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All happy families [linear systems] are alike, every unhappy family [nonlinear system] is unhappy [nonlinear] in its own way" L. TolstoiIt has been argued in the Introduction that a good starting point to develop a practically meaningful nonlinear control theory is to specialize the dass of systems under consideration. The main reason being, of course, that the vast array of nonlinear systems renders futile the quest of a monolithic theory applicable for all systems. In particular, it defies the approach of mimicking the, by now fairly complete, linear theory. Specializing the systems, on the other hand, intro duces additional constraints and structure, which may enable otherwise intractable problems to be answered. In this chapter we describe the dass of systems that we will consider throughout the book and which we call Euler-Lagrange (EL) systems. The most important reason for singling out the study of EL systems is that they capture a large dass of contemporary engineering problems, specially some which are intractable with linear control tools. Finally, by restricting ourselves to systems with physical constraints we believe we can contribute to reverse the tide of "find a plant for my controller" which still permeates most of the research on control of general nonlinear systems.What is an EL system? To answer this quest ion we will borrow inspiration from the definition of adaptive control quoted in the seminal book of Aström and Wittenmark [13] (i.e., "An adaptive system is a system that has been designed with an adaptive viewpoint"). Hence we will say: An EL system is a system whose motion is described by the EL equations. The logical question which arises next is "Wh at are the EL equations?" From a purely mathematical viewpoint they are a set of nonlinear ordinary differential equations with a certain specific structure. A far more interesting quest ion is "Where do they come from?" In contrast to the first two questions, the answer to the latter is far from simple and involves principles of minimization, calculus of variations and other tools from analytical dynamics. For the purposes of this book the EL equations are important because they are the outcome of a powerful modeling 15 R. Ortega et al., Passivity-based Control of Euler-Lagrange Systems

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