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The Calculus of Variations

The physical basis for the calculation of energy states in the various liquid crystal theories is that a physical system will always tend to its lowest potential energy state; this is manifested by changes to the system’s free energy, which will seek a minimum value. The states of the minimum free energy are calculated by the methods of the variational calculus, an extremely useful mathematical discipline, introduced below with some of its rather interesting history.

Maxima and minima (together extrema) calculations are commonly used in the natural and the engineering sciences. For example, in using Newton’s fluid dynamics to determine the very practical problem of the best shape of a ship’s hull to offer the lowest resistance to the water, it turns out unsurprisingly that it is just the minimum surface between two points, which result is easy to accept, if not so easy to prove. The extremum problem was also encountered in everyday life, as for instance in feudal Europe, land was ceded from father to sons according to how much land each son could mark off in one day given ropes of equal length, the objective of course being to encompass the maximum area possible.

Historically, this type of maximization has roots going back three thousand years (900 B.C.) to the era of the Phoenicians and their Princess Dido. In escaping from her tyrannical brother, the Princess sought asylum in what is today Tunisia on the northwestern shore of Africa. The king there ...