2/7/2024 0 Comments Variational calculus examplesTypical Problem: Consider a definite integral that depends on an unknown function \(y(x)\), as well as its derivative \(y'(x)=\frac \right]. A classic example of the calculus of variations is to find the brachistochrone, defined as that smooth curve joining two points A and B (not underneath one. The mathematical techniques developed to solve this type of problem are collectively known as the calculus of variations. One example is finding the curve giving the shortest distance between two points - a straight line, of course, in Cartesian geometry (but can you prove it?) but less obvious if the two points lie on a curved surface (the problem of finding geodesics.) Many problems involve finding a function that maximizes or minimizes an integral expression. When 0, the function I has an extremum and so dI d 0 when 0. 1 I(Y) depends on the value of, and we write I for the value of I(Y): I b a F(x, Y, Y ) dx.
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