What is meant by calculus of variations?
Definition of calculus of variations : a branch of mathematics concerned with applying the methods of calculus to finding the maxima and minima of a function which depends for its values on another function or a curve.
What is the story of calculus of variations?
Modern interest in the calculus of variations began in 1696 when Johann Bernoulli of Switzerland proposed a brachistochrone (“least-time”) problem as a challenge to his peers. Suppose that a thin wire in the shape of a curve joins two points at different elevations.
Why is calculus of variations important?
The calculus of variations is a powerful technique to solve some dynamic problems that are not intuitive to solve otherwise. It is the precursor to optimal control theory as it allows us to solve non-complex control systems.
Who worked on the calculus of variation *?
It was in his 1744 book, though, that Euler transformed a set of special cases into a systematic approach to general problems: the calculus of variations was born.
What is the difference between variation and differentiation?
variation (delta) is simply the change in a dependent variable due to a change in an independent variable (=delta y) while differentiation is the variation divided by a the change in the independent variable in a small range (=dy/dx).
Who invented calculus of variations?
What are the Extremals the functional?
A solution of the Euler-Lagrange equation is called an extremal of the functional. By considering y+g, where y is the solution from exercise 1 and g(x) is a variation in y(x) satisfying g(0)=g(1)=0, and then considering I(y+g), show explicitly that y(x) minimizes I(y) in Exercise 1 above.
Why is calculus called calculus?
In Latin, calculus means “pebble.” Because the Romans used pebbles to do addition and subtraction on a counting board, the word became associated with computation. Calculus has also been borrowed into English as a medical term that refers to masses of hard matter in the body, such as kidney stones.
What are the different kinds of variation in math?
Examples of types of variation include direct, inverse, joint, and combined variation.