I am familiar with basic calculus of variations. For example, how to find a function that makes some integral functional stationary (Euler-Lagrange Equations). Or for example, how to perform that same problem but with some additional holonomic constraint or with some integral constraint. The problem I am interested in, however, is how to find a function that makes some property stationary (let's say an integral functional) subject to(adsbygoogle = window.adsbygoogle || []).push({}); inequalityconstraints. In particular I am interested in solving the usual stationary integral functional problem, but where the solution is confined to some region of function space defined by simple inequalities.

A very simple example:

What is the shortest path f(x) that connects the point (-2,1) to the point (2,1) such that f(x)≤x²?

In this problem, it is as if the standard parabola is a wall that blocks one from taking the usual straight line path. Intuitively, I'd say that the shortest path is the one that starts as a straight line from (-2,1) that is tangent to the parabola, approaches the parabola and then begins following the parabola, then bends back off the parabola on the tangent that carries it to (2,1). However, I have no clue how to prove this mathematically. There are also more complicated functionals than arc length that I am interested in minimizing, but hopefully this example will suffice for me to understand the general principle for attacking these sorts of problems.

EDIT: Whoa! I typed this too hastily and originally used the points (-1,1) and (1,1) which are on the parabola itself ::facepalm::. Now it should make sense.

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# Calculus of Variation on Local Regions of Function Space

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