Brief Summary:

If you put a transparent straw in a glass filled with orange juice (for example), you will see that the surface of the orange juice in the straw is a little bit higher than in the rest of your glass. If you take a straw with a smaller diameter, the surface is located even higher. What's the reason for that behavior? The detailed answer will be given in the lecture - but here is the general recipe. You set up the total energy of the liquid in the straw consisting of potential energy in the gravitational field and a surface energy which is proportional to the size of the liquid surface. Writing things down, you see that the total energy depends on the height function h(x) which describes the liquid surface. According to a physical principle, the observed height is the one which minimizes the energy of the system - that means you have to find the minimum of the energy functional. Taking the derivative (which is called `variation' in this case) and setting it to zero, you find a necessary condition for the minimum. This condition turns out to be a partial differential equation with certain boundary conditions. Solving it, you can find the minimizer ...

The result of such a calculation can be seen in the following figure where the capillary surface has been calculated for three different straw diameters.

Here is an example from the theory of thin films:

If you take a wire hoop, bend it, and put it into soap liquid, you will observe a soap film in your wire frame once you remove it from the liquid. The shape of this surface is dictated again by the principle of least surface energy (it is surface of minimal area, i.e. a minimal surface). Going through the same process as described above: setting up the energy functional, taking its first variation and equating it to zero, you find a partial differential equation with boundary conditions as necessary condition for the surface to be minimal. The linearization of this PDE is the good old Laplace equation. Here is an exemplary solution: the graph on the right is the minimal surface for the given boundary curve (your wire hoop) and the graph on the left is the same calculation carried out with the simpler Laplace equation.

Here is an example from geometrical optics:

According to Fermat's principle, a ray of light travels from a light source to an observer in such a way that the total travel time is minimized (the total travel time is a functional  which assigns numbers to curves, i.e. functions - finding the physical ray requires minimization of the functional). Since the speed of light depends on the refraction index of the medium, the resulting path need not be a straight line if the refraction index is space dependent. For example, the density of the air (and thus also the refraction index) changes with height in the earth's atmosphere. Optical effects of this variation in the refraction index are most prominent for light rays which enter the atmosphere close to the horizon: just before sunset, the sun appears rather like an ellipse than a circle (flat), stars close to the horizon appear to be higher in the sky than they should be and so on...

Using the vanishing first variation as necessary condition for a minimum of the total time functional, one derives an ordinary differential equation for the light curve. Below, this equation is solved with several Neumann conditions on the left (prescribing the angle of the entering ray) and a Dirichlet condition on the right (the eye of the observer). The color indicates the refraction index which increases from blue to red.

The following example shows the typical form of light rays which enter the earth's atmosphere almost parallel to the horizon (sun set). The observer is located at the point (1,0) and the black line indicates the horizon (= tangent to the observer's position). The density is assumed to be radially symmetric with respect to the center of the earth. The corresponding refraction index is color coded in the graph below (blue=vacuum index, red=high index). One can see that, even if the sun is below the horizon, light rays can reach the observer's eye.

Now it is easy to explain the flat sun at sunset: The lower part of the sun is raised more than the upper part so that it becomes flat (rays entering at the same height are, of course treated in the same way so that the width of the sun is unchanged).

Here is a Hilbert-space example:

What happens if you project the constant one function onto the space of H¹(A) functions which vanish at the boundary of A? Using the standard H¹ scalar product, you get a function which is far away from the one function in the "picture norm". However, if you introduce a weight in the scalarproduct which reduces the importance of the L² norm of the derivatives the projection shows more resemblance with the constant one function. Below you see an example where A is an ellipse with two holes punched into it. The weights in front of the L² norm of the derivatives are (from left to right) 1/10, 1/100, 1/1000.

In the language of variational calculus: the functional is the square of the (weighted) H¹ distance to the one function and the necessary condition for a minimum (vanishing first variation) leads to a Helmholtz equation with unit source.