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Brachistochrone curve

In physics and mathematics, a brachistochrone curve, or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by Johann Bernoulli in 1696.

The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. However, the portion of the cycloid used for each of the two varies. More specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp. In contrast, the tautochrone problem can use only up to the first half rotation, and always ends at the horizontal. The problem can be solved using tools from the calculus of variations and optimal control.

The curve is independent of both the mass of the test body and the local strength of gravity. Only a parameter is chosen so that the curve fits the starting point A and the ending point B. If the body is given an initial velocity at A, or if friction is taken into account, then the curve that minimizes time differs from the tautochrone curve.

Galileo's problem
Earlier, in 1638, Galileo Galilei had tried to solve a similar problem for the path of the fastest descent from a point to a wall in his Two New Sciences. He draws the conclusion that the arc of a circle is faster than any number of its chords, From the preceding it is possible to infer that the quickest path of all [lationem omnium velocissimam], from one point to another, is not the shortest path, namely, a straight line, but the arc of a circle. ...

Consequently the nearer the inscribed polygon approaches a circle the shorter the time required for descent from A to C. What has been proven for the quadrant holds true also for smaller arcs; the reasoning is the same.

Just after Theorem 6 of Two New Sciences, Galileo warns of possible fallacies and the need for a "higher science". In this dialogue Galileo reviews his own work. Galileo studied the cycloid and gave it its name, but the connection between it and his problem had to wait for advances in mathematics.

Galileo’s conjecture is that “The shortest time of all [for a movable body] will be that of its fall along the arc ADB [of a quarter circle] and similar properties are to be understood as holding for all lesser arcs taken upward from the lowest limit B.”

In Fig.1, from the “Dialogue Concerning the Two Chief World Systems”, Galileo claims that the body sliding along the circular arc of a quarter circle, from A to B will reach B in less time than if it took any other path from A to B. Similarly, in Fig. 2, from any point D on the arc AB, he claims that the time along the lesser arc DB will be less than for any other path from D to B. In fact, the quickest path from A to B or from D to B, the brachistochrone, is a cycloidal arc, which is shown in Fig. 3 for the path from A to B, and Fig.4 for the path from D to B, superposed on the respective circular arc.

Introduction of the problem
Johann Bernoulli posed the problem of the brachistochrone to the readers of Acta Eruditorum in June, 1696. Solutions to Johann Bernoulli's problem of 1696:
 * Isaac Newton (January 1697) "De ratione temporis quo grave labitur per rectam data duo puncta conjungentem, ad tempus brevissimum quo, vi gravitatis, transit ab horum uno ad alterum per arcum cycloidis" (On a proof [that] the time in which a weight slides by a line joining two given points [is] the shortest in terms of time when it passes, via gravitational force, from one of these [points] to the other through a cycloidal arc), Philosophical Transactions of the Royal Society of London, 19 : 424-425.
 * G.G.L. (Gottfried Wilhelm Leibniz) (May 1697) "Communicatio suae pariter, duarumque alienarum ad edendum sibi primum a Dn. Jo. Bernoullio, deinde a Dn. Marchione Hospitalio communicatarum solutionum problematis curva celerrimi descensus a Dn. Jo. Bernoullio Geometris publice propositi, una cum solutione sua problematis alterius ab eodem postea propositi." (His communication together with [those] of two others in a report to him first from Johann Bernoulli, [and] then from the Marquis de l'Hôpital, of reported solutions of the problem of the curve of quickest descent, [which was] publicly proposed by Johann Bernoulli, geometer — one with a solution of his other problem proposed afterward by the same [person].), Acta Eruditorum, 19 : 201–205.
 * Johann Bernoulli (May 1697) "Curvatura radii in diaphanis non uniformibus, Solutioque Problematis a se in Actis 1696, p. 269, propositi, de invenienda Linea Brachystochrona, id est, in qua grave a dato puncto ad datum punctum brevissimo tempore decurrit, & de curva Synchrona seu radiorum unda construenda." (The curvature of [light] rays in non-uniform media, and a solution of the problem [which was] proposed by me in the Acta Eruditorum of 1696, p. 269, from which is to be found the brachistochrone line [i.e., curve], that is, in which a weight descends from a given point to a given point in the shortest time, and on constructing the tautochrone or the wave of [light] rays.), Acta Eruditorum, 19 : 206–211.
 * Jacob Bernoulli (May 1697) "Solutio problematum fraternorum, … " (A solution of [my] brother's problems, … ), Acta Eruditorum, 19 : 211–214.
 * Marquis de l'Hôpital (May 1697) "Domini Marchionis Hospitalii solutio problematis de linea celerrimi descensus" (Lord Marquis de l'Hôpital's solution of the problem of the line of fastest descent), Acta Eruditorum, 19 : 217-220.
 * reprinted: Isaac Newton (May 1697)  "Excerpta ex Transactionibus Philos. Anglic. M. Jan. 1697." (Excerpt from the English Philosophical Transactions of the month of January in 1697), Acta Eruditorum, 19 :  223–224. He said:

"I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise"

Bernoulli wrote the problem statement as:

"Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time''."

Johann and his brother Jakob Bernoulli derived the same solution, but Johann's derivation was incorrect, and he tried to pass off Jakob's solution as his own. Johann published the solution in the journal in May of the following year, and noted that the solution is the same curve as Huygens's tautochrone curve. After deriving the differential equation for the curve by the method given below, he went on to show that it does yield a cycloid. However, his proof is marred by his use of a single constant instead of the three constants, vm, 2g and D, below.

Bernoulli allowed six months for the solutions but none were received during this period. At the request of Leibniz, the time was publicly extended for a year and a half. At 4 p.m. on 29 January 1697 when he arrived home from the Royal Mint, Isaac Newton found the challenge in a letter from Johann Bernoulli. Newton stayed up all night to solve it and mailed the solution anonymously by the next post. Upon reading the solution, Bernoulli immediately recognized its author, exclaiming that he "recognizes a lion from his claw mark". This story gives some idea of Newton's power, since Johann Bernoulli took two weeks to solve it. Newton also wrote, "I do not love to be dunned [pestered] and teased by foreigners about mathematical things...", and Newton had already solved Newton's minimal resistance problem, which is considered the first of the kind in calculus of variations.

In the end, five mathematicians responded with solutions: Newton, Jakob Bernoulli, Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l'Hôpital. Four of the solutions (excluding l'Hôpital's) were published in the same edition of the journal as Johann Bernoulli's. In his paper, Jakob Bernoulli gave a proof of the condition for least time similar to that below before showing that its solution is a cycloid. According to Newtonian scholar Tom Whiteside, in an attempt to outdo his brother, Jakob Bernoulli created a harder version of the brachistochrone problem. In solving it, he developed new methods that were refined by Leonhard Euler into what the latter called (in 1766) the calculus of variations. Joseph-Louis Lagrange did further work that resulted in modern infinitesimal calculus.

Introduction
In a letter to L’Hôpital, (21/12/1696), Bernoulli stated that when considering the problem of the curve of quickest descent, after only 2 days he noticed a curious affinity or connection with another no less remarkable problem leading to an ‘indirect method’ of solution. Then shortly afterwards he discovered a ‘direct method’.

Direct method
In a letter to Henri Basnage, held at the University of Basel Public Library, dated 30 March 1697, Johann Bernoulli stated that he had found two methods (always referred to as "direct" and "indirect") to show that the Brachistochrone was the "common cycloid", also called the "roulette". Following advice from Leibniz, he included only the indirect method in the Acta Eruditorum Lipsidae of May 1697. He wrote that this was partly because he believed it was sufficient to convince anyone who doubted the conclusion, partly because it also resolved two famous problems in optics that "the late Mr. Huygens" had raised in his treatise on light. In the same letter he criticised Newton for concealing his method.

In addition to his indirect method he also published the five other replies to the problem that he received.

Johann Bernoulli's direct method is historically important as a proof that the brachistochrone is the cycloid. The method is to determine the curvature of the curve at each point. All the other proofs, including Newton's (which was not revealed at the time) are based on finding the gradient at each point.

In 1718, Bernoulli explained how he solved the brachistochrone problem by his direct method.

He explained that he had not published it in 1697, for reasons that no longer applied in 1718. This paper was largely ignored until 1904 when the depth of the method was first appreciated by Constantin Carathéodory, who stated that it shows that the cycloid is the only possible curve of quickest descent. According to him, the other solutions simply implied that the time of descent is stationary for the cycloid, but not necessarily the minimum possible.

Analytic solution

A body is regarded as sliding along any small circular arc Ce between the radii KC and Ke, with centre K fixed. The first stage of the proof involves finding the particular circular arc, Mm, which the body traverses in the minimum time.

The line KNC intersects AL at N, and line Kne intersects it at n, and they make a small angle CKe at K. Let NK = a, and define a variable point, C on KN extended. Of all the possible circular arcs Ce, it is required to find the arc Mm, which requires the minimum time to slide between the 2 radii, KM and Km. To find Mm Bernoulli argues as follows.

Let MN = x. He defines m so that MD = mx, and n so that Mm = nx + na and notes that x is the only variable and that m is finite and n is infinitely small. The small time to travel along arc Mm is $$ \frac{Mm}{MD^{\frac{1}{2}}} = \frac{n(x + a)}{(mx)^{\frac{1}{2}}} $$, which has to be a minimum (‘un plus petit’). He does not explain that because Mm is so small the speed along it can be assumed to be the speed at M, which is as the square root of MD, the vertical distance of M below the horizontal line AL.

It follows that, when differentiated this must give

\frac{(x - a)dx}{2x^{\frac{3}{2}}} = 0 $$ so that x = a.

This condition defines the curve that the body slides along in the shortest time possible. For each point, M on the curve, the radius of curvature, MK is cut in 2 equal parts by its axis AL. This property, which Bernoulli says had been known for a long time, is unique to the cycloid.

Finally, he considers the more general case where the speed is an arbitrary function X(x), so the time to be minimised is $$ \frac{(x + a)}{X} $$. The minimum condition then becomes $$ X = \frac{(x + a)dX}{dx} $$ which he writes as :$$ X = (x + a)\Delta x $$ and which gives MN (=x) as a function of NK (= a). From this the equation of the curve could be obtained from the integral calculus, though he does not demonstrate this.

Synthetic solution
He then proceeds with what he called his Synthetic Solution, which was a classical, geometrical proof, that there is only a single curve that a body can slide down in the minimum time, and that curve is the cycloid.

The reason for the synthetic demonstration, in the manner of the ancients, is to convince Mr de la Hire. He has little time for our new analysis, describing it as false (He claims he has found 3 ways to prove that the curve is a cubic parabola) – Letter from Johan Bernoulli to Pierre Varignon dated 27 Jul 1697.

Assume AMmB is the part of the cycloid joining A to B, which the body slides down in the minimum time. Let ICcJ be part of a different curve joining A to B, which can be closer to AL than AMmB. If the arc Mm subtends the angle MKm at its centre of curvature, K, let the arc on IJ that subtends the same angle be Cc. The circular arc through C with centre K is Ce. Point D on AL is vertically above M. Join K to D and point H is where CG intersects KD, extended if necessary.

Let $$ \tau $$ and t be the times the body takes to fall along Mm and Ce respectively.

 * $$ \tau \propto \frac{Mm}{MD^{\frac{1}{2}}} $$, $$ t \propto \frac{Ce}{CG^{\frac{1}{2}}} $$,

Extend CG to point F where, $$ CF = \frac{CH^2}{MD} $$ and since $$ \frac{Mm}{Ce}  = \frac{MD}{CH} $$, it follows that
 * $$ \frac {\tau}{t} = \frac{Mm}{Ce}.\left({\frac{CG}{MD}}\right)^{\frac{1}{2}} = \left({\frac{CG}{CF}}\right)^{\frac{1}{2}} $$

Since MN = NK, for the cycloid:
 * $$ GH = \frac{MD.HD}{DK} = \frac{MD.CM}{MK} $$, $$ CH = \frac{MD.CK}{MK} = \frac{MD.(MK + CM)}{MK} $$, and $$ CG = CH + GH = \frac{MD.(MK + 2CM)}{MK} $$

If Ce is closer to K than Mm then
 * $$ CH = \frac{MD.(MK - CM)}{MK} $$ and $$ CG = CH - GH = \frac{MD.(MK - 2CM)}{MK} $$

In either case,
 * $$ CF = \frac{CH^2}{MD} > CG $$, and it follows that $$ \tau < t $$

If the arc, Cc subtended by the angle infinitesimal angle MKm on IJ is not circular, it must be greater than Ce, since Cec becomes a right-triangle in the limit as angle MKm approaches zero.

Note, Bernoulli proves that CF > CG by a similar but different argument.

From this he concludes that a body traverses the cycloid AMB in less time than any other curve ACB.

Indirect method
According to Fermat’s principle, the actual path between two points taken by a beam of light is one that takes the least time. In 1697 Johann Bernoulli used this principle to derive the brachistochrone curve by considering the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration (that of gravity g).

By the conservation of energy, the instantaneous speed of a body v after falling a height y in a uniform gravitational field is given by:
 * $$v=\sqrt{2gy}$$,

The speed of motion of the body along an arbitrary curve does not depend on the horizontal displacement.

Bernoulli noted that the law of refraction gives a constant of the motion for a beam of light in a medium of variable density:
 * $$\frac{\sin{\theta}}{v}=\frac{1}{v}\frac{dx}{ds}=\frac{1}{v_m}$$,

where vm is the constant and $$\theta$$ represents the angle of the trajectory with respect to the vertical.

The equations above lead to two conclusions:
 * 1) At the onset, the angle must be zero when the particle speed is zero. Hence, the brachistochrone curve is tangent to the vertical at the origin.
 * 2) The speed reaches a maximum value when the trajectory becomes horizontal and the angle θ = 90°.

Assuming for simplicity that the particle (or the beam) with coordinates (x,y) departs from the point (0,0) and reaches maximum speed after falling a vertical distance D:
 * $$v_m=\sqrt{2gD}$$.

Rearranging terms in the law of refraction and squaring gives:
 * $$v_m^2 dx^2=v^2 ds^2=v^2 (dx^2+dy^2)$$

which can be solved for dx in terms of dy:
 * $$dx=\frac{v\, dy}{\sqrt{v_m^2-v^2}}$$.

Substituting from the expressions for v and vm above gives:
 * $$dx=\sqrt{\frac{y}{D-y}}\,dy\,,$$

which is the differential equation of an inverted cycloid generated by a circle of diameter D=2r, whose parametric equation is:
 * $$\begin{align}

x &= r(\varphi - \sin \varphi) \\ y &= r(1 - \cos \varphi). \end{align}$$ where φ is a real parameter, corresponding to the angle through which the rolling circle has rotated. For given φ, the circle's centre lies at $(x, y) = (rφ, r)$.

In the brachistochrone problem, the motion of the body is given by the time evolution of the parameter:
 * $$\varphi(t)=\omega t\,,\omega=\sqrt{\frac{g}{r}}$$

where t is the time since the release of the body from the point (0,0).

Jakob Bernoulli's solution
Johann's brother Jakob showed how 2nd differentials can be used to obtain the condition for least time. A modernized version of the proof is as follows. If we make a negligible deviation from the path of least time, then, for the differential triangle formed by the displacement along the path and the horizontal and vertical displacements,

 * $$ds^2=dx^2+dy^2$$.

On differentiation with dy fixed we get,

 * $$2ds\ d^2s=2dx\ d^2x$$.

And finally rearranging terms gives,

 * $$\frac{dx}{ds}d^2x=d^2s=v\ d^2t$$

where the last part is the displacement for given change in time for 2nd differentials. Now consider the changes along the two neighboring paths in the figure below for which the horizontal separation between paths along the central line is d2x (the same for both the upper and lower differential triangles). Along the old and new paths, the parts that differ are,

 * $$d^2t_1=\frac{1}{v_1}\frac{dx_1}{ds_1}d^2x$$
 * $$d^2t_2=\frac{1}{v_2}\frac{dx_2}{ds_2}d^2x$$

For the path of least times these times are equal so for their difference we get,
 * $$d^2t_2-d^2t_1=0=\bigg(\frac{1}{v_2}\frac{dx_2}{ds_2}-\frac{1}{v_1}\frac{dx_1}{ds_1}\bigg)d^2x$$

And the condition for least time is,
 * $$\frac{1}{v_2}\frac{dx_2}{ds_2}=\frac{1}{v_1}\frac{dx_1}{ds_1}$$

which agrees with Johann's assumption based on the law of refraction.

Introduction
In June 1696, Johann Bernoulli had used the pages of the Acta Eruditorum Lipsidae to pose a challenge to the international mathematical community: to find the form of the curve joining two fixed points so that a mass will slide down along it, under the influence of gravity alone, in the minimum amount of time. The solution was originally to be submitted within six months. At the suggestion of Leibniz, Bernoulli extended the challenge until Easter 1697, by means of a printed text called "Programma", published in Groningen, in the Netherlands.

The Programma is dated 1 January 1697, in the Gregorian Calendar. This was 22 December 1696 in the Julian Calendar, in use in Britain. According to Newton's niece, Catherine Conduitt, Newton learned of the challenge at 4 pm on 29 January and had solved it by 4 am the following morning. His solution, communicated to the Royal Society, is dated 30 January. This solution, later published anonymously in the Philosophical Transactions, is correct but does not indicate the method by which Newton arrived at his conclusion. Bernoulli, writing to Henri Basnage in March 1697, indicated that even though its author, "by an excess of modesty", had not revealed his name, yet even from the scant details supplied it could be recognised as Newton's work, "as the lion by its claw" (in Latin, ex ungue Leonem).

D. T. Whiteside notes that the letter in French has ex ungue Leonem preceded by the French word comme. The much quoted version tanquam ex ungue Leonem is due to David Brewster's 1855 book on the life and works of Newton. Bernoulli's intention was, Whiteside argues, simply to indicate he could tell the anonymous solution was Newton's, just as it was possible to tell that an animal was a lion given its claw; it was not meant to suggest that Bernoulli considered Newton to be the lion among mathematicians, as it has since come to be interpreted.

John Wallis, who was 80 years old at the time, had learned of the problem in September 1696 from Johann Bernoulli's youngest brother Hieronymus, and had spent three months attempting a solution before passing it in December to David Gregory, who also failed to solve it. After Newton had submitted his solution, Gregory asked him for the details and made notes from their conversation. These can be found in the University of Edinburgh Library, manuscript A $$78^1$$, dated 7 March 1697. Either Gregory did not understand Newton's argument, or Newton's explanation was very brief. However, it is possible, with a high degree of confidence, to construct Newton's proof from Gregory's notes, by analogy with his method to determine the solid of minimum resistance (Principia, Book 2, Proposition 34, Scholium 2). A detailed description of his solution of this latter problem is included in the draft of a letter in 1694, also to David Gregory. In addition to the minimum time curve problem, there was a second problem that Newton also solved at the same time. Both solutions appeared anonymously in Philosophical Transactions of the Royal Society, for January 1697.

The Brachistochrone problem

Fig. 1, shows Gregory’s diagram (except the additional line IF is absent from it, and Z, the start point has been added). The curve ZVA is a cycloid and CHV is its generating circle. Since it appears that the body is moving upward from e to E, it must be assumed that a small body is released from Z and slides along the curve to A, without friction, under the action of gravity.

Consider a small arc eE, which the body is ascending. Assume that it traverses the straight line eL to point L, horizontally displaced from E by a small distance, o, instead of the arc eE. Note, that eL is not the tangent at e, and that o is negative when L is between B and E. Draw the line through E parallel to CH, cutting eL at n. From a property of the cycloid, En is the normal to the tangent at E, and similarly the tangent at E is parallel to VH.

Since the displacement EL is small, it differs little in direction from the tangent at E so that the angle EnL is close to a right-angle. In the limit as the arc eE approaches zero, eL becomes parallel to VH, provided o is small compared to eE making the triangles EnL and CHV similar.

Also en approaches the length of chord eE, and the increase in length, $$eL - eE = nL= \frac{o.CH}{CV}$$, ignoring terms in $$ o^2 $$ and higher, which represent the error due to the approximation that eL and VH are parallel.

The speed along eE or eL can be taken as that at E, proportional to $$\sqrt{CB}$$, which is as CH, since $$CH=\sqrt{CB.CV}$$

This appears to be all that Gregory’s note contains.

Let t be the additional time to reach L,

$$t \propto \frac{nL}{\sqrt{CB}} =\frac{o.CH}{CV.\sqrt{CB}} = \frac{o}{\sqrt{CV}}$$

Therefore, the increase in time to traverse a small arc displaced at one endpoint depends only on the displacement at the endpoint and is independent of the position of the arc. However, by Newton’s method, this is just the condition required for the curve to be traversed in the minimum time possible. Therefore, he concludes that the minimum curve must be the cycloid.

He argues as follows.

Assuming now that Fig. 1 is the minimum curve not yet determined, with vertical axis CV, and the circle CHV removed, and Fig. 2 shows part of the curve between the infinitesimal arc eE and a further infinitesimal arc Ff a finite distance along the curve. The extra time, t, to traverse eL (rather than eE) is nL divided by the speed at E (proportional to $$\sqrt{CB}$$), ignoring terms in $$ o^2 $$ and higher:

$$t \propto \frac{o.DE}{eE.\sqrt{CB}}$$,

At L the particle continues along a path LM, parallel to the original EF, to some arbitrary point M. As it has the same speed at L as at E, the time to traverse LM is the same as it would have been along the original curve EF. At M it returns to the original path at point f. By the same reasoning, the reduction in time, T, to reach f from M rather than from F is

$$T \propto \frac{o.FG}{Ff.\sqrt{CI}}$$

The difference (t – T) is the extra time it takes along the path eLMf compared to the original eEFf :

$$(t - T) \propto \left(\frac{DE}{eE\sqrt{CB}} - \frac{FG}{Ff\sqrt{CI}}\right).o $$ plus terms in $$ o^2 $$ and higher  (1)

Because eEFf is the minimum curve, (t – T) is must be greater than zero, whether o is positive or negative. It follows that the coefficient of o in (1) must be zero:

$$ \frac{DE}{eE\sqrt{CB}} = \frac{FG}{Ff\sqrt{CI}} $$    (2) in the limit as eE and fF approach zero. Note since eEFf is the minimum curve it has to be assumed that the coefficient of $$ o^2 $$ is greater than zero.

Clearly there has to be 2 equal and opposite displacements, or the body would not return to the endpoint, A, of the curve.

If e is fixed, and if f is considered a variable point higher up the curve, then for all such points, f, $$ \frac{FG}{Ff\sqrt{CI}} $$ is constant (equal to $$ \frac{DE}{eE\sqrt{CB}} $$). By keeping f fixed and making e variable it is clear that $$ \frac{DE}{eE\sqrt{CB}} $$ is also constant.

But, since points, e and f are arbitrary, equation (2) can be true only if $$ \frac{DE}{eE\sqrt{CB}} = \text{constant} $$,  everywhere, and this condition characterises the curve that is sought. This is the same technique he uses to find the form of the Solid of Least Resistance.

For the cycloid, $$ \frac{DE}{eE} = \frac{BH}{VH} = \frac{CH}{CV} $$, so that $$  \frac{DE}{eE\sqrt{CB}} = \frac{CH}{CV.\sqrt{CB}} $$, which was shown above to be constant, and the Brachistochrone is the cycloid.

Newton gives no indication of how he discovered that the cycloid satisfied this last relation. It may have been by trial and error, or he may have recognised immediately that it implied the curve was the cycloid.

title: Brachistochrone curve


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Talk:Memphis Blues

Am I imagining things, or wasn't there an article under this title a short time ago? Wodering simply, -- Infrogmation 20:49 Jan 20, 2003 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Memphis Blues


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Calculus of variations

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.

Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet's principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

History
The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696). It immediately occupied the attention of Jacob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Lagrange was influenced by Euler's work to contribute significantly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum.

Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. Isaac Newton and Gottfried Leibniz also gave some early attention to the subject. To this discrimination Vincenzo Brunacci (1810), Carl Friedrich Gauss (1829), Siméon Poisson (1831), Mikhail Ostrogradsky (1834), and Carl Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), Otto Hesse (1857), Alfred Clebsch (1858), and Lewis Buffett Carll (1885), but perhaps the most important work of the century is that of Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The 20th and the 23rd Hilbert problem published in 1900 encouraged further development.

In the 20th century David Hilbert, Oskar Bolza, Gilbert Ames Bliss, Emmy Noether, Leonida Tonelli, Henri Lebesgue and Jacques Hadamard among others made significant contributions. Marston Morse applied calculus of variations in what is now called Morse theory. Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. The dynamic programming of Richard Bellman is an alternative to the calculus of variations.

Extrema
The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps functions to scalars, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements $$y$$ of a given function space defined over a given domain. A functional $$J[y]$$ is said to have an extremum at the function $$f$$ if $$\Delta J = J[y] - J[f]$$ has the same sign for all $$y$$ in an arbitrarily small neighborhood of $$f.$$ The function $$f$$ is called an extremal function or extremal. The extremum $$J[f]$$ is called a local maximum if $$\Delta J \leq 0$$ everywhere in an arbitrarily small neighborhood of $$f,$$ and a local minimum if $$\Delta J \geq 0$$ there. For a function space of continuous functions, extrema of corresponding functionals are called strong extrema or weak extrema, depending on whether the first derivatives of the continuous functions are respectively all continuous or not.

Both strong and weak extrema of functionals are for a space of continuous functions but strong extrema have the additional requirement that the first derivatives of the functions in the space be continuous. Thus a strong extremum is also a weak extremum, but the converse may not hold. Finding strong extrema is more difficult than finding weak extrema. An example of a necessary condition that is used for finding weak extrema is the Euler–Lagrange equation.

Euler–Lagrange equation
Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions for which the functional derivative is equal to zero. This leads to solving the associated Euler–Lagrange equation.

Consider the functional $$J[y] = \int_{x_1}^{x_2} L\left(x,y(x),y'(x)\right)\, dx \, .$$ where
 * $$x_1, x_2$$ are constants,
 * $$y(x)$$ is twice continuously differentiable,
 * $$y'(x) = \frac{dy}{dx},$$
 * $$L\left(x, y(x), y'(x)\right)$$ is twice continuously differentiable with respect to its arguments $$x, y,$$ and $$y'.$$

If the functional $$J[y]$$ attains a local minimum at $$f,$$ and $$\eta(x)$$ is an arbitrary function that has at least one derivative and vanishes at the endpoints $$x_1$$ and $$x_2,$$ then for any number $$\varepsilon$$ close to 0, $$J[f] \le J[f + \varepsilon \eta] \, .$$

The term $$\varepsilon \eta$$ is called the variation of the function $$f$$ and is denoted by $$\delta f.$$

Substituting $$f + \varepsilon \eta$$ for $$y$$ in the functional $$J[y],$$ the result is a function of $$\varepsilon,$$

$$\Phi(\varepsilon) = J[f+\varepsilon\eta] \, .$$ Since the functional $$J[y]$$ has a minimum for $$y = f$$ the function $$\Phi(\varepsilon)$$ has a minimum at $$\varepsilon = 0$$ and thus, $$\Phi'(0) \equiv \left.\frac{d\Phi}{d\varepsilon}\right|_{\varepsilon = 0} = \int_{x_1}^{x_2} \left.\frac{dL}{d\varepsilon}\right|_{\varepsilon = 0} dx = 0 \, .$$

Taking the total derivative of $$L\left[x, y, y'\right],$$ where $$y = f + \varepsilon \eta$$ and $$y' = f' + \varepsilon \eta'$$ are considered as functions of $$\varepsilon$$ rather than $$x,$$ yields $$\frac{dL}{d\varepsilon}=\frac{\partial L}{\partial y}\frac{dy}{d\varepsilon} + \frac{\partial L}{\partial y'}\frac{dy'}{d\varepsilon}$$ and because $$\frac{dy}{d \varepsilon} = \eta$$ and $$\frac{d y'}{d \varepsilon} = \eta',$$ $$\frac{dL}{d\varepsilon}=\frac{\partial L}{\partial y}\eta + \frac{\partial L}{\partial y'}\eta'.$$

Therefore, $$\begin{align} \int_{x_1}^{x_2} \left.\frac{dL}{d\varepsilon}\right|_{\varepsilon = 0} dx & = \int_{x_1}^{x_2} \left(\frac{\partial L}{\partial f} \eta + \frac{\partial L}{\partial f'} \eta'\right)\, dx \\ & = \int_{x_1}^{x_2} \frac{\partial L}{\partial f} \eta \, dx + \left.\frac{\partial L}{\partial f'} \eta \right|_{x_1}^{x_2} - \int_{x_1}^{x_2} \eta \frac{d}{dx}\frac{\partial L}{\partial f'} \, dx \\ & = \int_{x_1}^{x_2} \left(\frac{\partial L}{\partial f} \eta - \eta \frac{d}{dx}\frac{\partial L}{\partial f'} \right)\, dx\\ \end{align}$$ where $$L\left[x, y, y'\right] \to L\left[x, f, f'\right]$$ when $$\varepsilon = 0$$ and we have used integration by parts on the second term. The second term on the second line vanishes because $$\eta = 0$$ at $$x_1$$ and $$x_2$$ by definition. Also, as previously mentioned the left side of the equation is zero so that $$\int_{x_1}^{x_2} \eta (x) \left(\frac{\partial L}{\partial f} - \frac{d}{dx}\frac{\partial L}{\partial f'} \right) \, dx = 0 \, .$$

According to the fundamental lemma of calculus of variations, the part of the integrand in parentheses is zero, i.e. $$\frac{\partial L}{\partial f} -\frac{d}{dx} \frac{\partial L}{\partial f'}=0$$ which is called the Euler–Lagrange equation. The left hand side of this equation is called the functional derivative of $$J[f]$$ and is denoted $$\delta J/\delta f(x).$$

In general this gives a second-order ordinary differential equation which can be solved to obtain the extremal function $$f(x).$$ The Euler–Lagrange equation is a necessary, but not sufficient, condition for an extremum $$J[f].$$  A sufficient condition for a minimum is given in the section Variations and sufficient condition for a minimum.

Example
In order to illustrate this process, consider the problem of finding the extremal function $$y = f(x),$$ which is the shortest curve that connects two points $$\left(x_1, y_1\right)$$ and $$\left(x_2, y_2\right).$$ The arc length of the curve is given by $$A[y] = \int_{x_1}^{x_2} \sqrt{1 + [ y'(x) ]^2} \, dx \, ,$$ with $$y'(x) = \frac{dy}{dx} \,, \ \ y_1=f(x_1) \, , \ \ y_2=f(x_2) \, .$$ Note that assuming $y$ is a function of $x$ loses generality; ideally both should be a function of some other parameter. This approach is good solely for instructive purposes.

The Euler–Lagrange equation will now be used to find the extremal function $$f(x)$$ that minimizes the functional $$A[y].$$ $$\frac{\partial L}{\partial f} -\frac{d}{dx} \frac{\partial L}{\partial f'}=0$$ with $$L = \sqrt{1 + [ f'(x) ]^2} \, .$$

Since $$f$$ does not appear explicitly in $$L,$$ the first term in the Euler–Lagrange equation vanishes for all $$f(x)$$ and thus, $$\frac{d}{dx} \frac{\partial L}{\partial f'} = 0 \, .$$ Substituting for $$L$$ and taking the derivative, $$\frac{d}{dx} \ \frac{f'(x)} {\sqrt{1 + [f'(x)]^2}} \ = 0 \, .$$

Thus $$\frac{f'(x)}{\sqrt{1+[f'(x)]^2}} = c \, ,$$ for some constant $$c.$$ Then $$\frac{[f'(x)]^2}{1+[f'(x)]^2} = c^2 \, ,$$ where $$0 \le c^2<1.$$ Solving, we get $$[f'(x)]^2=\frac{c^2}{1-c^2}$$ which implies that $$f'(x)=m$$ is a constant and therefore that the shortest curve that connects two points $$\left(x_1, y_1\right)$$ and $$\left(x_2, y_2\right)$$ is $$f(x) = m x + b \qquad \text{with} \ \ m = \frac{y_2 - y_1}{x_2 - x_1} \quad \text{and} \quad b = \frac{x_2 y_1 - x_1 y_2}{x_2 - x_1}$$ and we have thus found the extremal function $$f(x)$$ that minimizes the functional $$A[y]$$ so that $$A[f]$$ is a minimum. The equation for a straight line is $$y = f(x).$$ In other words, the shortest distance between two points is a straight line.

Beltrami's identity
In physics problems it may be the case that $$\frac{\partial L}{\partial x} = 0,$$ meaning the integrand is a function of $$f(x)$$ and $$f'(x)$$ but $$x$$ does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the Beltrami identity $$L - f' \frac{\partial L}{\partial f'} = C \, ,$$ where $$C$$ is a constant. The left hand side is the Legendre transformation of $$L$$ with respect to $$f'(x).$$

The intuition behind this result is that, if the variable $$x$$ is actually time, then the statement $$\frac{\partial L}{\partial x} = 0$$ implies that the Lagrangian is time-independent. By Noether's theorem, there is an associated conserved quantity. In this case, this quantity is the Hamiltonian, the Legendre transform of the Lagrangian, which (often) coincides with the energy of the system. This is (minus) the constant in Beltrami's identity.

Euler–Poisson equation
If $$S$$ depends on higher-derivatives of $$y(x),$$ that is, if $$S = \int_{a}^{b} f(x, y(x), y'(x), \dots, y^{(n)}(x)) dx,$$ then $$y$$ must satisfy the Euler–Poisson equation, $$\frac{\partial f}{\partial y} - \frac{d}{dx} \left( \frac{\partial f}{\partial y'} \right) + \dots + (-1)^{n} \frac{d^n}{dx^n} \left[ \frac{\partial f}{\partial y^{(n)}} \right]= 0.$$

Du Bois-Reymond's theorem
The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral $$J$$ requires only first derivatives of trial functions. The condition that the first variation vanishes at an extremal may be regarded as a weak form of the Euler–Lagrange equation. The theorem of Du Bois-Reymond asserts that this weak form implies the strong form. If $$L$$ has continuous first and second derivatives with respect to all of its arguments, and if $$\frac{\partial^2 L}{\partial f'^2} \ne 0,$$ then $$f$$ has two continuous derivatives, and it satisfies the Euler–Lagrange equation.

Lavrentiev phenomenon
Hilbert was the first to give good conditions for the Euler–Lagrange equations to give a stationary solution. Within a convex area and a positive thrice differentiable Lagrangian the solutions are composed of a countable collection of sections that either go along the boundary or satisfy the Euler–Lagrange equations in the interior.

However Lavrentiev in 1926 showed that there are circumstances where there is no optimum solution but one can be approached arbitrarily closely by increasing numbers of sections. The Lavrentiev Phenomenon identifies a difference in the infimum of a minimization problem across different classes of admissible functions. For instance the following problem, presented by Manià in 1934: $$L[x] = \int_0^1 (x^3-t)^2 x'^6,$$ $${A} = \{x \in W^{1,1}(0,1) : x(0)=0,\ x(1)=1\}.$$

Clearly, $$x(t) = t^{\frac{1}{3}}$$minimizes the functional, but we find any function $$x \in W^{1, \infty}$$ gives a value bounded away from the infimum.

Examples (in one-dimension) are traditionally manifested across $$W^{1,1}$$ and $$W^{1,\infty},$$ but Ball and Mizel procured the first functional that displayed Lavrentiev's Phenomenon across $$W^{1,p}$$ and $$W^{1,q}$$ for $$1 \leq p < q < \infty.$$ There are several results that gives criteria under which the phenomenon does not occur - for instance 'standard growth', a Lagrangian with no dependence on the second variable, or an approximating sequence satisfying Cesari's Condition (D) - but results are often particular, and applicable to a small class of functionals.

Connected with the Lavrentiev Phenomenon is the repulsion property: any functional displaying Lavrentiev's Phenomenon will display the weak repulsion property.

Functions of several variables
For example, if $$\varphi(x, y)$$ denotes the displacement of a membrane above the domain $$D$$ in the $$x,y$$ plane, then its potential energy is proportional to its surface area: $$U[\varphi] = \iint_D \sqrt{1 +\nabla \varphi \cdot \nabla \varphi} \,dx\,dy.$$ Plateau's problem consists of finding a function that minimizes the surface area while assuming prescribed values on the boundary of $$D$$; the solutions are called minimal surfaces. The Euler–Lagrange equation for this problem is nonlinear: $$\varphi_{xx}(1 + \varphi_y^2) + \varphi_{yy}(1 + \varphi_x^2) - 2\varphi_x \varphi_y \varphi_{xy} = 0.$$ See Courant (1950) for details.

Dirichlet's principle
It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by $$V[\varphi] = \frac{1}{2}\iint_D \nabla \varphi \cdot \nabla \varphi \, dx\, dy.$$ The functional $$V$$ is to be minimized among all trial functions $$\varphi$$ that assume prescribed values on the boundary of $$D.$$ If $$u$$ is the minimizing function and $$v$$ is an arbitrary smooth function that vanishes on the boundary of $$D,$$ then the first variation of $$V[u + \varepsilon v]$$ must vanish: $$\left.\frac{d}{d\varepsilon} V[u + \varepsilon v]\right|_{\varepsilon=0} = \iint_D \nabla u \cdot \nabla v \, dx\,dy = 0.$$ Provided that u has two derivatives, we may apply the divergence theorem to obtain $$\iint_D \nabla \cdot (v \nabla u) \,dx\,dy = \iint_D \nabla u \cdot \nabla v + v \nabla \cdot \nabla u \,dx\,dy = \int_C v \frac{\partial u}{\partial n} \, ds,$$ where $$C$$ is the boundary of $$D,$$ $$s$$ is arclength along $$C$$ and $$\partial u / \partial n$$ is the normal derivative of $$u$$ on $$C.$$ Since $$v$$ vanishes on $$C$$ and the first variation vanishes, the result is $$\iint_D v\nabla \cdot \nabla u \,dx\,dy =0 $$ for all smooth functions $$v$$ that vanish on the boundary of $$D.$$ The proof for the case of one dimensional integrals may be adapted to this case to show that $$\nabla \cdot \nabla u= 0 $$in $$D.$$

The difficulty with this reasoning is the assumption that the minimizing function $$u$$ must have two derivatives. Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy. Riemann named this idea the Dirichlet principle in honor of his teacher Peter Gustav Lejeune Dirichlet. However Weierstrass gave an example of a variational problem with no solution: minimize $$W[\varphi] = \int_{-1}^{1} (x\varphi')^2 \, dx$$ among all functions $$\varphi$$ that satisfy $$\varphi(-1)=-1$$ and $$\varphi(1)=1.$$ $$W$$ can be made arbitrarily small by choosing piecewise linear functions that make a transition between −1 and 1 in a small neighborhood of the origin. However, there is no function that makes $$W=0.$$ Eventually it was shown that Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998).

Generalization to other boundary value problems
A more general expression for the potential energy of a membrane is $$V[\varphi] = \iint_D \left[ \frac{1}{2} \nabla \varphi \cdot \nabla \varphi + f(x,y) \varphi \right] \, dx\,dy \, + \int_C \left[ \frac{1}{2} \sigma(s) \varphi^2 + g(s) \varphi \right] \, ds.$$ This corresponds to an external force density $$f(x,y)$$ in $$D,$$ an external force $$g(s)$$ on the boundary $$C,$$ and elastic forces with modulus $$\sigma(s)$$acting on $$C.$$ The function that minimizes the potential energy with no restriction on its boundary values will be denoted by $$u.$$ Provided that $$f$$ and $$g$$ are continuous, regularity theory implies that the minimizing function $$u$$ will have two derivatives. In taking the first variation, no boundary condition need be imposed on the increment $$v.$$ The first variation of $$V[u + \varepsilon v]$$ is given by $$\iint_D \left[ \nabla u \cdot \nabla v + f v \right] \, dx\, dy + \int_C \left[ \sigma u v + g v \right] \, ds = 0. $$ If we apply the divergence theorem, the result is $$\iint_D \left[ -v \nabla \cdot \nabla u + v f \right] \, dx \, dy + \int_C v \left[ \frac{\partial u}{\partial n} + \sigma u + g \right] \, ds =0. $$ If we first set $$v = 0$$ on $$C,$$ the boundary integral vanishes, and we conclude as before that $$- \nabla \cdot \nabla u + f =0 $$ in $$D.$$ Then if we allow $$v$$ to assume arbitrary boundary values, this implies that $$u$$ must satisfy the boundary condition $$\frac{\partial u}{\partial n} + \sigma u + g =0, $$ on $$C.$$ This boundary condition is a consequence of the minimizing property of $$u$$: it is not imposed beforehand. Such conditions are called natural boundary conditions.

The preceding reasoning is not valid if $$\sigma$$ vanishes identically on $$C.$$ In such a case, we could allow a trial function $$\varphi \equiv c,$$ where $$c$$ is a constant. For such a trial function, $$V[c] = c\left[ \iint_D f \, dx\,dy + \int_C g \, ds \right].$$ By appropriate choice of $$c,$$ $$V$$ can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless $$\iint_D f \, dx\,dy + \int_C g \, ds =0.$$ This condition implies that net external forces on the system are in equilibrium. If these forces are in equilibrium, then the variational problem has a solution, but it is not unique, since an arbitrary constant may be added. Further details and examples are in Courant and Hilbert (1953).

Eigenvalue problems
Both one-dimensional and multi-dimensional eigenvalue problems can be formulated as variational problems.

Sturm–Liouville problems
The Sturm–Liouville eigenvalue problem involves a general quadratic form $$Q[y] = \int_{x_1}^{x_2} \left[ p(x) y'(x)^2 + q(x) y(x)^2 \right] \, dx, $$ where $$y$$ is restricted to functions that satisfy the boundary conditions $$y(x_1)=0, \quad y(x_2)=0. $$ Let $$R$$ be a normalization integral $$R[y] =\int_{x_1}^{x_2} r(x)y(x)^2 \, dx.$$ The functions $$p(x)$$ and $$r(x)$$ are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio $$Q/R$$ among all $$y$$ satisfying the endpoint conditions, which is equivalent to minimizing $$Q[y]$$ under the constraint that $$R[y]$$ is constant. It is shown below that the Euler–Lagrange equation for the minimizing $$u$$ is $$-(p u')' +q u -\lambda r u = 0, $$ where $$\lambda$$ is the quotient $$\lambda = \frac{Q[u]}{R[u]}. $$ It can be shown (see Gelfand and Fomin 1963) that the minimizing $$u$$ has two derivatives and satisfies the Euler–Lagrange equation. The associated $$\lambda$$ will be denoted by $$\lambda_1$$; it is the lowest eigenvalue for this equation and boundary conditions. The associated minimizing function will be denoted by $$u_1(x).$$ This variational characterization of eigenvalues leads to the Rayleigh–Ritz method: choose an approximating $$u$$ as a linear combination of basis functions (for example trigonometric functions) and carry out a finite-dimensional minimization among such linear combinations. This method is often surprisingly accurate.

The next smallest eigenvalue and eigenfunction can be obtained by minimizing $$Q$$ under the additional constraint $$\int_{x_1}^{x_2} r(x) u_1(x) y(x) \, dx = 0. $$ This procedure can be extended to obtain the complete sequence of eigenvalues and eigenfunctions for the problem.

The variational problem also applies to more general boundary conditions. Instead of requiring that $$y$$ vanish at the endpoints, we may not impose any condition at the endpoints, and set $$Q[y] = \int_{x_1}^{x_2} \left[ p(x) y'(x)^2 + q(x)y(x)^2 \right] \, dx + a_1 y(x_1)^2 + a_2 y(x_2)^2, $$ where $$a_1$$ and $$a_2$$ are arbitrary. If we set $$y = u + \varepsilon v$$, the first variation for the ratio $$Q/R$$ is $$V_1 = \frac{2}{R[u]} \left( \int_{x_1}^{x_2} \left[ p(x) u'(x)v'(x) + q(x)u(x)v(x) -\lambda r(x) u(x) v(x) \right] \, dx + a_1 u(x_1)v(x_1) + a_2 u(x_2)v(x_2) \right), $$ where λ is given by the ratio $$Q[u]/R[u]$$ as previously. After integration by parts, $$\frac{R[u]}{2} V_1 = \int_{x_1}^{x_2} v(x) \left[ -(p u')' + q u -\lambda r u \right] \, dx + v(x_1)[ -p(x_1)u'(x_1) + a_1 u(x_1)] + v(x_2) [p(x_2) u'(x_2) + a_2 u(x_2)]. $$ If we first require that $$v$$ vanish at the endpoints, the first variation will vanish for all such $$v$$ only if $$-(p u')' + q u -\lambda r u =0 \quad \hbox{for} \quad x_1 < x < x_2.$$ If $$u$$ satisfies this condition, then the first variation will vanish for arbitrary $$v$$ only if $$-p(x_1)u'(x_1) + a_1 u(x_1)=0, \quad \hbox{and} \quad p(x_2) u'(x_2) + a_2 u(x_2)=0.$$ These latter conditions are the natural boundary conditions for this problem, since they are not imposed on trial functions for the minimization, but are instead a consequence of the minimization.

Eigenvalue problems in several dimensions
Eigenvalue problems in higher dimensions are defined in analogy with the one-dimensional case. For example, given a domain $$D$$ with boundary $$B$$ in three dimensions we may define $$Q[\varphi] = \iiint_D p(X) \nabla \varphi \cdot \nabla \varphi + q(X) \varphi^2 \, dx \, dy \, dz + \iint_B \sigma(S) \varphi^2 \, dS, $$ and $$R[\varphi] = \iiint_D r(X) \varphi(X)^2 \, dx \, dy \, dz.$$ Let $$u$$ be the function that minimizes the quotient $$Q[\varphi] / R[\varphi],$$ with no condition prescribed on the boundary $$B.$$ The Euler–Lagrange equation satisfied by $$u$$ is $$-\nabla \cdot (p(X) \nabla u) + q(x) u - \lambda r(x) u=0,$$ where $$\lambda = \frac{Q[u]}{R[u]}.$$ The minimizing $$u$$ must also satisfy the natural boundary condition $$p(S) \frac{\partial u}{\partial n} + \sigma(S) u = 0,$$ on the boundary $$B.$$ This result depends upon the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998) for details. Many extensions, including completeness results, asymptotic properties of the eigenvalues and results concerning the nodes of the eigenfunctions are in Courant and Hilbert (1953).

Optics
Fermat's principle states that light takes a path that (locally) minimizes the optical length between its endpoints. If the $$x$$-coordinate is chosen as the parameter along the path, and $$y=f(x)$$ along the path, then the optical length is given by $$A[f] = \int_{x_0}^{x_1} n(x,f(x)) \sqrt{1 + f'(x)^2} dx, $$ where the refractive index $$n(x,y)$$ depends upon the material. If we try $$f(x) = f_0 (x) + \varepsilon f_1 (x)$$ then the first variation of $$A$$ (the derivative of $$A$$ with respect to ε) is $$\delta A[f_0,f_1] = \int_{x_0}^{x_1} \left[ \frac{ n(x,f_0) f_0'(x) f_1'(x)}{\sqrt{1 + f_0'(x)^2}} + n_y (x,f_0) f_1 \sqrt{1 + f_0'(x)^2} \right] dx.$$

After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation $$-\frac{d}{dx} \left[\frac{ n(x,f_0) f_0'}{\sqrt{1 + f_0'^2}} \right] + n_y (x,f_0) \sqrt{1 + f_0'(x)^2} = 0. $$

The light rays may be determined by integrating this equation. This formalism is used in the context of Lagrangian optics and Hamiltonian optics.

Snell's law
There is a discontinuity of the refractive index when light enters or leaves a lens. Let $$n(x,y) = \begin{cases} n_{(-)} & \text{if} \quad x<0, \\ n_{(+)} & \text{if} \quad x>0, \end{cases}$$ where $$n_{(-)}$$ and $$n_{(+)}$$ are constants. Then the Euler–Lagrange equation holds as before in the region where $$x < 0$$ or $$x > 0,$$ and in fact the path is a straight line there, since the refractive index is constant. At the $$x = 0,$$ $$f$$ must be continuous, but $$f'$$ may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form $$\delta A[f_0,f_1] = f_1(0)\left[ n_{(-)}\frac{f_0'(0^-)}{\sqrt{1 + f_0'(0^-)^2}} - n_{(+)}\frac{f_0'(0^+)}{\sqrt{1 + f_0'(0^+)^2}} \right].$$

The factor multiplying $$n_{(-)}$$ is the sine of angle of the incident ray with the $$x$$ axis, and the factor multiplying $$n_{(+)}$$ is the sine of angle of the refracted ray with the $$x$$ axis. Snell's law for refraction requires that these terms be equal. As this calculation demonstrates, Snell's law is equivalent to vanishing of the first variation of the optical path length.

Fermat's principle in three dimensions
It is expedient to use vector notation: let $$X = (x_1,x_2,x_3),$$ let $$t$$ be a parameter, let $$X(t)$$ be the parametric representation of a curve $$C,$$ and let $$\dot X(t)$$ be its tangent vector. The optical length of the curve is given by $$A[C] = \int_{t_0}^{t_1} n(X) \sqrt{ \dot X \cdot \dot X} \, dt. $$

Note that this integral is invariant with respect to changes in the parametric representation of $$C.$$ The Euler–Lagrange equations for a minimizing curve have the symmetric form $$\frac{d}{dt} P = \sqrt{ \dot X \cdot \dot X} \, \nabla n, $$ where $$P = \frac{n(X) \dot X}{\sqrt{\dot X \cdot \dot X} }.$$

It follows from the definition that $$P$$ satisfies $$P \cdot P = n(X)^2. $$

Therefore, the integral may also be written as $$A[C] = \int_{t_0}^{t_1} P \cdot \dot X \, dt.$$

This form suggests that if we can find a function $$\psi$$ whose gradient is given by $$P,$$ then the integral $$A$$ is given by the difference of $$\psi$$ at the endpoints of the interval of integration. Thus the problem of studying the curves that make the integral stationary can be related to the study of the level surfaces of $$\psi.$$In order to find such a function, we turn to the wave equation, which governs the propagation of light. This formalism is used in the context of Lagrangian optics and Hamiltonian optics.

Connection with the wave equation
The wave equation for an inhomogeneous medium is $$u_{tt} = c^2 \nabla \cdot \nabla u, $$ where $$c$$ is the velocity, which generally depends upon $$X.$$ Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy $$\varphi_t^2 = c(X)^2 \, \nabla \varphi \cdot \nabla \varphi. $$

We may look for solutions in the form $$\varphi(t,X) = t - \psi(X). $$

In that case, $$\psi$$ satisfies $$\nabla \psi \cdot \nabla \psi = n^2, $$ where $$n=1/c.$$ According to the theory of first-order partial differential equations, if $$P = \nabla \psi,$$ then $$P$$ satisfies $$\frac{dP}{ds} = n \, \nabla n,$$ along a system of curves (the light rays) that are given by $$\frac{dX}{ds} = P. $$

These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification $$\frac{ds}{dt} = \frac{\sqrt{ \dot X \cdot \dot X} }{n}. $$

We conclude that the function $$\psi$$ is the value of the minimizing integral $$A$$ as a function of the upper end point. That is, when a family of minimizing curves is constructed, the values of the optical length satisfy the characteristic equation corresponding the wave equation. Hence, solving the associated partial differential equation of first order is equivalent to finding families of solutions of the variational problem. This is the essential content of the Hamilton–Jacobi theory, which applies to more general variational problems.

Mechanics
In classical mechanics, the action, $$S,$$ is defined as the time integral of the Lagrangian, $$L.$$ The Lagrangian is the difference of energies, $$L = T - U, $$ where $$T$$ is the kinetic energy of a mechanical system and $$U$$ its potential energy. Hamilton's principle (or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral $$S = \int_{t_0}^{t_1} L(x, \dot x, t) \, dt$$ is stationary with respect to variations in the path $$x(t).$$ The Euler–Lagrange equations for this system are known as Lagrange's equations: $$\frac{d}{dt} \frac{\partial L}{\partial \dot x} = \frac{\partial L}{\partial x}, $$ and they are equivalent to Newton's equations of motion (for such systems).

The conjugate momenta $$P$$ are defined by $$p = \frac{\partial L}{\partial \dot x}. $$ For example, if $$T = \frac{1}{2} m \dot x^2, $$ then $$p = m \dot x. $$ Hamiltonian mechanics results if the conjugate momenta are introduced in place of $$\dot x$$ by a Legendre transformation of the Lagrangian $$L$$ into the Hamiltonian $$H$$ defined by $$H(x, p, t) = p \,\dot x - L(x,\dot x, t).$$ The Hamiltonian is the total energy of the system: $$H = T + U.$$ Analogy with Fermat's principle suggests that solutions of Lagrange's equations (the particle trajectories) may be described in terms of level surfaces of some function of $$X.$$ This function is a solution of the Hamilton–Jacobi equation: $$\frac{\partial \psi}{\partial t} + H\left(x,\frac{\partial \psi}{\partial x},t\right) = 0.$$

Further applications
Further applications of the calculus of variations include the following:

 * The derivation of the catenary shape
 * Solution to Newton's minimal resistance problem
 * Solution to the brachistochrone problem
 * Solution to the tautochrone problem
 * Solution to isoperimetric problems
 * Calculating geodesics
 * Finding minimal surfaces and solving Plateau's problem
 * Optimal control
 * Analytical mechanics, or reformulations of Newton's laws of motion, most notably Lagrangian and Hamiltonian mechanics;
 * Geometric optics, especially Lagrangian and Hamiltonian optics;
 * Variational method (quantum mechanics), one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states;
 * Variational Bayesian methods, a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning;
 * Variational methods in general relativity, a family of techniques using calculus of variations to solve problems in Einstein's general theory of relativity;
 * Finite element method is a variational method for finding numerical solutions to boundary-value problems in differential equations;
 * Total variation denoising, an image processing method for filtering high variance or noisy signals.

Variations and sufficient condition for a minimum
Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation is defined as the linear part of the change in the functional, and the second variation is defined as the quadratic part.

For example, if $$J[y]$$ is a functional with the function $$y = y(x)$$ as its argument, and there is a small change in its argument from $$y$$ to $$y + h,$$ where $$h = h(x)$$ is a function in the same function space as $$y,$$ then the corresponding change in the functional is $$\Delta J[h] = J[y+h] - J[y].$$

The functional $$J[y]$$ is said to be differentiable if $$\Delta J[h] = \varphi [h] + \varepsilon \|h\|,$$ where $$\varphi[h]$$ is a linear functional, $$\|h\|$$ is the norm of $$h,$$ and $$\varepsilon \to 0$$ as $$\|h\| \to 0.$$ The linear functional $$\varphi[h]$$ is the first variation of $$J[y]$$ and is denoted by, $$\delta J[h] = \varphi[h].$$

The functional $$J[y]$$ is said to be twice differentiable if $$\Delta J[h] = \varphi_1 [h] + \varphi_2 [h] + \varepsilon \|h\|^2,$$ where $$\varphi_1[h]$$ is a linear functional (the first variation), $$\varphi_2[h]$$ is a quadratic functional, and $$\varepsilon \to 0$$ as $$\|h\| \to 0.$$ The quadratic functional $$\varphi_2[h]$$ is the second variation of $$J[y]$$ and is denoted by, $$\delta^2 J[h] = \varphi_2[h].$$

The second variation $$\delta^2 J[h]$$ is said to be strongly positive if $$\delta^2J[h] \ge k \|h\|^2,$$ for all $$h$$ and for some constant $$k > 0$$.

Using the above definitions, especially the definitions of first variation, second variation, and strongly positive, the following sufficient condition for a minimum of a functional can be stated.

title: Calculus of variations


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Talk:Thing -- 1982

What does 1982 have to do with anything that this page is redirected to? -- Zoe

 * The release date of John Carpenter's remake. --Brion 21:04 Jan 20, 2003 (UTC)

 * Should be under The Thing (1982 movie), since there's more than one film by that name. Koyaanis Qatsi

title: Talk:Thing -- 1982


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Eduard Zeller

Eduard Gottlob Zeller (22 January 1814, Kleinbottwar – 19 March 1908, Stuttgart) was a German philosopher and Protestant theologian of the Tübingen School of theology. He was well known for his writings on Ancient Greek philosophy, especially Pre-Socratic Philosophy, and most of all for his celebrated, multi-volume historical treatise The Philosophy of Greeks in their Historical Development (1844–52). Zeller was also a central figure in the revival of neo-Kantianism.

Life
Eduard Zeller was born at Kleinbottwar in Württemberg, the son of a government official. He was educated first at the Evangelical Seminaries of Maulbronn and Blaubeuren starting in 1831, and later at the University of Tübingen (the Tübinger Stift), then much under the influence of Hegel. He received his doctorate in 1836 with a thesis on Plato's Laws. In 1840 he was Privatdozent of theology at Tübingen, in 1847 professor of theology at Berne, and in 1849 professor of theology at Marburg, where he soon shifted to the philosophy faculty as the result of disputes with the Clerical party. He became professor of philosophy at the University of Heidelberg in 1862, moved to Berlin in 1872, and retired around 1895. He remained best known for his The Philosophy of Greeks in their Historical Development (1844–52). He continued to expand and improve this work to reflect new research, and the last edition appeared in 1902. It was translated into most European languages and became the standard textbook on Greek philosophy.

Zeller also published many works on theology and three volumes of philosophical essays. He was also one of the founders of the Theologische Jahrbücher (Theological Yearbooks), a periodical which became well known as the exponent of the historical method of David Strauss and Christian Baur. He wrote much on the debate about whether theology was a kind of science (Wissenschaft). Like most of his contemporaries, including Friedrich Theodor Vischer, he began with Hegelianism, but subsequently developed a system of his own. He felt the necessity of going back to Kant and critically reconsidering the epistemological problems which, he believed, Kant had only partially resolved.

Philosophical work
Nonetheless, his accomplishments in the history of philosophy were far more influential than his contributions as an original thinker. Zeller's conception of the history of Greek thought was influenced by the dialectical philosophy of Hegel. Some critics maintain that Zeller was not alive enough to cultural context and to the idiosyncrasies of individual thinkers. Some hold that he laid too much stress upon Hegel's notion of "concept", and relied too much on the Hegelian antithesis of subject and object, though his history of Greek philosophy was nonetheless influential and highly regarded. He received the highest recognition, not only from philosophers and learned societies all over the world, but also from the German emperor and German people. In 1894 the Emperor Wilhelm II made him a "Wirklicher Geheimrat" with the title of "Excellenz," and his bust, along with that of Helmholtz, was set up at the Brandenburg Gate near the statues erected to the Emperor and Empress Frederick.

The Philosophie der Griechen has been translated into English by S. F. Alleyne (2 vols, 1881) in sections: S. F. Alleyne, History of Greek Philosophy to the time of Socrates (1881) Volume 1 and Volume 2; O. J. Reichel, Socrates and the Socratic Schools (1868; 2nd ed. 1877; 3rd ed. 1885); S. F. Alleyne and A. Goodwin, Plato and the Older Academy (1876); Benjamin Francis Conn Costelloe and J. H. Muirhead, Aristotle and the Earlier Peripatetics (1897)Volume 1 and Volume 2; O. J. Reichel, Stoics, Epicureans and Sceptics (1870 and 1880); S. F. Alleyne, History of Eclecticism in Greek Philosophy (1883).

Zeller was also, in his Philosophie der Griechen, one of the first to use the word 'Superhuman' (übermensch), later central in Nietzsche and the propaganda of the Nazi Party, in adjectival form as a technical term in philosophy. He said "... thus the happiness in her can be designated superhuman (übermenschliche) while in contrast the happiness flowing from ethical virtues is merely a characteristic human good.'

Works
The Philosophie appeared in an abbreviated form as Grundriss der Geschichte der Griechischen Philosophie (1883; 5th ed. 1898); English transl. by Alleyne and Evelyn Abbott (1886), under the title, Outlines of the History of Greek Philosophy.

Among his other works are:

 * Platonische Studien (1839)
 * Die Apostelgeschichte kritisch untersucht (1854; English translation J Dare, 1875–76: Volume 1 and Volume 2)
 * Entwickelung des Monotheismus bei den Griechen (1862)
 * Strauss und Renan (1864); (English translation 1866)
 * Geschichte der christlichen Kirche (1898)
 * Geschichte der deutschen Philosophie seit Leibniz (1873, ed. 1875)
 * Staat und Kirche (1873)
 * Strauss in seinen Leben und Schriften 1874; (English translation 1874)
 * Über Bedeutung und Aufgabe der Erkenntnisstheorie (1862)
 * Über teleologische und mechanische Naturerklärung (1876)
 * Vorträge und Abhandlungen (1865–84)
 * Religion und Philosophie bei den Römern (1866, ed. 1871)
 * Philosophische Aufsätze (1887).

title: Eduard Zeller


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Family farm

A family farm is generally understood to be a farm owned and/or operated by a family. It is sometimes considered to be an estate passed down by inheritance.

Although a recurring conceptual and archetypal distinction is that of a family farm as a smallholding versus corporate farming as large-scale agribusiness, that notion does not accurately describe the realities of farm ownership in many countries. Family farm businesses can take many forms, from smallholder farms to larger farms operated under intensive farming practices. In various countries, most farm families have structured their farm businesses as corporations (such as limited liability companies) or trusts, for liability, tax, and business purposes. Thus, the idea of a family farm as a unitary concept or definition does not easily translate across languages, cultures, or centuries, as there are substantial differences in agricultural traditions and histories between countries and between centuries within a country. For example, in U.S. agriculture, a family farm can be of any size, as long as the ownership is held within a family. A 2014 USDA report shows that family farms operate 90 percent of the nation's farmland, and account for 85 percent of the country's agricultural production value. However, that does not at all imply that corporate farming is a small presence in U.S. agriculture; rather, it simply reflects the fact that many corporations are closely held. In contrast, in Brazilian agriculture, the official definition of a family farm (agricultura familiar) is limited to small farms worked primarily by members of a single family; but again, this fact does not imply that corporate farming is a small presence in Brazilian agriculture; rather, it simply reflects the fact that large farms with many workers cannot be legally classified under the family farm label because that label is legally reserved for smallholdings in that country.

Farms that would not be considered family farms would be those operated as collectives, non-family corporations, or in other institutionalised forms. At least 500 million of the world's [estimated] 570 million farms are managed by families, making family farms predominant in global agriculture.

Definitions
An "informal discussion of the concepts and definitions" in a working paper published by Food and Agriculture Organization of the United Nations in 2014 reviewed English, Spanish and French definitions of the concept of "family farm". Definitions referred to one or more of labor, management, size, provision of family livelihood, residence, family ties and generational aspects, community and social networks, subsistence orientation, patrimony, land ownership and family investment. The disparity of definitions reflects national and geographical differences in cultures, rural land tenure, and rural economies, as well as the different purposes for which definitions are coined.

The 2012 United States Census of Agriculture defines a family farm as "any farm where the majority of the business is owned by the operator and individuals related to the operator, including relatives who do not live in the operator’s household"; it defines a farm as "any place from which $1,000 or more of agricultural products were produced and sold, or normally would have been sold, during a given year."

The Food and Agriculture Organization of the United Nations defines a "family farm" as one that relies primarily on family members for labour and management.

In some usages, "family farm" implies that the farm remains within the ownership of a family over a number of generations.

Being special-purpose definitions, the definitions found in laws or regulations may differ substantially from commonly understood meanings of "family farm". For example, In the United States, under federal Farm Ownership loan regulations, the definition of a "family farm" does not specify the nature of farm ownership, and management of the farm is either by the borrower, or by members operating the farm when a loan is made to a corporation, co-operative or other entity. The complete definition can be found in the US Code of Federal Regulations 7 CFR 1943.4.

History
In the Roman Republic, latifundia, great landed estates, specialised in agriculture destined for export, producing grain, olive oil, or wine, corresponding largely to modern industrialized agriculture but depending on slave labour instead of mechanization, developed after the Second Punic War and increasingly replaced the former system of family-owned small or intermediate farms in the Roman Empire period. The basis of the latifundia in Spain and Sicily was the ager publicus that fell to the dispensation of the state through Rome's policy of war in the 1st century BC and the 1st century AD.

In the collapse of the Western Roman Empire, the largely self-sufficient villa-system of the latifundia remained among the few political-cultural centres of a fragmented Europe. These latifundia had been of great importance economically, until the long-distance shipping of wine and oil, grain and garum disintegrated, but extensive lands controlled in a single pair of hands still constituted power: it can be argued that the latifundia formed part of the economic basis of the European social feudal system, taking the form of Manorialism, the essential element of feudal society, and the organizing principle of rural economy in medieval Europe. Manorialism was characterised by the vesting of legal and economic power in a Lord of the Manor, supported economically from his own direct landholding in a manor (sometimes called a fief), and from the obligatory contributions of a legally subject part of the peasant population under the jurisdiction of himself and his manorial court. Manorialism died slowly and piecemeal, along with its most vivid feature in the landscape, the open field system. It outlasted serfdom as it outlasted feudalism: "primarily an economic organization, it could maintain a warrior, but it could equally well maintain a capitalist landlord. It could be self-sufficient, yield produce for the market, or it could yield a money rent." The last feudal dues in France were abolished at the French Revolution. In parts of eastern Germany, the Rittergut manors of Junkers remained until World War II. The common law of the leasehold estate relation evolved in medieval England. That law still retains many archaic terms and principles pertinent to a feudal social order. Under the tenant system, a farm may be worked by the same family over many generations, but what is inherited is not the farm's estate itself but the lease on the estate. In much of Europe, serfdom was abolished only in the modern period, in Western Europe after the French Revolution, in Russia as late as in 1861.

In contrast to the Roman system of latifundia and the derived system of manoralism, the Germanic peoples had a system based on heritable estates owned by individual families or clans. The Germanic term for "heritable estate, allodium" was *ōþalan (Old English ēþel), which incidentally was also used as a rune name; the gnomic verse on this term in the Anglo-Saxon rune poem reads:
 * [Ēðel] byþ oferleof æghwylcum men, gif he mot ðær rihtes and gerysena on brucan on bolde bleadum oftast.
 * "[An estate] is very dear to every man, if he can enjoy there in his house whatever is right and proper in constant prosperity."

In the inheritance system known as Salic patrimony (also gavelkind in its exceptional survival in medieval Kent) refers to this clan-based possession of real estate property, particularly in Germanic context. Terra salica could not be sold or otherwise disposed; it was not alienable. Much of Germanic Europe has a history of overlap or conflict between the feudal system of manoralism, where the estate is owned by noblemen and leased to the tenants or worked by serfs, and the Germanic system of free farmers working landed estates heritable within their clan or family. Historical prevalence of the Germanic system of independent estates or Höfe resulted in dispersed settlement (Streusiedlung)  structure, as opposed to the village-centered settlements of manoralism.

In German-speaking Europe, a farmyard is known as a Hof; in modern German this word designates the area enclosed by the farm buildings, not the fields around them, and it is also used in other everyday situations for courtyards of any type (Hinterhof = 'back yard', etc.). The recharacterized compound Bauernhof was formed in the early modern period to designate family farming estates and today is the most common word for 'farm', while the archaic Meierhof designated a manorial estate. Historically, the unmarked term Hof was increasingly used for the royal or noble court. The estate as a whole is referred to by the collective Gehöft (15th century); the corresponding Slavic concept being Khutor. Höfeordnung is the German legal term for the inheritance laws regarding family farms, deriving from inheritance under medieval Saxon law. In England, the title of yeoman was applied to such land-owning commoners from the 15th century.

In the early modern and modern period, the dissolution of manoralism went parallel to the development of intensive farming parallel to the Industrial Revolution. Mechanization enabled the cultivation of much larger areas than what was typical for the traditional estates aimed at subsistence farming, resulting in the emergence of a smaller number of large farms, with the displaced population partly contributing to the new class of industrial wage-labourers and partly emigrating to the New World or the Russian Empire (following the 1861 emancipation of the serfs). The family farms established in Imperial Russia were again collectivized under the Soviet Union, but the emigration of European farmers displaced by the Industrial Revolution contributed to the emergence of a system of family estates in the Americas (Homestead Act of 1862).

Thomas Jefferson's argument that a large number of family estates are a factor in ensuring the stability of democracy was repeatedly used in support of subsidies.

Perceptions of the family farm
In developed countries the family farm is viewed sentimentally, as a lifestyle to be preserved for tradition's sake, or as a birthright. It is in these nations very often a political rallying cry against change in agricultural policy, most commonly in France, Japan, and the United States, where rural lifestyles are often regarded as desirable. In these countries, strange bedfellows can often be found arguing for similar measures despite otherwise vast differences in political ideology. For example, Pat Buchanan and Ralph Nader, both candidates for the office of President of the United States, held rural rallies together and spoke for measures to preserve the so-called family farm. On other economic matters they were seen as generally opposed, but found common ground on this one.

The social roles of family farms are much changed today. Until recently, staying in line with traditional and conservative sociology, the heads of the household were usually the oldest man followed closely by his oldest sons. The wife generally took care of the housework, child rearing, and financial matters pertaining to the farm. However, agricultural activities have taken on many forms and change over time. Agronomy, horticulture, aquaculture, silviculture, and apiculture, along with traditional plants and animals, all make up aspects of today's family farm. Farm wives often need to find work away from the farm to supplement farm income and children sometimes have no interest in farming as their chosen field of work.

Bolder promoters argue that as agriculture has become more efficient with the application of modern management and new technologies in each generation, the idealized classic family farm is now simply obsolete, or more often, unable to compete without the economies of scale available to larger and more modern farms. Advocates argue that family farms in all nations need to be protected, as the basis of rural society and social stability.

Viability
According to the United States Department of Agriculture, ninety-eight percent of all farms in the U.S. are family farms. Two percent of farms are not family farms, and those two percent make up fourteen percent of total agricultural output in the United States, although half of them have total sales of less than $50,000 per year. Overall, ninety-one percent of farms in the United States are considered "small family farms" (with sales of less than $250,000 per year), and those farms produce twenty-seven percent of U.S. agricultural output.

Depending on the type and size of independently owned operation, some limiting factors are:
 * Economies of scale: Larger farms are able to bargain more competitively, purchase more competitively, profit from economic highs, and weather lows more readily through monetary inertia than smaller farms.
 * Cost of inputs: fertilizer and other agrichemicals can fluctuate dramatically from season to season, partially based on oil prices, a range of 25% to 200% is common over a period of a few years.
 * oil prices: Directly (for farm machinery) and somewhat less directly (long-distance transport; production cost of agrichemicals), the cost of oil significantly impacts the year-to-year viability of all mechanized conventional farms.
 * commodity futures: the predicted price of commodity crops, hogs, grain, etc., can determine ahead of a season what seems economically viable to grow.
 * technology user agreements: a less publicly known factor, patented GE seed that is widely used for many crops, like cotton and soy, comes with restrictions on use, which can even include who the crop can be sold to.
 * wholesale infrastructure: A farmer growing larger quantities of a crop than can be sold directly to consumers has to meet a range of criteria for sale into the wholesale market, which include harvest timing and graded quality, and may also include variety, therefore, the market channel really determines most aspects of the farm decisionmaking.
 * availability of financing: Larger farms today often rely on lines of credit, typically from banks, to purchase the agrichemicals, and other supplies needed for each growing year. These lines are heavily affected by almost all of the other constraining factors.
 * government economic intervention: In some countries, notably the US and EU, government subsidies to farmers, intended to mitigate the impact on domestic farmers of economic and political activities in other areas of the economy, can be a significant source of farm income. Bailouts, when crises such as drought or the "mad cow disease" problems hit agricultural sectors, are also relied on. To some large degree, this situation is a result of the large-scale global markets farms have no alternative but to participate in.
 * government and industry regulation: A wide range of quotas, marketing boards and legislation governing agriculture impose complicated limits, and often require significant resources to navigate. For example, on the small farming end, in many jurisdictions, there are severe limits or prohibitions on the sale of livestock, dairy and eggs. These have arisen from pressures from all sides: food safety, environmental, industry marketing.
 * real estate prices: The growth of urban centers around the world, and the resulting urban sprawl have caused the price of centrally located farmland to skyrocket, while reducing the local infrastructure necessary to support farming, putting effectively intense pressure on many farmers to sell out.

Over the 20th century, the people of developed nations have collectively taken most of the steps down the path to this situation. Individual farmers opted for successive waves of new technology, happily "trading in their horses for a tractor", increasing their debt and their production capacity. This in turn required larger, more distant markets, and heavier and more complex financing. The public willingly purchased increasingly commoditized, processed, shipped and relatively inexpensive food. The availability of an increasingly diverse supply of fresh, uncured, unpreserved produce and meat in all seasons of the year (oranges in January, freshly killed steers in July, fresh pork rather than salted, smoked, or potassium-impregnated ham) opened an entirely new cuisine and an unprecedented healthy diet to millions of consumers who had never enjoyed such produce before. These abilities also brought to market an unprecedented variety of processed foods, such as corn syrup and bleached flour. For the family farm this new technology and increasingly complex marketing strategy has presented new and unprecedented challenges, and not all family farmers have been able to effectively cope with the changing market conditions.

Local food and the organic movement
In the last few decades there has been a resurgence of interest in organic and free range foods. A percentage of consumers have begun to question the viability of industrial agriculture practices and have turned to organic groceries that sell products produced on family farms including not only meat and produce but also such things as wheat germ breads and natural lye soaps (as opposed to bleached white breads and petroleum based detergent bars). Others buy these products direct from family farms. The "new family farm" provides an alternative market in some localities with an array of traditionally and naturally produced products.

Such "organic" and "free-range" farming is attainable where a significant number of affluent urban and suburban consumers willingly pay a premium for the ideals of "locally produced produce" and "humane treatment of animals". Sometimes, these farms are hobby or part-time ventures, or supported by wealth from other sources. Viable farms on a scale sufficient to support modern families at an income level commensurate with urban and suburban upper-middle-class families are often large scale operations, both in area and capital requirements. These farms, family owned and operated in a technologically and economically conventional manner, produce crops and animal products oriented to national and international markets, rather than to local markets. In assessing this complex economic situation, it is important to consider all sources of income available to these farms; for instance, the millions of dollars in farm subsidies which the United States government offers each year. As fuel prices rise, foods shipped to national and international markets are already rising in price.

United States
In 2012, the United States had 2,039,093 family farms (as defined by USDA), accounting for 97 percent of all farms and 89 percent of census farm area in the United States. In 1988 Mark Friedberger warned, "The farm family is a unique institution, perhaps the last remnant, in an increasingly complex world, of a simpler social order in which economic and domestic activities were inextricably bound together. In the past few years, however, American agriculture has suffered huge losses, and family farmers have seen their way of life threatened by economic forces beyond their control." However, by 1981 Ingolf Vogeler argued it was too late—the American family farm had been replaced by large agribusiness corporations pretending to be family operated.

A USDA survey conducted in 2011 estimated that family farms account for 85 percent of US farm production and 85 percent of US gross farm income. Mid-size and larger family farms account for 60 percent of US farm production and dominate US production of cotton, cash grain and hogs. Small family farms account for 26 percent of US farm production overall, and higher percentages of production of poultry, beef cattle, some other livestock and hay.

Several kinds of US family farms are recognized in USDA farm typology:

Small family farms are defined as those with annual gross cash farm income (GCFI) of less than $350,000; in 2011, these accounted for 90 percent of all US farms. Because low net farm incomes tend to predominate on such farms, most farm families on small family farms are extremely dependent on off-farm income. Small family farms in which the principal operator was mostly employed off-farm accounted for 42 percent of all farms and 15 percent of total US farm area; median net farm income was $788. Retirement family farms were small farms accounting for 16 percent of all farms and 7 percent of total US farm area; median net farm income was $5,002.

The other small family farm categories are those in which farming occupies at least 50 percent of the principal operator's working time. These are:

Low-sales small family farms (with GCFI less than $150,000); 26 percent of all US farms, 18 percent of total US farm area, median net farm income $3,579.

Moderate-sales small family farms (with GCFI of $150,000 to $349,999); 5.44 percent of all US farms, 13 percent of total US farm area, median net farm income $67,986.

Mid-size family farms (GCFI of $350,000 to $999,999); 6 percent of all US farms, 22 percent of total US farm area; median net farm income $154,538.

Large family farms (GCFI $1,000,000 to $4,999,999); 2 percent of all US farms, 14 percent of total US farm area; median net farm income $476,234.

Very large family farms (GCFI over $5,000,000); <1 percent of all US farms, 2 percent of total US farm area; median net farm income $1,910,454.

Family farms include not only sole proprietorships and family partnerships, but also family corporations. Family-owned corporations account for 5 percent of all farms and 89 percent of corporate farms in the United States. About 98 percent of US family corporations owning farms are small, with no more than 10 shareholders; average net farm income of family corporate farms was $189,400 in 2012. (In contrast, 90 percent of US non-family corporations owning farms are small, having no more than 10 shareholders; average net cash farm income for US non-family corporate farms was $270,670 in 2012.)

Canada
In Canada, the number of "family farms" cannot be inferred closely, because of the nature of census data, which do not distinguish family and non-family farm partnerships. In 2011, of Canada's 205,730 farms, 55 percent were sole proprietorships, 25 percent were partnerships, 17 percent were family corporations, 2 percent were non-family corporations and <1 percent were other categories. Because some but not all partnerships involve family members, these data suggest that family farms account for between about 73 and 97 percent of Canadian farms. The family farm percentage is likely to be near the high end of this range, for two reasons. The partners in a [Canadian] farm partnership are typically spouses, often forming the farm partnership for tax reasons. Also, as in the US, family farm succession planning can use a partnership as a means of apportioning family farm tenure among family members when a sole proprietor is ready to transfer some or all of ownership and operation of a farm to offspring. Conversion of a sole proprietorship family farm to a family corporation may also be influenced by legal and financial, e.g. tax, considerations. The Canadian Encyclopedia estimates that more than 90 percent of Canadian farms are family operations. In 2006, of Canadian farms with more than one million dollars in annual gross farm receipts, about 63 percent were family corporations and 13 percent were non-family corporations.

Europe
Analysis of data for 59,000 farms in the 12 member states of the European Community found that in 1989, about three-quarters of the farms were family farms, producing just over half of total agricultural output.

As of 2010, there were approximately 139,900 family farms in Ireland, with an average size of 35.7 hectares per holding. (Nearly all farms in Ireland are family farms.)  In Ireland, average family farm income was 25,483 euros in 2012. Analysis by Teagasc (Ireland's Agriculture and Food Development Authority) estimates that 37 percent of Irish farms are economically viable and an additional 30 percent are sustainable due to income from off-farm sources; 33 percent meet neither criterion and are considered economically vulnerable.

Newly industrialized countries
In Brazil, there are about 4.37 million family farms. These account for 84.4 percent of farms, 24.3 percent of farmland area and 37.5 percent of the value of agricultural production.

Developing countries
In sub-Saharan Africa, 80% of farms are family owned and worked.

Sub-Saharan agriculture was mostly defined by slash-and-burn subsistence farming, historically spread by the Bantu expansion. Permanent farming estates were established during colonialism, in the 19th to 20th century. After decolonisation, white farmers in some African countries have tended to be attacked, killed or evicted, notably in South Africa and Zimbabwe.

In southern Africa, "On peasant family farms ..., cash input costs are very low, non‐household labour is sourced largely from communal work groups through kinship ties, and support services needed to sustain production are minimal." On commercial family farms, "cash input costs are high, little non‐family labour is used and strong support services are necessary."

International Year of Family Farming
At the 66th session of the United Nations General Assembly, 2014 was formally declared to be the "International Year of Family Farming" (IYFF). The Food and Agriculture Organization of the United Nations was invited to facilitate its implementation, in collaboration with Governments, International Development Agencies, farmers' organizations and other relevant organizations of the United Nations system as well as relevant non-governmental organizations.

The goal of the 2014 IYFF is to reposition family farming at the centre of agricultural, environmental and social policies in the national agendas by identifying gaps and opportunities to promote a shift towards a more equal and balanced development. The 2014 IYFF will promote broad discussion and cooperation at the national, regional and global levels to increase awareness and understanding of the challenges faced by smallholders and help identify efficient ways to support family farmers.

title: Family farm


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Lizzie Miles

Elizabeth Mary Landreaux (March 31, 1895 – March 17, 1963), known by the stage name Lizzie Miles, was an Afro-Creole blues singer in the United States.

Biography
Miles was born in the Faubourg Marigny neighborhood of New Orleans, Louisiana, in an Afro-Creole Kouri-Vini (Louisiana Creole) speaking family. As a child, she sang in her Catholic church and performed at parties and dances. She worked with Joe Oliver, Kid Ory, Bunk Johnson, and A.J. Piron from 1909 to 1911. She then toured the South, performing in theaters, circuses, and with minstrel shows owned/managed by J. Augustus Jones, Elmer H. Jones and their family. In 1917 she sang in Chicago with Manuel Manetta, and then, in 1921 with Freddie Keppard, Charlie Elgar, and again with Oliver. She moved to New York and made her first phonograph recordings in 1922. They were blues songs, but she did not like to be referred to as a blues singer since she sang a wide repertoire.

Miles toured Europe in 1924 and 1925 and then returned to New York and worked in clubs from 1926 to 1931. During this time she worked with her half-brother, Herb Morand. Miles recorded as leader of a trio with Oliver, and in a duo with Jelly Roll Morton. There is uncertainty in that some sources suggest that several of the Miss Frankie recordings were the work of Lizzie Miles. This particularly applies to the tracks "When You Get Tired of Your New Sweetie", and "Shooting Star Blues", issued on Conqueror Records (January 1928).

She suffered a serious illness and retired from the music industry in the 1930s, not before she recorded "My Man o' War", described by one music journalist as "a composition stuffed with rococo suggestiveness". Despite her illness, Miles appeared in two films in the early 1930s. She began working regularly again in 1935, performing with Paul Barbarin at the Strollers Club in New York. She sang with Fats Waller in 1938 and then worked in Chicago until she left music in 1942.

In 1950, Miles lived in California where she sang with George Lewis in 1953 and 1954. She performed and spent time with Bob Scobey in Las Vegas, Nevada, from 1955 to 1957. She sang with Joe Darensbourg in Chicago in 1958 and 1959. She returned to New Orleans, where she appeared with Freddie Kohlman and Paul Barbarin. She recorded with several Dixieland and traditional jazz bands, appeared at the Monterey Jazz Festival in 1958, and made regular radio broadcasts before retiring in 1959.

In 1959 she quit singing, except for gospel music. She began working closely with the Sisters of the Holy Family, an order of Black religious in the city, declaring that she had decided "to live the life of a nun". She died of a heart attack, in March 1963, at the sisters' Lafon Nursing Home in New Orleans and was buried in the city at Saint Louis Cemetery No. 3.

Woody Allen included her version of "A Good Man Is Hard to Find" on the soundtrack of his 2013 film Blue Jasmine.

Her half-sister, Edna Hicks, was also a blues singer.

Personal life
She married August Pajaud in New Orleans, Louisiana on May 9, 1912. The marriage certificate indicates that she was 19 [sic].

She married John C. Miles, from whom she took her stage name, in Norfolk, Virginia in 1914. He was a bandleader also working for the Jones brothers. J.C. Miles died of Spanish flu in Shreveport, Louisiana on October 19, 1918 while on tour and was buried in Crown Hill Cemetery, Indianapolis, Indiana.

Singles released in 1922
The following singles were all released in 1922 by Okeh Records:

 * "Wicked Blues"
 * "Take It 'Cause It's All Yours"
 * "Lonesome Monday Morning Blues"
 * "Please Don't Tickle Me, Babe"
 * "He May Be Your Man, but He Comes to See Me Sometimes"
 * "Muscle Shoals Blues"
 * "She Walked Right Up and Took My Man"

title: Lizzie Miles