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Bible Encyclopedias
Calculus of Variations
1911 Encyclopedia Britannica
or In mathematics. The calculus of variations arose from the attempts that were made by Origin mathematicians in the 17th century to solve problems of the of which the following are typical examples. (i) It Calculus. 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 determine 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. served in the writings of Pappus of Alexandria and Theon history of Smyrna. He proved that of all curves of given peri meter 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 x1F(x, y, y')dx, J X0 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 xly(1+y")3dx xo must be a minimum, while the integral x1 (1 +y' 2) 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 (1690), 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 Leonhard Euler in 1736, and was more fully developed by him in his Euier. treatise Methodus inveniendi.. . published in 1 744. .
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 x 1F (x, y, y', y",... y(n)) 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 (9F d OF, OF do aF EuOx ay axay'd? 2 Oy"
+(- 1 ') n dx n ayk n?J =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 211 arbitrary constants which are contained in the complete primitive must be adjusted to satisfy the conditions that y, y', y",. yo- ) 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 43 instead of F, may have a prescribed value, Euler achieved his purpose by replacing F in the differential equation by F-{-Acb, 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 new theory was provided with a special symbolism by Joseph Louis de la Grange (commonly called Lagrange) in a series of memoirs published in 1760-62. This symbolism Lagrange. was afterwards adopted by Euler (1764), and 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-ordinates, 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 expressed in terms of x, y, z, dx, dy, dz, d 2 x, ... when the co-ordinates x, y, z are changed by " infinitely " small increments. This change he denoted by SZ, and regarded as the variation of Z. He expressed the rules of operation with S by the equations SdZ=dIZ, SfZ =fSZ. By means of these equations fSZ 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 The transformed integral contains no differentials of varia- symbol s. tions. 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 Lagrange, 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 Exten- other mathematicians to various extensions of the cal- sions culus of variations. These include problems concerning of Laintegrals of which the limits are variable in accordance g range's with assigned condit i ons, 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 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 (x i, y,) so that the line integral f x1 F(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 maximum. 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 case tion 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 e is a constant which may be taken as small as we please. The function u is independent of is differentiable, and its differential coefficient is continuous within the interval of Weak integration. It must vanish at =xo and at 1 This W method of variation has the property that, when the varia- ordinate of the curve is but slightly changed, the direction lions. of the tangent is but slightly changed. Such variations are called weak variations. By such a variation the integral is changed into f F(x, y+fu, y'+Eu')dx, xo and the increment, or variation of the integral, is xl {F(x, y +eu, y'+En') - F(x, y, y')}dx. 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 The first term of the expansion contributes to the variation the term e xo(? y u + ayu') dx. OF, This term is called the first variation. The variation of the integral cannot be one-signed unless the first variation vanishes. On transforming the first variation by integration by parts, and observing that u vanishes at =xo and at x = x1, we find a necessary condition for an extremum in the form "x l aF d aF udx =0. xo (ay ax ay') It is a fundamental theorem that this equation cannot hold for all admissible functions u, unless the differential equation d OF OF = dx ay' ay =° is satisfied at every point of the curve along which the integral is taken. This is the principal equation for this problem. The curves that are determined by it are called the stationary curves, or the extremals, of the integral. We learn that 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 B. The integral is then of the form ° l f(x, y, ±, y)de, where the dots denote differentiation with respect to 0, and f is a homogeneous function of x, j, 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 B. 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 It is then found that the principal equations of the problem are d at af_ d at of _ dO ax ax °' dB -- ay O' 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, y 1 d Of at 1 d at _Of a2f a2f y I de ax - ax _ x de a y ay=ayaxaxay +fl(xy-yx ), where 1 a 2 f a 2 r fl - y 2 a2 - - xy axay - x 2 ay2 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 of the j same side of the axis of x, it is required to find a curve ("He oining them, so that this curve may generate, by revolution, about the axis of x, a surface of minimum area. The integral to be made a minimum is 01Y(2+Ý2)ide, equation is d yx 2 2 integral is y x(x 2 , 2) -2 or c - 1+ ( d ) 2 x and the stationary curves are the catenaries y=c cosh{ ( x - a)/c}. 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- path of tions of the path of a ray of light in a variable medium.
According to Fermat's principle, the integral Lids is a a ray. minimum, ds representing the element of arc of a ray, and p the refractive index. Thus the integral to be made a minimum is Oi(2+)2+2) ido. The equations are found at once in forms of the type _ 2 2 d6 I? (x 2 +y 2 +$ 2) i ax (x o; and, since ( x 2 - 1 - 51 2 + 2)id0=ds, these equations can be written in the usual forms of the type ds (1 dx al c = ` ds 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 5x, Sy,.. . 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 mul- scribed value, is formed in this way, and if it vanishes, the tipsier. 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 E l 211+E2u2, where E 1 and 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, yl ) is movable on a given guiding curve ,(x i, y l) =0, the first variation of the integral can be written e?ul af +vaJ x=xl, y=y1 - J:: ? d at _af S u+ da y _a y ] v de, ax ayd6 ax ax dB a 'where (xi+Eul, Y1-I-evl) is on the curve cb(x i, yi) =o, and u 1, vl denote the values of u, v at (x i, y l). It follows that the required curve must be a stationary curve, and that the condition at a A at _ ° ax ay l - a-5;axl The corresponding condition in the case of Variable limits. at _, aF af _ aF ax a - y „ ay,- ay, to be F (x, y, (ax? - y and the principal of which the first The first tion. ary curves. Parametric method. z, must hold at (' x1, yl)
the integral F(x, y, y')dx is found from the equations 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 are near together, and let the line PQ be of length v, and make an angle w with the axis of x (fig. 1). 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 A aF
cos w F(x, y, y) -}- (tan ? - y) y, w In this formula x, y are the co-ordinates of x P, and y has the value belonging to the FIG. I. 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 transversal of those curves. In the problem of variable limits, when a terminal point moves on a given guiding curve, the integral cannot be an extremum unless the stationary curve along which it is taken is cut transversely 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 Alterna- (xi, yi) move on a fixed guiding curve C i. Now, whatever tive 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 i 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 i, y i) in terms of the co-ordinates x i, y i , and then making this expression an extremum, in regard to variations of x i, y i , by the methods of the differential calculus, subjecting ( x 1 , y i) to the condition of moving on the curve Ci.
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 qi, of the co-ordinates qi, 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 Principle T+V = E, of leas action. where E is a constant. A course of the system is defined when the co-ordinates q are expressed as functions of a single parameter e. The action A of the system is defined as the integral 2Tdt, taken along a course from the initial position (q(°)) to the final position (q')), but to and t i are not fixed. The equations of motion are the principal equations answering to this integral. To obtain them it is most convenient to write. 1)(q) for T, and to express the integral in the form f ©'2 (E - V) { .13(g') } 2de, where q' denotes the differential coefficient of a co-ordinate q with respect to 0, 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 aT aT aV dt 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 ( q(o )) 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 Principle of varying action. additive to A; and the courses of the system which are compatible with the equations of motion are determined by equations of the form aAOA aA aa = b i, aa = b 2 , ... aa - i = bn_i i 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 transformation; 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, y', ... y(n)) should be the differential coefficient of another expression of the form Fi (x, y, y ',... y(n - i)) is the identical vanishing of the expression aF d aF d 2 aF do aF ay - Tx ay' + dx 2 ay" -
+(-1) n ay(n). The result was first found by Euler (1744). A differential equation 4)(x, y, y ', y ") = o is the principal equation answering to an integral of the form f y, y')dx d a4 a4 _ dx ay" ay' is satisfied identically. In the more general case of an equation of the form.
vg5 (x, y, y',... y(2n)) = 0 the corresponding condition is that thedifferential obtained by Lagrange's process of variation, viz., 4 a¢ dsy 4 d 2 " 3y ay ° Y -r ay' dx +... +a (2n ) dx2n must be identical with the " adjoint " differential expression act. s d(a? s) d2 (a?) d2n (a¢ S) ay y ay' y + dx2 ay" Sy - .. + d x 2n a y (2n) y 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 ff,)(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 are ff (a4' a4) aw a4 aw a4 a 2 w a4 a 2 w 4 a 2 w ! az w+ ap ax + aq ay + ar ax: 2 + as axay + at ay 2 j dxdy. 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 3 2 w/ax e. We have ff a ? ZdxdY= a a? aw a a,?. aw ff ? ax (ar ax) ax (ar ?? dxdy _ a aw _ a a a,? a2 a,? - aY ax) w (aY) +w ax 2 dxdy. The first two terms are transformed into a line integral taken round the boundary s', and we thus find fJ -rax2d x d y ?cos(x,v ) a y -u'axM ds' + J fwax 2 (aY) 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 Ow aw Ow ax - 5; cos (x, v)-- (975; cos ( y, v), cos(x, v)cos(y, v) a as' ds' = - f a, ) cos(x, v)cos(y, v)- 8 - 7 wds'. if the equation Transversals of stationary curves. aA aT aq = aq' When the kinetic energy T is expressed as a homogeneous quadratic function of the momenta aT/aq, say T = 2?r, (bT8aT -aT), ( b = aqs 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 (brs 8 J +V =E. aqs A complete integral of this equation would yield an expression for A as a function of the q's containing n arbitrary constants, al, a n, of which one a n is merely Principle of varying action ized. Condition of integrability. Condition that a ential equation may arise from a problem of the calculus of variations. expression First variation of a double integral. cos'(x, v). 4 -- ,, ' P av ds' ., 5 / cos(x, v)cos(y, v)-5--r - - cos(x, v)- a 1% F) ] wds'. In forming the first term within the square brackets we then use the relations as'cos(x, v) _ - P ,cos(Y, v),a s ,cos(y, v) = P ,cos(x, v), a al a aq, a as' ar = - cos(y, where p denotes the radius of curvature of the curve s'. The necessity of freeing the calculus of variations from dependence upon the notion of infinitely small quantities was realized by Lagrange, and the process of discarding such quantities was partially carried out by him in his Theorie des fonctions analytiques (1797). In accordance with the interpretation of differentials which he made in that treatise, he interpreted the variation of an integral, as expressed by means of his symbol S, as the first term, or the sum of the terms of the first order, in the development in series of the complete expression for the change that is made in the value of the integral when small finite changes are made in the variables. The quantity which had been regarded as the variation of the integral came to be regarded as the first variation, and the discrimination between maxima and minima came to be regarded as requiring the investigation variation of the second variation. The first step in this theory had been taken by A. M. Legendre in 1786. In the case of an integral of the form y, Legendre defined the second variation as the integral rx,1 a2F 2 02F , 3°F 1, 2 x072a(8Y y 2) +2a y a y ,s y s y +a,2(Sy) To this expression he added the term [cy)2], which vanishes identically because ay vanishes at x =x 0 and at x =x1. He took a to satisfy the equation 0°F /0°F d a l / 8°F 2 a y, 2 a ye+dx - (93,831+a ), and thus transformed the expression for the second variation to o(y '+ niy)2dx, 8 2 F 3°F m a y ,2 = ayay1+ a.
From this investigation Legendre deduced a new condition for the existence of an extremum. It is necessary, not only that the variation should vanish, but also that the second variation should be one-signed. In the case of the First Problem Legendre concluded that this cannot happen unless 82F/ay'2 has the same sign at all points of the stationary curve between the end points, and that the sign must be +for a minimum and - for a maximum. In the application of the perametric method the function which has been denoted by f i takes the place of a2F/ay'2. The transformation of the second variations of integrals of various types into forms in which their signs can be determined by inspection subsequently became one of the leading problems of the calculus of variations. This result came about chiefly through the publica- Jacobi. tion in 1837 of a memoir by C. G. J. Jacobi. He trans formed Legendre's equation for the auxiliary function a into a linear differential equation of the second order by the substitution 0°F 8°F 1 dw ayay ,+a = - ay' 2 w and he pointed out that Legendre's transformation of the second variation cannot be effected if the function w vanishes between the limits of integration. He pointed out further, that if the stationary curves of the integral are given by an equation of the form y=0(x, a, b ), where a, b are arbitrary constants, the complete primitive of the equation for w is of the form W = A aa + B ab, where A, B are new arbitrary constants. Jacobi stated these propositions without proof, and the proof of them, and the extension of the results to more general problems, became the object of numerous investigations. These investigations were, for the most part, and for a long time, occupied almost exclusively with analytical developments; and the geometrical interpretation which Jacobi had given, and which he afterwards emphasized in his Vorlesungen fiber Dynamik, was neglected until rather recent times. According to this interpretation, the stationary curves which start from a point (xo, yo ) have an envelope; and the integral of F, taken along such a curve, cannot be an extremum if the point (60, no) where the curve touches the envelope lies on the arc between the end points. Pairs of points such as (x o, yo) and (60, no) were afterwards called conjugate points by Weierstrass. The proof that the in tegral cannot be an extremum if the arc of the curve between the fixed end points contains a pair of conjugate points was first published by G. Erdmann (1878). Examples of conjugate points are afforded by antipodal points on a sphere, the conjugate foci of geometrical optics, the kinetic foci of analytical dynamics. If the terminal points are a pair of conjugate points, the integral is not in general an extremum; but there is an exceptional case, of which a suitably chosen arc of the equator of an oblate spheroid may serve as an example. In the problem of the catenoid a pair of conjugate points on any of the catenaries, which are the stationary curves of the problem, is such that the tangents to the catenary at the two points A and A' meet on the axis of revolution (fig. 2). When both the end points of the required curve move on fixed guiding curves Co, C 1, a stationary curve C, joining a point Ao of Co to a point Ap of C 1, cannot yield an extremum unless it is cut transversely by Coo at Ao and by C 1 at A l. The envelope of stationary curves which set out from Co towards C 2, and are cut transversely by Co at points near Ao, meets C at a point Do; and the envelope of stationary curves which proceed from Co to C1, and are cut transversely by C 1 at points near A1, meets C at a point D 1. The curve C, drawn from Ao to A1, cannot yield an extremum if Do or D 1 lies between Ao and A1, or if Do lies between A 1 and D1. These results are due to G. A. Bliss (1903). A simple example is afforded by the shortest line on a sphere drawn from one small circle to another. In fig. 3 Do is that pole of the small circle A0B0 which occurs first on great circles. cutting A0B0 at right angles, and proceeding towards A i B 1; D 1 is that pole of the small circle A1B1 which occurs first on great circles cutting. A 1 B 1 at right angles, and drawn from points of A0B0. towards A 1 B 1. The arc AoA 1 is the required shortest line, and it is distinguished from BoB1 by the above criterion.
Jacobi's introduction of conjugate points is one of the germs. from which the modern theory of the calculus of variations has sprung. Another is a remark made by Legendre (1786) in regard to the solution of Newton's problem of the solid of least resistance. This problem requires that a curve be found for which the integral -yy ' 2 (1 should be a minimum. The stationary curves are given by the equation yy'3(i +y' 2)- 2 = const., a result equivalent to Newton's solution of the problem; but Legendre observed that, if the integral is taken along a broken line, consisting of two straight lines equally inclined to the axis of x in opposite senses, the integral can be made as small as we please by sufficiently diminishing the angle of inclination. Legendre's remark amounts to admitting a variation of Newton's curve, which is not a weak variation. Variations which are not weak are such that, while the points of a curve are but slightly displaced, the tangents. undergo large changes of direction. They are distinguished as strong variations. A general theory of strong variations in connexion with the First Problem, and of the conditions which are sufficient to secure that the integral taken along a stationary curve may be an extremum, was given by Weierstrass in lectures. He delivered courses of lectures on the calculus of variations in several years between 1865 and 1889, and his chief discoveries in the subject seem to have been included in the course for 1879. Through these lectures his theory became known to some students and teachers in Europe and America, and there have been published a few treatises and memoirs devoted to the exposition of his ideas.
| A | In the First Problem the following conditions are known to be necessary for an extremum. I. The path of integration must be a stationary curve. II. The expression a 2 F/8y' 2 , or the expression denoted by fi in the application of the parametric method, must not change sign at any point of this curve between the end points. III. The arc of the curve between the end points must not contain a pair of conjugate points. All these results are obtained by using weak variations. Additional v) ax ar +cos(x, v)-a--yar ' where FIG. 2. Missing image Variationscalculusof-1.jpg FIG. 3. results, relating to strong as well as weak variations, are obtained by a method which permits of the expression of the variation of an integral as a line integral taken along the varied curve. Let A, B be the end points, and let the stationary curve AB be drawn. If the end points A, B are not a pair of conjugate points, and if the point conjugate to A does not lie on the arc AB, then we may find a point A', on the backward continuation of the stationary curve BA beyond A, so near to A that the point conjugate to A' lies on the forward continuation of the arc AB beyond B. This being the case, it is possible to delimit an area of finite breadth, so that the arc AB of the stationary curve joining A, B lies entirely within the area, and no two stationary curves drawn through A' intersect within F the area. Through any point of such an area it is possible to draw one, and only one, stationary curve which passes curves through A'. This family of stationary curves is said to con stitute afield of stationary curves about the curve AB. We suppose that such a field exists, and that the varied curve Aqpb lies entirely within the delimited area. The variation of the integral fF(x, y, y')dx is identical with the line integral of F taken round a contour consisting of the varied curve Aqpb and the stationary curve AB, in the sense AqpbA. The line integral may, as usual, be replaced by the sum of line integrals taken round a series of cells, the external boundaries of the set of cells being identical with the Q, given contour, and the in ternal boundaries of ad B jacent cells being traversed twice in opposite senses. We may choose a suitable FIG. 4. set of cells as follows. Let Q, P be points on the varied curve, and let A'Q, A'P be the stationary curves of the field which pass through Q, P. Let P follow Q in the sense Aqpb in which the varied curve is described. Then the contour consisting of the stationary curve A'Q, expresfrom A' to Q, the varied curve QP, from Q to P, and the stationary curve A'P, from P to A', is the boundary of a cell (fig. 4). Let us denote the integral of F taken along a stationary curve by round brackets, thus and the integral of F taken along any other curve by square brackets, thus [PQ]. If the varied curve is divided into a number of arcs such as QP we have the result [[[Aqpb]]]-(AB) =Z{ (A 'Q) +[Q P ] - (A' P) }, and the right-hand member can be expressed as a line integral taken along the varied curve Aqpb. To effect this transformation we seek an approximate expression for the term (A'Q).-1-[QP] - (A'P) when Q, P are near together. Let as denote the arc QP, and ly the angle which the tangent at P to the varied curve, in the sense from A to B, makes with the axis of x (fig. 5). Also let 4 ) be the angle which the tangent at P to the stationary curve A'P, in the sense from A' to P, makes with x the axis of x. We evaluate (A'Q) - (A'P) approximately by means of a result which we obtained in connexion with the problem of variable limits. Observing that the angle here denoted by V, is equivalent to the angle formerly denoted by (cf. fig. I), while tan 49 is equivalent to the quantity formerly denoted by y', we obtain the approximate equation (A'Q) - (A'P) = - Os. cos V F(x,y,p) d -(tan - p) ap p= which is correct to the first order in Os. Also we have [QP] =0s. cos 1/iF(x,y, tan 4.) correctly to the same order. Hence we find that, correctly to the first order in As, (A'Q)+[QP] - (A'P) =E(x,y, tan 4 ), tan 4))os, taken along the varied curve. We can therefore state a sufficient (but not necessary) condition for the existence of an extremum in the form: - When the integral is taken along a stationary curve, and there is no pair of conjugate points on the arc of the curve terminated by the given end points, the integral is certainly an extremum if the excess function has the same sign at all points of a finite area containing the whole of this arc within Sufficient it. Further, we may specialize the excess function by identifying A' with A, and calculating the function for a point P on the arc AB of the stationary curve AB, and an arbitrary direction of the tangent at P to the varied curve. This process is equivalent to the introduction of a particular type of strong variation. We may in fact take, as a varied curve, the arc AQ of a neighbouring stationary curve, the straight line QP drawn from Q to a point of the arc AB, and the arc PB of the stationary curve AB (fig. 6). The sign of the variation is then the same as that of the function E(x, y, tan 4 ), tan 1P), where (x, y ) is Q the point P, is the angle which the straight line QP makes with the FIG. 6. axis of x, and 0 is the angle which the tangent at P to the curve APB makes with the same axis. We thus arrive at a new necessary (but not sufficient) condition for the existence of an extremum of the integral fFds, viz. the specialized excess function, so calculated, must not change sign between A and B. The sufficient condition, and the new necessary condition, associated with the excess function, as well as the expression for the variation as fEds, are due to Weierstrass. In applications to special problems it is generally permissible to identify A' with A, and to regard QP as straight. The direction of QP must be such that the integral of F taken along it is finite and real. We shall describe such directions as admissible. In the statement of the sufficient condition, and the new necessary condition, it is of course understood that the direction specified by V, is admissible. The excess function generally vanishes if V. =49, but it does not change sign. It can be shown without difficulty that, when 4) is very nearly equal to 0, the sign of E is the same as that of ( ay a2F/ tan ? - tan ?) 2 cos(2 y ' =tan and thus the necessary condition as to the sign of the excess function includes Legendre's condition as to the sign of a1F/ay'2. Weierstrass's conditions have been obtained by D. Hilbert from the observation that, if p is a function of l x and y, the integral F(x,y,p) +(y' - p) y, dx, taken along a curve joining two fixed points, has the same value for all such curves, provided that there is a field of stationary curves, and that p is the gradient at the point ( x, y ) of that stationary curve of the field which passes through this point. An instructive example of the excess function, and the conditions connected with it, is afforded by the integral f y2y " dx or f y23y-2(10. The first integral of the principal equation is y 2x2y z =const., and the stationary curves include the axis of x, straight lines parallel to the axis of y, and the family of exponential curves y =ae cx . A field of stationary curves is expressed by the equation y = yo exp { c (x - xo) }, and, as these have no envelope other than the initial point (xo, yo), there are no conjugate points. The function f l is 6g4, and this is positive for curves going from the initial point in the positive direction of the axis of x. The value of the excess function is y 2 cos 4)(cot 2 1L-3 cot 2 q5 +2 tan 1P cot34)). The directions 4) = o and V, = it are inadmissible. On putting 4) = fir we get 2y 2 cot 3 4; and on putting 1,1, = 2,r we get - 2y 2 cot 3 4. Hence the integral taken along AQ'PB is greater than that taken along APB, and the integral taken along Aqpb is less than that taken along APB, when Q'Q are sufficiently near to P on the ordinate of P (fig. 7). It follows that the integral is neither a maximum 'y nor a minimum. Missing image Variationscalculusof-2.jpg It has been proved by Weierstrass that the excess function cannot be one-signed if the function f of the parametric method is a rational function of a and y. This result includes the above example, and the problem of the solid of least resistance, for which, as Legendre had seen, there can be no solu tion if strong variations are admitted. As another example of the calculation of excess functions, it may be noted that the value of the excess function in the problem of the catenoid is 2y sin22(41 - 0). FIG. 5. where E(x, y, tan 43, tan 4)) =cos V, F(x,y, tanF x,F 'G) - (y p) - (tantan - p)ap p= When the parametric method is used the function E takes the form of 3f of of where A, µ are the direction cosines of the tangent at P to the curve Aqpb, in the sense from A to B, and 1, m are the direction cosines of the tangent at P to the stationary curve A'P, in the sense from A' to P.
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