Main
Contributors to the Calculus of Variations
Bernoulli Brothers:
The first problem in
the Calculus of Variations is attributed to the Bernoulli Brothers:
Jakob (1654-1705) and Johann (1667 - 1748). Both studied with Leibnitz.
Johann is considered to be the Inventor of the Calculus of Variations,
being the first to formulate and solve the Brachystochrone problem.
Johann accepted the Chair of Mathematics at Basel University in
1705 vacated by the death of his brother and taught there until
his death.
Brachistochrone Problem: What Should be the shape of the rope such
that the cart travels from the top of the tower to the ground in
least amount of time? Answer: A Cycloid
Leonhard Euler (1707
- 1783)
Leonhard Euler (Basel, Switzerland) is considered to be the most
productive mathematician of all time. He studied with the Bernoulli
brothers, and was profoundly influenced by Johann Bernoulli. From
1725 through 1741 he stayed at the St.Petersburg Academy. From 1741
to 1766 Euler was at the Berlin Academy under the tutelage of Frederick
the Great. From 1766-83 he was again in St.Petersburg, this time
under the sponsorship of the empress Catherine. Although he lost
one eye in 1735 and the other eye in 1766, he continued his remarkable
career in Mathematics. He published 530 books and papers during
his life. St.Petersburg Academy continued publishing the manuscripts
he left at death for the next forty Seven years.
Joseph Louis Lagrange
( 1736 - 1813)
Lagrange was born in Turin. At the age of 19 he became the professor
of mathematics in the artillery school of Turin. In 1766 when Euler
left for St. Petersburg, Frederick the great invited Lagrange to
come to Berlin. After the death of Frederick the great in 1786,
he went to Paris and participated in the French revolution. He assisted
in introducing the Metric System, and later became a Professor at
the Ecole Polytechnique (1797). His most well known contributions
are in the fields of Differential Calculus, Calculus of Variations
and Analytical Mechanics.
Adrien Marie Legendre
(1752-1833)
Legendre taught at the Military School in Paris from 1775-1780.
Later he worked in several Government positions, such as Professor
at Ecole Normale, Examiner at Ecole Polytechnique and Geodetic Surveyor.
Legendre's name is associated with one of the Second-Order Necessary
Conditions for a Weak Local Minimum in the Calculus of Variations.
The other Second-Order
Necessary Condition for a Weak Local Minimum is due to Carl Gustav
Jacob Jacobi (1805 - 1851).
Karl Weierstrass
(1815-1897)
Karl Weierstrass was a teacher at the Prussian Gymnasia for many
years. In 1856 he became the Professor of Mathematics at the University
of Berlin where he taught for 30 years. Most of Weierstrass's ideas
come to us through his lecture notes published by his students.
The Necessary
Condition for a Strong Local Minimum in the Calculus of Variations
is one of his noted contributions.
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