Functionals: An Introduction to Variational Calculus by way of Variation of Parameters: Approximating the Geodesic
The introductory form of calculus of variations involves finding an unknown function, y = y(x), which maximizes or minimizes a specified definite integral of y(x).
As an introductory problem, we might as a question such as, "of all paths that traverse a rolling landscape, which one is the shortest?" The y(x) that minimizes the total length is called the geodesic.
The total length of a path over a fixed surface can be written as an integral of the unknown y(x) (this is worked out in the accompanying lecture notes)
L[y(x)] = 
= 
Suppose that the "hill" has the shape h(x) = 
.
Here, the "real" problem of finding the geodesic from the infinite set of all paths connecting y(x=0) = 0 to y(x=1) = 1 will be replaced by looking at all second-order polynomials: y(x) = a + bx + c
and then minimizing over the (constrained) parameters. This will be compared with the exact calculation which we will work out below via calculus of variations.
We specify a landscape and an arbitrary quadratic path (a function of three parameters)
Illustrate this surface and identify the starting and ending points.
The general path must satisfy the boundary conditions:
There is one remaining free variable, it can be determined by minimizing the integral
To get an idea of how the path' s distance compares with its shape, we can manipulate the path and display that path' s length
Interactive Path Modification and Length Calculation
