Bernoulli's lightray solution for the brachistochrone problem from ...

Bernoulli’s lightray solutionfor the brachistochrone problemfrom Hamilton’s viewpointHenk BroerJohann Bernoulli Institute for Mathematics and Computer ScienceRijksuniversiteit Groningen

Summaryi. Introductionii. Bernoulli’s solutioniii. Translation into Hamilton’s termsvi. Concluding remarksA study in anachronism...H.H. Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century.Studies in the History of Mathematics and Physical Sciences 5, Springer 1980J.A. van Maanen, Een Complexe Grootheid, leven en werk van Johann Bernoulli. 1667–1748.Epsilon Uitgaven 34 1995D.J. Struik (ed.), A Source Book in Mathematics 1200–1800. Harvard University Press 1969;Princeton University Press 1986

IntroductionJohann Bernoulli (1667 - 1748)‘his’ brachistochrone problem- Groningen period 1695 – 1705- Acta Eruditorum 1697Opera Johannis Bernoullii. G. Cramer (ed.; 4 delen), Genève 1742

Bernoulli’s lightray solutionFermat principle of least time for light rays optical solution in mechanical setting- Discrete approximation in horizontal layerseach layer homogeneous and isotropic refraction index n = 1/v (c = 1)- Fermat principle⇒- inside layer:ray = straight line with velocity v = 1/n- at boundary: Snell’s lawG.W. Leibniz, Nova methodus pro maximis et minimis. Acta Eruditorum 1684. In: D.J. Struik, Asource book in mathematics 1200–1800. Harvard University Press 1969, 271-281

Fermat and Snelln 1 sinα = n 2 sinβProof: Let x = x C indicate position of Ct AC = n 1 |A−C| andt CB = n 2 |C −B|

Fermat and Snell, ctdBy Pythagoras theorem:|A−C| = √ x 2 +b 2 , |C −B| =√(a−x) 2 +c 2Differentiatingt AC and t CB w.r.t. x :ddx t AC(x) =ddx t AC(x) = − n 2(a−x)√(a−x) 2 +c 2n 1 x√x 2 +b 2 = n 1 sinα= −n 2 sinβ:ddx (t AC +t CB )(x) = 0 ⇔ n 1 sinα = n 2 sinβ□- locally minimum- globally caustics (think of focal points ellipse)

Bernoulli and Snell- Snell’s law: n j sinα j = n j+1 sinα ′ j- Euclidean geometry: α ′ j = α j+1 Conservation law: n j sinα j = n j+1 sinα j+1just take limits ... inclination α with verticaland conserved quantity nsinα

The cycloid as brachistochroneTwo conserved quantities as a function ofyS = n(y)sinα(y) (1)E = 1 2 m(v(y))2 +mgy (2)Better look for profile α ↦→ (x(α),y(α)) !Theorem.x(α) = x 0 − 14S 2 g (2α−sin(2α))y(α) = y 0 + 14S 2 g cos(2α)Note: E = 1 2 m(v(y 0)) 2 +mgy 0 ,Possible choices: v(y 0 ) = 0 and even y 0 = 0 falling ratev(y) = √ −2gy

A few high school computationsLemma. (abbreviating v ′ = dv/dy)v ′ = − gv(y)andcosα = Sv ′ (y) dydα ,Proof: Differentiate (2) with respect toy and (1) with respect toα□Thusdydαlemma=1Sv ′ (y)cosαlemma= − v(y)Sgvcosα(1)= − 12S 2 gsin(2α) (3)while:dxdα= tanαdydα(3)= − v(y)Sgsinα(1)= − 12S 2 g (1−cos(2α))Integration then gives the desired result.Cycloid with radius14S 2 gand rolling angle 2α□

Radius and rolling angleRoll wheel ( radius̺) along ceilingcycloid:x(ϕ) = ̺(ϕ+sinϕ),y(ϕ) = ̺(1−cosϕ)parameterϕcalled rolling angle

Christiaan Huygens (1629-1695)Christiaan Huygens (by Gaspar Netscher)and page from Horologium Oscillatorium 1673

Commentary- Challenge in Acta Eruditorum (1697)Many contemporaries published solutionsNewton’s was anonymous, however,ex ungue leonem cognavi- Isochrony and tautochrony cycloid dynamicsby harmonic oscillationsJ.L. Lagrange, Mécanique Analytique. 4 ed., 2 vols. Gauthier-Villars et fils, Paris, 1888-89

William Rowan Hamilton- Many contributions to mathematics,physics and astronomy ,principle of least actionW.R. Hamilton, Theory of systems of rays. Trans. Roy. Irish Academy 15 (1828), 69-174—, On a general method in dynamics. Phil. Trans. Roy. Soc. Vol. 124 (1834), 247-308; Secondessay on a general method in dynamics. Phil. Trans. Roy. Soc. Vol. 125 (1835), 95-144

Fermat principle revisited- Given curve τ ↦→ q(τ) consider ˙q = dq/dτ anddt = n(q(τ))||˙q(τ)||dτ magic formula- To (locally) minimize ∫ τ 2τ 1n(q(τ))||˙q(τ)||dτ or,rather and equivalently,I(q) :=∫ τ2τ 112 n2 (q(τ))||˙q(τ)|| 2 dτ- Lagrangian L(q, ˙q) = 1 2 n2 (q)||˙q|| 2(= energy = kinetic energy)

Calculus of variations- If q = (x,y) then Euler–Lagrange equationsddτddτ∂L∂ẋ = ∂L∂x∂L∂ẏ = ∂L∂ynecessary & sufficient for local optimality ofI- under small variations ofq,while keeping endpointsq(τ 1 ) andq(τ 2 ) fixed- in current optical setting also minimalL. Euler, Leonhardi Euleri Opera Omnia. 72 vols., Bern 1911-1975J.L. Lagrange, Œuvres. A. Serret and G. Darboux eds., Paris 1867-1892

Hamilton’s canonical formalism- Legendre transformation:R 2 × R 2 → R 2 × R 2 ,(q, ˙q) ↦→ (q,p) with p = n 2 (q)˙q- Hamiltonian H(q,p) = L(q,p/n 2 (q)) = 1n 2 (q) ||p||2Theorem. The lightrays are the projections of thesolutions of the canonical equationsto R 2 = {q}˙q j = ∂H∂p j, ṗ j = − ∂H∂q j(j = 1,2)- modern lingo in terms of (co-) tangent bundles

Laws, properties- Energy conservationḢ≡ 0, say H(q,p) = E- q = (x,y) and p x = n 2 (y)ẋ,p y = n 2 (y)ẏH(x,y,p x ,p y ) = 12n 2 (y) (p2 x +p 2 y)- H independent of x (cyclic variable): ṗ x ≡ 0conservation of momentum, sayp x = I- translation symmetry, Noether principle- conservation ofI √2E(x,y,ẋ,ẏ) = n(y)ẋ√ẋ2+ẏ 2 = n(y)sinα(ẋ,ẏ)(= S for Snell, this is familiar !)

Reduction of the symmetryTheorem. Given p x = I can reduce to planar systemH I (y,p y ) = 12n 2 (y) p2 y +V I (y) with V I (y) = I22n 2 (y)(effective potential)Reduced system:TakeI ≠ 0ẏ =1n 2 (y) p yṗ y = − n′ (y)n 3 (y) (I2 +p 2 y)

Gravitational refraction indexBack to brachistochrone problemGravitational energy 1 2 (v(y))2 +gy = 0 (n(y)) 2 = 1−2gy , fory < 0 reduced canonical equationsẏ = −2gyp yṗ y = g(p 2 y +I 2 )Level curve H I (y,p y ) = E has formy = − 1 ( ) Eg p 2 y +I 2(4)

Reduced phase-portraitReduced phase-portraitbrachistochrone lightray dynamics

Reconstruction ITime parametrization reduced system,for given I ≠ 0 completely determined by E,N- Time parametrizationdτ =dp yg(p 2 y +I 2 ) τ = 1 (gI arctan py)I- Gives parametrization (withp y (0) = 0):p y = Itan(gIτ) (4) y = − EgI 2 cos2 (gIτ) (5)PS: Similar action for pendulum system leadsto elliptic integrals ...

Reconstruction IICycloids recovered- I = p x = n 2 (y)ẋ ⇒ẋ =1n 2 (y)(5)I = −2Igy =2EI cos2 (gIτ) = E I (1+cos(2gIτ))- And sox = x 0 + E2gI2(2gIτ +sin(2gIτ))y = − E2gI 2(1+cos(2gIτ))Cycloid radius̺ = E/2gI 2 , rolling angleϕ = 2gIτInclination α = gIτ (proportionality)

Scholium I- Reminds of how Newton ‘principia’ yield theKeplerian orbits in central force field- Nowadays brachistochrone problem excercise incourse calculus of variations:Look for curvey = y(x)arclengthds = √ dx 2 +dy 2 = √ 1+(y ′ ) 2 dxenergy conservation in gravitational fieldso have to optimize... dsdt = √ −2gy

Scholium I: ctd- ... timedt =√1+(y′ ) 2√ −2gy L(y,y ′ ) =√− 1+(y′ ) 22gy- Euler–Lagrange equations ( 1+(y ′ ) 2) y = constantsubstitution of(x,y) = (x(θ),y(θ))again gives cycloid resultV.I. Arnold, Mathematical methods of classical mechanics. GTM 60, Springer 1978, 1989H. Goldstein, Classical mechanics 2nd ed. Addison Wesley 1950, 1980

Scholium II: lightrays geodesicsGeorg Friedrich Bernhard Riemann 1826-1866- Riemannian metricG q (˙q 1 , ˙q 2 ) = n 2 (q)〈˙q 1 , ˙q 2 〉,where latter brackets are EuclideannoteL(q, ˙q) = 1 2 G q(˙q, ˙q)- lightrays are geodesics of metricGM. Spivak, Differential geometry, Vols. I, II, III. Publish or Perish 1970

Atmospherical optics- blank strip in the setting Sun- illusions: fata morgana’s, Nova Zembla phenomenaM.G.J. Minnaert, De Natuurkunde van ’t vrije veld. Vol. 1, Vijfde Editie, ThiemeMeulenhoff1996; The Nature of Light & Colour in the Open Air, Dover 1954H.W. Broer, Aardse en hemelse luchtspiegelingen, met Bernoulli,Wegener en Minnaert. To bepublished Epsilon Uitgaven 2013

Arclength cycloidarclengths = s(ϕ) (using Pythagoras):ds = √ dx 2 +dy 2 == √ (dx/dϕ) 2 +(dy/dϕ) 2 dϕ == ̺√2 √ 1+cosϕ dϕ = 2̺cos ϕ 2 dϕ s(ϕ) = 4̺sin ϕ 2

Cycloidal wire profileVertical heighty(ϕ) = 2̺sin 2 ϕ 2 = 18̺(s(ϕ))2 potential energy“V = mgy” : V(s) = mg8̺s2−→ equation of motion beads ′′ = − g4̺s :a harmonic oscillator with ω = √ g/(4̺)Conclusion: cycloid isochronous curve(also tautochronous...)