Post on 19-Dec-2015
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
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