Applied Mathematical Methods in Theoretical Physics || Calculus of Variations: Fundamentals

✍ Scribed by Masujima, Michio

• Publisher: Wiley-VCH Verlag GmbH & Co. KGaA
• Year: 2009
• Tongue: German
• Weight: 287 KB
• Edition: 2
• Category: Article
• ISBN: 352740936X
• DOI: 10.1002/9783527627745.ch9

✦ Synopsis

Calculus of Variations: Fundamentals

9.1 Historical Background

The calculus of variations was first found in the late 17th century soon after calculus was invented. The main figures involved are Newton, the two Bernoulli brothers, Euler, Lagrange, Legendre, and Jacobi.

Isaac Newton (1642-1727) formulated the fundamental laws of motion. The fundamental quantities of motion were established as the momentum and the force. Newton's laws of motion state:

(1) In the inertial frame, every body remains at rest or in uniform motion unless acted on by a force F. The condition F = 0 implies a constant velocity v and a constant momentum p = m v.

(2) In the inertial frame, application of force F alters the momentum p by an amount specified by

(3) To each action of a force, there is an equal and opposite action of a force. Thus if F 21 is the force exerted on particle 1 by particle 2, then

and these forces act along the line separating the particles.

Contrary to the common belief that he discovered the gravitational force by observing that the apple dropped from the tree at Trinity College, he actually deduced Newton's laws of motion from the careful analysis of Kepler's laws. He also invented the calculus, named methodus fluxionum in 1666, about 10 year ahead of Leibniz. In 1687, Newton published Philosophiae naturalis principia mathematica, often called Principia. It consists of three parts: Newton's laws of motion, laws of the gravitational force, and laws of motion of the planets. The Bernoulli brothers, Jacques (1654-1705) and Jean (1667-1748), came from the family of mathematicians in Switzerland. They solved the problem of Brachistochrone. They established the principle of virtual work as a general principle of