Bernoulli’s lightray solution

for the brachistochrone problem

from Hamilton’s viewpoint

Henk Broer

Johann Bernoulli Institute for Mathematics and Computer Science

Rijksuniversiteit Groningen

Summary

i. Introduction

ii. Bernoulli’s solution

iii. Translation into Hamilton’s terms

vi. Concluding remarks

A 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 1980

J.A. van Maanen, Een Complexe Grootheid, leven en werk van Johann Bernoulli. 1667–1748.

Epsilon Uitgaven 34 1995

D.J. Struik (ed.), A Source Book in Mathematics 1200–1800. Harvard University Press 1969;

Princeton University Press 1986

Introduction

Johann Bernoulli (1667 - 1748) ‘his’ brachistochrone problem

- Groningen period 1695 – 1705

- Acta Eruditorum 1697

Opera Johannis Bernoullii. G. Cramer (ed.; 4 delen), Genève 1742

Bernoulli’s lightray solution

Fermat 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 law

G.W. Leibniz, Nova methodus pro maximis et minimis. Acta Eruditorum 1684. In: D.J. Struik, A

source book in mathematics 1200–1800. Harvard University Press 1969, 271-281

Fermat and Snell

n1 sinα = n2 sin β

Proof: Let x = xC indicate position of C

tAC = n1 |A− C| and tCB = n2 |C − B|

where | − | denotes Euclidean metric

Fermat and Snell, ctd

By Pythagoras theorem:

|A− C| =√

x2 + b2, |C −B| =√

(a− x)2 + c2

Differentiating tAC and tCB w.r.t. x :

d

dxtAC(x) =

n1x√x2 + b2

= n1 sinα

d

dxtAC(x) = − n2(a− x)

√

(a− x)2 + c2= −n2 sinβ

:d

dx(tAC + tCB)(x) = 0 ⇔ n1 sinα = n2 sinβ �

- locally minimum

- globally caustics (think of focal points ellipse)

Bernoulli and Snell

- Snell’s law: nj sinαj = nj+1 sinα′

j

- Euclidean geometry: α′

j = αj+1

Conservation law: nj sinαj = nj+1 sinαj+1

just take limits . . . inclination α with verticaland conserved quantity n sinα

The cycloid as brachistochrone

Two conserved quantities as a function of y

S = n(y) sinα(y) (1)

E = 12m(v(y))2 +mgy (2)

Better look for profile α 7→ (x(α), y(α)) !

Theorem.

x(α) = x0 −1

4S2g(2α− sin(2α))

y(α) = y0 +1

4S2gcos(2α)

Note: E =12m(v(y0))2 +mgy0,

Possible choices: v(y0) = 0 and even y0 = 0 falling rate v(y) =√−2gy

A few high school computationsLemma. (abbreviating v′ = dv/dy)

v′ = − g

v(y)and cosα = Sv′(y)

dy

dα,

Proof: Differentiate (2) with respect to y and (1) with respect to α �

Thus

dy

dα

lemma=

1

Sv′(y)cosα

lemma= −v(y)

Sgv cosα

(1)= − 1

2S2gsin(2α) (3)

while:dx

dα= tanα

dy

dα

(3)= −v(y)

Sgsinα

(1)= − 1

2S2g(1− cos(2α))

Integration then gives the desired result. �

Cycloid with radius 14S2g

and rolling angle 2α

Radius and rolling angle

Roll wheel ( radius ) along ceiling cycloid:

x(ϕ) = (ϕ+ sinϕ),

y(ϕ) = (1− cosϕ)

parameter ϕ called 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 solutions

Newton’s was anonymous, however,ex ungue leonem cognavi

- Isochrony and tautochrony cycloid dynamicsby harmonic oscillations

J.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 action

W.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; Second

essay on a general method in dynamics. Phil. Trans. Roy. Soc. Vol. 125 (1835), 95-144

Fermat principle revisited

- Given curve τ 7→ q(τ) consider q = dq/dτ and

dt = n(q(τ))||q(τ)|| dτ magic formula

- To (locally) minimize∫ τ2τ1

n(q(τ))||q(τ)|| dτ or,

rather and equivalently,

I(q) :=∫ τ2

τ1

12n2(q(τ))||q(τ)||2 dτ

- Lagrangian L(q, q) = 12n2(q)||q||2

(= energy = kinetic energy)

Calculus of variations- If q = (x, y) then Euler–Lagrange equations

d

dτ

∂L

∂x=

∂L

∂xd

dτ

∂L

∂y=

∂L

∂y

necessary & sufficient for local optimality of I- under small variations of q,

while keeping endpoints q(τ1) and q(τ2) fixed

- in current optical setting also minimal

L. Euler, Leonhardi Euleri Opera Omnia. 72 vols., Bern 1911-1975

J.L. Lagrange, Œuvres. A. Serret and G. Darboux eds., Paris 1867-1892

Hamilton’s canonical formalism- Legendre transformation:

R2 × R2 → R2 × R2,(q, q) 7→ (q, p) with p = n2(q)q

- Hamiltonian H(q, p) = L(q, p/n2(q)) = 1n2(q) ||p||2

Theorem. The lightrays are the projections of thesolutions of the canonical equations

qj =∂H

∂p j

, pj = −∂H

∂q j

(j = 1, 2)

to R2 = {q}

- modern lingo in terms of (co-) tangent bundles

Laws, properties

- Energy conservation H ≡ 0, say H(q, p) = E

- q = (x, y) and px = n2(y)x, py = n2(y)y

H(x, y, px, py) =1

2n2(y)(p2x + p2y)

- H independent of x (cyclic variable): px ≡ 0conservation of momentum, say px = I

- translation symmetry, Noether principle

- conservation ofI√2E(x, y, x, y) = n(y) x√

x2+y2= n(y) sinα(x, y)

(= S for Snell, this is familiar !)

Reduction of the symmetry

Theorem. Given px = I can reduce to planar system

HI(y, py) =1

2n2(y)p2y + VI(y) with VI(y) =

I2

2n2(y)

(effective potential)Reduced system:

y =1

n2(y)py

py = −n′(y)

n3(y)(I2 + p2y)

Take I 6= 0

Gravitational refraction indexBack to brachistochrone problem

Gravitational energy 12(v(y))2 + gy = 0

(n(y))2 =1

−2gy, for y < 0

reduced canonical equations

y = −2gypy

py = g(p2y + I2)

Level curve HI(y, py) = E has form

y = −1

g

(

E

p2y + I2

)

(4)

Reduced phase-portrait

Reduced phase-portraitbrachistochrone lightray dynamics

Reconstruction ITime parametrization reduced system,for given I 6= 0 completely determined by E,N

- Time parametrization

dτ =dpy

g(p2y + I2) τ =

1

gIarctan

(pyI

)

- Gives parametrization (with py(0) = 0):

py = I tan(gIτ)(4) y = − E

gI2cos2(gIτ) (5)

PS: Similar action for pendulum system leadsto elliptic integrals . . .

Reconstruction IICycloids recovered

- I = px = n2(y)x ⇒

x =1

n2(y)I = −2Igy

(5)=

2E

Icos2(gIτ) =

E

I(1 + cos(2gIτ))

- And so

x = x0 +E

2gI2(2gIτ + sin(2gIτ))

y = − E

2gI2(1 + cos(2gIτ))

Cycloid radius = E/2gI2, rolling angle ϕ = 2gIτInclination α = gIτ (proportionality)

Scholium I- Reminds of how Newton ‘principia’ yield the

Keplerian orbits in central force field

- Nowadays brachistochrone problem excercise incourse calculus of variations:

Look for curve y = y(x)arclength

ds =√

dx2 + dy2 =√

1 + (y′)2 dx

energy conservation in gravitational field

ds

dt=

√

−2gy

so have to optimize . . .

Scholium I: ctd

- . . . time

dt =

√

1 + (y′)2√−2gy L(y, y′) =

√

−1 + (y′)2

2gy

- Euler–Lagrange equations

(

1 + (y′)2)

y = constant

substitution of (x, y) = (x(θ), y(θ))again gives cycloid result

V.I. Arnold, Mathematical methods of classical mechanics. GTM 60, Springer 1978, 1989

H. Goldstein, Classical mechanics 2nd ed. Addison Wesley 1950, 1980

Scholium II: lightrays geodesics

Georg Friedrich Bernhard Riemann 1826-1866

- Riemannian metric Gq(q1, q2) = n2(q)〈q1, q2〉,where latter brackets are Euclideannote

L(q, q) = 12Gq(q, q)

- lightrays are geodesics of metric G

M. 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 phenomena

M.G.J. Minnaert, De Natuurkunde van ’t vrije veld. Vol. 1, Vijfde Editie, ThiemeMeulenhoff

1996; The Nature of Light & Colour in the Open Air, Dover 1954

H.W. Broer, Aardse en hemelse luchtspiegelingen, met Bernoulli,Wegener en Minnaert. To be

published Epsilon Uitgaven 2013

Arclength cycloid

arclength s = s(ϕ) (using Pythagoras):

ds =√

dx2 + dy2 =

=√

(dx/dϕ)2 + (dy/dϕ)2 dϕ =

= √2√

1 + cosϕ dϕ = 2 cosϕ

2dϕ

s(ϕ) = 4 sin ϕ2

Cycloidal wire profile

Vertical height

y(ϕ) = 2 sin2ϕ

2=

1

8(s(ϕ))2

potential energy “V = mgy” : V (s) = mg8 s

2

−→ equation of motion bead

s′′ = − g

4s :

a harmonic oscillator with ω =√

g/(4)

Conclusion: cycloid isochronous curve(also tautochronous . . .)