Abstract
In this chapter we present the early solutions of the brachistochrone problem in the same mathematical words of their authors, if not simply in their words. We restate the problem.
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- 1.
In Latin, Opera Omnia, t. I, p. 155 and p.166,
Problema Novum,
ad cujus Solutionem Mathematici invitantur.
Datis in plano verticali duobus punctis A et B assignare mobili M viam AMB, per quam gravitate sua descendens et moveri incipiens a puncto A, brevissimo tempore perveniat ad alterum punctum B.
- 2.
The translation of the original Latin sounds as: “The curvature of a ray in nonuniform media, and the solution of the proposed problem in Acta [Eruditorum] 1696, p. 269, to find the brachistochrone line, that is, the curve on which a heavy point falls from a given position to another given position in the shortest time, as well as on the construction of the synchrone or the wave of the rays”.
- 3.
Johann refers to [127] Proposition XXV, [129] vol. 18; for a presentation see [203] [130] [110].
- 4.
We shall discuss the second direct method in Section 2.5. A direct discover of the agreement of the isochronous curve and the brachistochrone could go along the following lines. We may think of the idea of Huygens as of modifying the circle of the standard pendulum in such a way that the accelerating force becomes proportional to the arc length s. This way, the movement of the pendulum would be described by
$$\ddot{s}+ ks=0,$$which has oscillations independent of the amplitude.
The accelerating force is \(f=-dy/ds,\) hence our requirement \(f=-ks\) becomes
$$dy=ksds.$$Integrating (\(s=0\) for \(y=0\))
$$y=\frac{k}{2} s^2 \text { or } s=\sqrt{\frac{2y}{k}}.$$Therefore, for our curve the height is proportional to the square of the arc length and, in terms of the variables x, y one finds
$$\sqrt{\frac{c-y}{y}}dy=dx$$with \(c=1/(2k).\) A part from the shift in y, as we shall see, this is precisely the equation of the brachistochrone.
- 5.
See [96] vol. II.
- 6.
See [160].
- 7.
See [128].
- 8.
The translation is taken from [187] p. 392-3.
- 9.
The translation is taken from [187] p. 393. The original passage in Latin is:
Si nunc concipiamus medium non uniformiter densum, sed velut per infinitas lamellas horizontaliter interjectas distinctum, quarum interstitia sint repleta materia diaphana raritatis certa ratione accrescentis vel decrescentis; manifestum est, radium, quem ut globulum consideramus, non emanaturum in linea recta, sed in curva quadam [...]. Constat quoque, cum sinus refractionum in singulis punctis sint respective ut raritates medii vel celeritates globuli, curvam habere eam proprietatem, ut sinus inclinationum suarum ad lineam verticalem sint ubique in eadem ratione celeritatum. Quibus praemissis nullo negotio perspicitur, curvam brachystochronam illam ipsam esse, quam formaret radius transiens per medium, cujus raritates essent in ratione velocitatum, quas grave verticaliter cadendo acquireret. Sive enim velocitatum incrementa dependeant a natura medii magis minusve resistentis, ut in radio; sive abstrahatur a medio, & ab alia causa acceleratio eadem tamen lege generari intelligatur,
ut in gravi: cum utroque in casu curva brevissimo tempore percurri supponatur, quid vetat, quo minus altera in alterius locum substitui possit?
Sic generaliter solvere licet problema nostrum, quamcunque statuamus accelerationis legem. Eo enim reductum est, ut quaeratur curvatura radii in medio secundum raritates, prout libuerit, variante.
- 10.
In principle, the choice of the factor \(\sqrt{a}\) is arbitrary and, as we know, a more reasonable value is \(t=\sqrt{2g x},\) where g is the gravity constant, but it is easily seen that this leads to the same equation (2.4), of course with a different constant as a, that we may again name a.
- 11.
Thus
$$y=\sqrt{ax-x^2}+\frac{1}{2} \int \frac{a}{\sqrt{ax-x^2}}$$that is the equation of the cycloid that Leibniz had found in 1686.
- 12.
This is easily seen analytically or as consequence of Euclid’s theorem ; in fact,
$$\begin{aligned} \sqrt{ax-x^{2}}= & {} \sqrt{GK\cdot GO-GO^{2}}=\sqrt{GO(GK-GO)}= \\= & {} \sqrt{GO\cdot KO}=LO, \end{aligned}$$since KLG is a right-angle triangle.
- 13.
As for instance,
$$\begin{aligned} d[\mathrm {arc}\,GL]= & {} \sqrt{dLO^{2}+dOG^{2}}= \\= & {} \sqrt{[d(\sqrt{ax-x^{2}})]^{2}+dx^{2}}= \sqrt{\big [ \frac{adx-2xdx}{2\sqrt{ax-x^{2}}}\big ]^{2}+dx^{2}}= \\= & {} \sqrt{\frac{a^{2}dx^{2}+4x^{2}dx^{2}-4axdx^{2}+4axdx^{2}-4x^{2}dx^{2}}{(2\root 2 \of {ax-x^{2}})^{2}}}= \\= & {} \frac{adx}{2\sqrt{ax-x^{2}}}. \end{aligned}$$ - 14.
This is also seen analytically as in [117] p.41 (footnote 49): let \(b=a/2\), \(\mathrm {arc}\,LK=b\varphi \) and \(\mathrm {arc}\,GL=b\pi -b\varphi =bt\). Then \(\sin t=\sin \varphi =LO/b\) and \(CO=AG=b\pi =y+ML+LO=y+\mathrm {arc}\,LK+b\sin t\); hence \(y=b\pi -\mathrm {arc}\,LK-b\sin t=b\pi -b\varphi -b\sin t=b(t-\sin t).\) To find x, note that \(\cos t=-\cos \varphi =-(b-OK)\) and that \(x= AC= GK-OK= b(1-\cos t).\) Note that b is the radius of the generating circle.
In the remainder of this section we shall see how, in the light of the common knowledge of the time about the geometrical properties of the cycloid, there are several ways of inferring from (2.1) that the solution is a cycloid. Such a geometrical understanding most likely was the way to the solution for the scholars of the time including Leibniz, Newton, Johann and Jakob Bernoulli. It is also likely that the mathematicians of the Continent stressed the analytical aspects in view of the generality of the approach and in contrast to the geometric character of the Newtonian calculus.
- 15.
For modern standard, if a quickest time descent curve exists.
- 16.
Given a cycloid ARS on the horizontal line AS, the cycloid ABL defined as above is the quickest time descent curve from A to B. This is Newton’s solution to the brachistochrone problem without any motivation of why the solution has to be a cycloid, see Section 2.4.
- 17.
This may be regarded as an anticipation of the transversality condition that will play an important role in the Nineteenth century.
- 18.
Previously we mentioned that this is often called Euler’s lemma.
- 19.
This is essential for the following and involves the second differentials.
- 20.
In fact
$$\begin{aligned} CL&= \sqrt{CM^2+ML^2}= CM\, \sqrt{1+\frac{ML^2}{CM^2}} \\&\backsimeq CM+\frac{1}{2} \frac{ML^2}{CM} \backsimeq CM. \end{aligned}$$ - 21.
[166], III/1, n. XXVIII and XXIX, pp. 277-290. See also [140] [139] [138].
- 22.
Gerhardt edited as Beitrage to the letter, [166], III/1 pp. 290-295.
- 23.
It is not proved that the addendum was sent together with the letter or later or ever. But, there are reasons to believe that it was written before the letter was sent, see [117].
- 24.
It is the first time an extremal is shown to yield actually a minimum. Indeed, giving sufficient conditions so that such a claim holds was a central problem in the calculus of variations until the beginning of last century, and several approaches were developed: Jacobi’s theory, the approaches of Weierstrass , Kneser , Hilbert , and Mayer , and the quickest and probably most elegant approach of Carathéodory, known as the royal road to calculus of variations, all via field theory. The interested reader is referred, for instance, to [33] [112] [117] [193].
The modern reader might be surprised that the simple remark “extremals of a strictly convex functional are indeed minimizers” did not occur for so long. However, though already Archimedes investigated convex curves, in the Eighteenth and Nineteenth centuries the notion of convexity appeared only sporadically; the foundation of the geometry of convex bodies is due to Brunn and Minkowski around 1900.
- 25.
In [109] he deals with geodesic fields and transversal surfaces.
- 26.
Here he refers to Huygens principle on light rays and wave fronts.
- 27.
Quotation from [112] I, pp. 396-397.
- 28.
See [30] 1, p. 348 and [166] III.1, p. 370. See also [22] in [118] p. 285.
- 29.
The following is the original passage in French:
Pour mettre fin à ce Memoire l’y vas ajouter ma Methode directe de resoudre le fameux Problème de la plus vîte descente, n’ayant point encore publié cette Methode, quoi-que je l’aye communiquée à plusieurs de mes Amis dès 1697 que je publiai mon autre indirecte. L’incomparable M.Leibnitz, à qui je les avois communiquées toutes deux, comme il l’a temoigné lui-meme dans les Actes de Leipsick de cette meme année 1697 [G.W. Leibniz, Communicatio suae pariter, duarumque alienarum [...] solutionum problematis curvae celerrimi descensus, AE, Maji 1697 , pp.201-205], pag 204, trouva cette Methode directe d’une beauté si singuliere, qu’il me conseilla de ne la pas publier, pour de raisons qui étoient alors, & qui ne subsistent plus. J’espere qu’elle plaira aussi d’autant plus au Lecteur que, quoi-que l’Analyse n’en conduise qu’au rayon de la curvité ou du cercle osculateur de la Courbe cherchée, laquelle se trouve ainsi être la cycloïde ordinaire qu’on sçait avoir seule, en quelque point que ce soit, un tel rayon da sa curvité ou de son cercle osculateur; cette Methode me fournit cependant aussi une démonstration synthetique, qui avec une facilité surprenante & agreeable fait voir que cette cycloïde est effectivement la Courbe cherchée de la plus vîte descente.
- 30.
We have replaced the original letters x, m, n with \(\xi \), p, and q, respectively, to avoid possible confusions.
- 31.
Chosing horizontal coordinate x and downwards vertical coordinate y, the cycloid generated by a circle of radius r has parametric equations
$$\begin{aligned} \left\{ \begin{array}{c} x=r(\varphi -\sin \varphi ) \\ y=r(1-\cos \varphi ). \end{array} \right. \end{aligned}$$We can easily compute
$$\frac{dy}{dx}=\cot \frac{\varphi }{2}, \quad \frac{ds}{d \varphi }=2a \sin \frac{\varphi }{2},$$and show that \(MK=y\, ds/dx\) where s is the arc length. The radius of curvature (as inverse of the curvature) is \(ds/d \theta \) where \(\theta \) is \(\arctan dy/dx.\) Since \(\cot \varphi /2=dy/dx=\tan \theta \) we see that \(\theta = \pi /2-\varphi /2\) and \(ds/d \theta =2ds/d \varphi ,\) i.e. \(MK=2NK.\) Actually, this is characteristic of a cycloid.
- 32.
Cc is the hypotenuse of the right triangle Cec.
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Freguglia, P., Giaquinta, M. (2016). The Brachistochrone Problem: Johann and Jakob Bernoulli. In: The Early Period of the Calculus of Variations. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-38945-5_2
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