by Stefan Müller
Calculus of variations - mathematics of optimal shapes
Many questions in geometry or physics can be stated in terms of maximization or minimization of certain quantities. Calculus of variations studies exactly such minimization problems. Some relevant geometrical questions, as for example the search of the curve which, under fixed length, contains the maximal area, have been already studied by the ancients. A breakthrough in the calculus of variations was the work of Leonard Euler (1707-1783), that developed systematically a calculus which allows to formulate necessary conditions for minimizers and maximizers, the so-called Euler-Lagrange equations. Armed with these equations, one was lead into expecting the optimal shape to be particularly smooth and esthetically pleasing. One can think to spheres, crystals or minimal surfaces (surfaces which take the smallest surface area with fixed boundary conditions, as e.g. soap films). Not only for their static properties, but also for their aestethic appeal minimal surfaces have been often employed in architecture, for example in the Olympiastadion in Munich (see www.olympiapark.de).
The rough, small-scale microstructures do not at first glance conform to the expectation that optimal shapes are smooth. Their existence is strongly connected to a crisis that the calculus of variations experienced at the end of the XIX century, when its methods did not seem able to comply with the new requirements in terms of rigor. Only through the work of Weierstrass, Hilbert, Tonelli and Carathéodory a new foundation arose, which strongly influenced the general evolution of mathematics in the following century. L. C. Young understood, in the 30ies, that also the problem of very rough (microstructured) optimal shapes admits a very satisfactory solution, if one widens the mathematical space. He often explained his idea using the following example from the "theory of sailing" (Figure 6). A sailer tries to sail against the wind, but along the flow, in a wide canal. On the one hand, he should pick a direction at a certain optimal angle with respect to the direction of the wind, to optimize the push. On the other hand, he should remain as close as possible to the center of the canal, to make optimal use of the water flow, which is stronger in the middle. If one neglects the time required to change direction, one should choose a path which corresponds both to a zigzag path (because of the wind), and to a straight line (because of the flow). To satisfy both at the same time is of course impossible, and this is the mathematical problem. Inspired by the contemporary rapid developments in Quantum Mechanics, L. C. Young overcame this difficulty introducing the so-called "generalized curves", which at any point in time do not have a definite direction, but can take instead many directions with different probabilities.
Figure 6 The sailer should sail at a given angle with the direction of the wind, at the same time he should remain in the center of the canal, to make optimal use of the flow. The result is a strongly oscillating path.
Figure 7 In a rough energy landscape it is much more difficult to locate the lowest point as in a smooth one.
The example of the sailer can seem idealized, but the situation we described, in which one must change direction very often, is typical for many applications when it comes to optimal control. The theory of Young revolutionized this field, and allowed to apply also here the powerful calculus tools based on Euler-Lagrange equations.
How does one capture the essentials?
Mathematical objects as Young generalized curves (also called Young measures) make it possible to capture the essence of microstructures, without having to resolve all the details. Only this way the problem can be solved reliably. Which properties of the microstructure are relevant, depends on the problem considered. The Young measure is the appropriate quantity in many cases. In other situations one has to consider different objects, some of which have been discovered by mathematicians only in the last decades, and which include e.g. additional information about the spatial orientation of the microstructure. For a series of relevant problems we do not yet know which is the relevant object. The difference between the attempt to understand microstructure though resolution of details and the reduction to the Young measure can be represented as follows (Figure 7). The first attempt corresponds to the situation in which one seeks the lowest point in a very irregular mountain landscape. One has always to take long detours around obstacles and, being situated in a deep valley, it is very difficult to recognize if the ones behind the mountains are even deeper. In the second approach one first averages over the rapid oscillation between highs and lows, and obtains a much smoother and clearer landscape, in which the lowest point can easily be found.
Such averaging or coarse-graining procedures have been intensely studied in theoretical physics and in many applications. It is however very important to consider in any particular case the specific properties of the considered system, e.g. of the shape-memory alloys.