'F(x, y, y')dx is found from the equations a1 =F—y', af= ax ay ay ay Va, fable limits. to be F(x, y, y')+ (dx'—y) ay =o. Trans- transversal of those curves. In the problem of variable versals limits, when a terminal point moves on a given guiding ofsta- curve, the integral cannot be an extremum unless the tionary stationary curve along which it is taken is cut transversely curves. by the guiding curve at the terminal point. A simple example is afforded by the shortest line, drawn on a surface, from a point to a given curve, lying on the surface. The required curve must be a geodesic, and it must cut the given curve at right angles. The problem of variable limits may always be treated by a method of which the following is the principle: In the First Problem let the initial point (xo, yo) be fixed, and let the terminal point Anemia- (xi, y,) move on a fixed guiding curve C,. Now, whatever rive the terminal point may be, the integral cannot be an method. extremum unless it is taken along a stationary curve. We have then to choose among those stationary curves which are drawn from (xo, yo) to points of C, that one which makes the integral an extremum. This can be done by expressing the value of the integral taken along a stationary curve from the point (xo, yo) to the point (x,, y,) in terms of the co-ordinates y,, and then making this expression an extremum, in regard to variations of x,, y,, by the methods of the differential calculus, subjecting (x1, y,) to the condition of moving on the curve C,. An important example of the first variation of integrals is afforded by the Principle of Least Action in dynamics. The kinetic energy T is a homogeneous function of the second degree in the differential coefficients q,, q2, . q„ of the co-ordinates q,, q2, . . . q,, with respect to the time t, and the: potential energy V is a function of these co-ordinates. The energy equation is of the form least aactn where E is a constant. A course of the system is defined when the co-ordinates qare expressed as functions of a single parameter O. The action A of the system is defined as the integral f to?Tdt, taken along a course from the initial position (q(°)) to the final position (¢1)), but to and t, are not fixed. The equations of motion are the principal equations answering to this integral. To obtain them it is most convenient to write t'(q) for T, and to express the integral in the form f B2(E—V)i 4'(q'))Ide, where q' denotes the differential coefficient of a co-ordinate q with respect to 8, and, in accordance with the parametric method, the limits of integration are fixed, and the integrand is a homogeneous function of the q"s of the first degree. There is then no difficulty in deducing the Lagrangian equations of motion of the type d aTaT+aV = o. tit aq aq aq These equations determine the actual course of the system. Now if the system, in its actual course, passes from a given initial position (¢°)) to a variable final position (q), the action A becomes a function of the q's, and the first method used in the problem of variable limits shows that, for every q OA aT aq — aq' When the kinetic energy T is expressed as a homogeneous quadratic function of the momenta aT/aq, say T 12 ,, , (b,. aq, aT) , (be, = br,), W. and the differential coefficients of A are introduced instead of those of T, the energy equation becomes a non-linear partial differential equation of the first order for the determination of A as a function of the q's. This equation is Principle ( aA OA) of vary- b"aqr aq. +V = E. g action. A complete integral of this equation would yield an expression for A as a function of the q's containing n arbitrary constants, al, az, . . . as, of which one as is merely additive to A; and the courses of the system which are compatible with the equations of motion are determined by equations of the form 0A A aa1=b1, a -=b2,...aa—~=LI where the b's are new arbitrary constants. It is noteworthy that the differential equations of the second order by which the geodesics on an ellipsoid are determined were first solved by this method (C. G. J. Jacobi, 1839). It has been proved that every problem of the calculus of variations, in which the integral to be made an extremum contains only one independent variable, admits of a similar trans-formation; that is to say, the integrals of the principal equations can always be obtained, in the way described above, from a complete integral of a partial differential equation of the first order, and this partial differential equation can always be formed by a process of elimination. These results were first proved by A. Clebsch (1858). Among other analytical developments of the theory of the first variation we may note that the necessary and sufficient condition that an expression of the form F(x, y°)) should be the differential coefficient of another expression of the form Fi(x, Y, Y', •y(°-'1)) is the identical vanishing of the expression OF d OF d2 OF n d* OF ay—axay'+dx2 ay"— ... +(—1) dx^_ay<^>. The result was first found by Euler (1744). A differential equation 0(x, y, y', y") =o is the principal equation answering to an integral of the form f F(x, y, y')dx d =act. ax ay"—ay' is satisfied identically. In the more general case of an equation of the form (x, ... y(2n)) = 0 the corresponding condition is that the differential expression obtained by Lagrange's process of variation, viz., dsy a4' d2i~sy ~y Sy+ay d +... +ay(2a1 dx2n ' must be identical with the " adjoint " differential expression ay "Y—a (ay Sy) +a 22 (ay sy) . +d (ay(2n)Sy). This matter has been very fully investigated by A. Hirsch (1897). To illustrate the transformation of the first variation of multiple integrals we consider a double integral of the form ff4G(x, y, z, p, q, r, s, t)dxdy, taken over that area of the z plane which is bounded by a closed curve s'. Here p, q, . t denote the partial differential coefficients of z with respect to x and y of the first and second orders, according to the usual notation. When z is changed into z+ew, the terms of the first order in a are 1f (a,~w+a4' aw+aw+a,' a2w+a>G a2w +a,' a2w\ dxdy. as ap ax aq ay Or ax2 Os axay at aye f Each term must be transformed so that no differential coefficients of w are left under the sign of double integration. We exemplify the process by taking the term containing a2w/axe. We have J/'a,' a2w S a (a4' aw _ a O aw .f ft ax2dxdy= ff t ax (Br 3x c3x (ar . ax dxdy = ff Lax (~3r ax) — w ax (M) +wax2 (ar) ] dxdy. The first two terms are transformed into a line integral taken round the boundary s', and we thus find a axsa2w 14,11- a s a2 a ff a dxdy =fcos(x,v) ) ds'+ffwax2 (a) dxdy, where v denotes the direction of the normal to the edge s' drawn outwards. The double integral on the right-hand side contributes a term to the principal equation, and the line integral contributes terms to the boundary conditions. The line integral admits of further transformation by' means of the relations aw Ow Ow ax = av cos (x, v) — as, cos (y, v), a~ Ow , a) a,~ fcos(x, v)cos(y, v)ar as ds = — f as cos(x, v)cos(y, v)ar This discussion yields an important result, which may be stated as follows: Let two stationary curves of the integral be drawn from the same initial point A to points P, Q, which near together, and let the line PQ be 4/are of length v, and make an angle w with the axis of x (fig. I). The excess of the integral taken along AQ, from A to Q, above the integral taken along AP, from A to P, is expressed, correctly to the first order in v, by the formula v cos w F(x, y, y') + (tan w — aF Y')ay' In this formula x, y are the co-ordinates of Fin 1. x P, and y' has the value belonging to the point P and the stationary curve AP. When the coefficient of v cos w in the formula vanishes, the curve AP is said to be cut transversely by the line PQ, and a curve which cuts a family of stationary curves transversely is described as a A Principle T+V = E, Principle of varying action generalized. if the equation Condition ofintegrability. Condition that a differential equation may arise from a problem of the calculus of variations. First variation of a double integral. It becomes
End of Article: CALCULUS OF VARIATIONS
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