The idea of writing a book on the principle of least action came to us after many conversations over coffee, while we pondered ways of communicating to students the ideas of mechanics with an historical flavor. We chose the principle of least action because we think that its importance and aesthetic value as a unifying idea in physics is not sufficiently emphasized in regular courses. To the general public, even to those interested in science at a popular level, the beautiful notion that the fundamental laws of physics can be expressed as the minimum (or an extremum) of something often seems foreign. Nature loves extremes. Soap films seek to minimize their surface area, and adopt a spherical shape; a large piece of matter tends to maximize the gravitational attraction between its parts, and as a result the planets are also spherical; light rays refracting in a glass window bend and follow the path of least time; the orbits of the planets are those that minimize something called the “action;” and the path that a relativistic particle chooses to follow between two events in space-time is the one that maximizes the time measured by a clock on the particle.

Our initial intention was to write a popular book, but the project morphed into a more technical presentation. Nevertheless we have tried to keep sophisticated mathematics to a minimum: nothing more than freshman calculus is needed for most of the book, and a good part of the book requires only high school algebra. Some familiarity with differential equations would be useful in certain sections. While the different sections have various levels of difficulty, the book does not need to be read in a linear fashion. It is quite feasible to browse through this book, as most of the chapters and many of the sections are relatively self-contained. Sections and subsections that are a bit more technical and that can easily be omitted on a first reading include 1.1, 2.5, 3.2.2, 3.2.5, 4.3, 5.7, 6.2 to 6.6, 7.7 and 8.8. These are marked in the text with an asterisk.