1. The Bernoulli family The brachistochrone problem Willem Dijkstra February 2006, Eindhoven

2. Nicolaus JohannJacob Nicolaus I Nicolaus IIDanielJohann II Johann IIIDaniel IIJacob II Family tree L’Hospital’s Rule Differentiation, integration Brachistochrone problem Differentiation, integration Calculus of variations Probability theory Bernoulli numbers: Bernoulli’s law: Bernoulli polynomials:

3. Brachistochrone problem

4. Solution: cycloid

5. A B y x Time to travel from A to B: Energy balance: Arclength: Modern derivation

6. Beltrami identity: Non-linear ODE: Solution:

7. Bernoulli’s solution Snellius’ law:In each layer: is constant in infinitesimal time

8. Johann’s solution Use:and Jacob Bernoulli: more general applicable Leibniz: more or less the same Newton: does not show derivation

9. 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