variation

Variation of a Function in Classical Mechanics

ADV-MATHEMATICS

Calculus of variations, a branch of mathematics concerned with the problem of finding a function for which the value of a certain integral is either the largest or the smallest possible.

The problem of Calculus of variations was first solved by Jacob Bernoulli in 1696 but a general method of solving such problem was given by Euler.

Recapitulate:

If y=f(x) is a function, then necessary condition for y to be maximum or minimum is

dy/dx=0

§If  (d^2 y)/(dx^2 )>0 then that point is minimum point.

§If  (d^2 y)/(dx^2 )<0 then that point is maximum point.

If z=f(x,y) is a function, then necessary condition for z to be maximum or minimum is

∂z/∂x=0, ∂z/∂y=0

§If  (∂^2 z)/(∂x^2 )  (∂^2 z)/(∂y^2 )-((∂^2 z)/∂x∂y)^2>0 and (∂^2 z)/(∂x^2 )>0  then that point is minimum point.

§If  (∂^2 z)/(∂x^2 )  (∂^2 z)/(∂y^2 )-((∂^2 z)/∂x∂y)^2>0 and (∂^2 z)/(∂y^2 )<0  then that point is minimum point.

variation of function
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