The Bernoulli familyThe brachistochrone problem

Willem Dijkstra

February 2006, Eindhoven

Nicolaus

Nicolaus JohannJacob

Nicolaus I

Nicolaus II Daniel Johann II

Johann III Daniel II Jacob II

Family tree

•L’Hospital’s Rule

•Differentiation, integration

•Brachistochrone problem

•Differentiation, integration

•Calculus of variations

•Probability theory

•Bernoulli numbers:

1 !

n

nx

x xB

e n

•Bernoulli’s law:

•Bernoulli polynomials:

2 21 11 1 2 22 2p v p v

( ) n jn j

nb x B x

j

Brachistochrone problem

Solution: cycloid

A

B

y

x

Time to travel from A to B:

B

AB

A

dst

v

Energy balance:21

2 ( ) ( ) 2mv x mgy x v gy

Arclength: 2 2 21 ( ')ds dx dy y dx

21 ( ')

2

B

AB

A

yt dx

gy

Modern derivation

21 ( ')min ( , , ') with ( , , ')

2

B

A

yF x y y dx F x y y

gyy

Modern derivation21 ( ')

min ( , , ') with ( , , ')2

B

A

yF x y y dx F x y y

gyy

Beltrami identity: ''

FF y C

y

2

21dy

y kdx

212

212

( sin )

(1 cos )

x k

y k

Non-linear ODE:

Solution:

Bernoulli’s solution

Snellius’ law:sinsin r

rv v

In each layer: sindx

ds

1 1 r

r r

dxdx

v ds v ds

1 r

r r

dx

v dsis constant in infinitesimal time

dxcv

ds

dxcv

ds

Johann’s solution

Use: 2 2 2ds dx dy 2v gyand

221

dyy k

dx

• Jacob Bernoulli: more general applicable

• Leibniz: more or less the same

• Newton: does not show derivation

Conclusions

• Bernoulli familiy contributed to many fields in mathematics

• Brachistochrone problem marked the beginning of Calculus of Variations and discretisations.

…Although these problems seem to be difficult, I immediately started working on them. And what a succes I had! Instead of the proposed three months to get a flavor of the problems, instead of the remaining of the year to

solve them, I did not even use three minutes to explore the problem, to start working on them, and to completely solve them. And I even went further than that! For I will provide with solutions that are 1000 times more general than the

problems!

Johann Bernoulli, 1697