Variational methods may be seen as a new language leading to new formulations and new methods of solution of equations, where the central concept is the physical notion of work. In this language, we do not say that a quantity is null, but we say that its work is null for all the possible values of a connected variable. For instance, we do not say that a displacement is null, but that the associated work is null for any force. However, we do not say that a force is null, but that its work is null for any displacement. In the simplest situation, we do not say that x = 0, but we say that “x is a real number such that its product for any other real number is equal to zero”, e.g. ∈ and xy = 0, ∀y ∈ . Although this modification and the equivalence between both the formulations seem trivial, it implies a deep conceptual change, as it will be seen in the following. Moreover, despite the view expressed by Richard Feynman – from the standpoint of Physics – in [FEY 85], these two formulations may not be equivalent – from the standpoint of Mathematics – in certain situations: it may become necessary to adopt a complex theoretical framework in order to obtain such an equivalence – that will never be complete.

The beginning of the history of variational methods is often brought to the works of the Greek philosopher Aristotle regarding the lever problem. Nowadays, lever analysis is an exercise for students, which automatically applies the physical principles of momentum and force equilibrium in order to explain how a lever works. So, it may be difficult to conceive that such knowledge was built up patiently over more than a millenium and has intrigued mankind for a long time. For more than ten centuries, the reason why a small force was able to move a large load has remained obscure and has been the subject of passionate discussions. For instance, the text Problemata Mechanica (Problems of Mechanics, attributed to Aristotle, but whose real author is uncertain) starts by saying that [ARI 38]:

Among the problems included in this class are included those concerned with the lever. For, it is strange that a great weight can be moved by a small force, and that, too, when a greater weight is involved. For the very same weight, which a man cannot move without a lever, he quickly moves by applying the weight of the lever.

In this text, the author studies the lever by considering fictitious possible motions – e.g. virtual motions – of the system. An explanation is proposed by considering these possible motions and selecting a particular motion among all: the analysis of fictitious movements furnishes the real one. This approach contains the seeds of the modern variational methods: define the virtual motions and find the real motion by analyzing the virtual motions. It is interesting to notice that restrictions are taken into account: it is observed that the extremities of the lever move on a circle – there are natural and unnatural motions: the natural motion of a load placed on the extremity of the lever corresponds to the tangent direction, while, in fact, the load moves on a circle.

A second ancient author is Heron Alexandrius, who formulated a principle of economy (or of minimum violence of Nature on itself). This principle leads to the principle that light rays follow the shortest path between two points. In the Middle Ages, Jordanus Nemorarius studied the motion of bodies on inclined planes and associated it with the statics of levers having arms of unequal lengths and obtained a result of equality of virtual works [RAD 98, SIM 12].

These ancient texts do not consider the notion of work, unknown in ancient times and which needed many centuries to emerge and to be formalized. Tools such as linear functionals, limits and differential calculus were not available at those times, which limited the development and the formalization of variational methods.

The situation changed radically with the invention of differential and integral calculus. Gottfried Wilhelm Leibniz, Jacques Bernoulli, Jean Bernoulli, Leonhard Euler, Pierre Varignon, Pierre Louis de Maupertuis, Charles-Augustin Coulomb, Jean le Rond D’Alembert and others seized these new tools and used them to develop a new theory of mechanics, which culminated with the publication of Joseph-Louis Lagrange’s Analytical Mechanics and the formalization of the Principle of Virtual Works.

Lagrange’s formalism was extended by William Rowan Hamilton, who introduced a unified point of view connecting optics and dynamics. Hamilton’s approach led to new variational formulations in Physics, usually referred to as Hamiltonian Mechanics, which is a highly reducing expression – in fact, the works of Hamilton led to formulations involving many fields, such as, for instance, classical mechanics, quantum mechanics, thermodynamics, electromagnetism, etc. It furnished a unified point of view, which may be considered as not completely exploited at this date. Moreover, significant works by Charles Jacobi, Joseph Liouville, Henri Poincaré, Alexandre Lyapounov, Lev Pontryagin, Andrei Kolmogorov and many other researchers extended Hamilton’s theory, namely to applications in automatics and control.

The development of information technology has popularized numerical methods based on variational formulations such as, for instance, finite elements, finite volumes, fundamental solutions, smooth particle hydrodynamics and so on. A number of contributions led to these developments, namely the works of Boris Galerkin, Paul Dirac, Maurice Fréchet, Norbert Wiener, Sergei Sobolev, Solomon Bochner, Laurent Schwarz among a large set of researchers.

The revolution of variational methods is still under development, since new connections between variational formulations and probabilities have been highlighted by recent developments. The exploration of these relationships opens up new perspectives and gives us a glimpse of exciting prospects for the future.