In his first-year undergraduate lecture on the Principle of Least Action [Feynman Lectures on Physics (1964), Vol. II, Chap.19], Richard Feynman says that after his high school teacher told him about this principle, he found it “…absolutely fascinating, and have, since then, always found fascinating. Every time the subject comes up, I work on it. In fact when I began to prepare this lecture I found myself making more analyses on the thing.…” Feynman wrote his Ph.D. thesis on the subject [The Principle of Least Action in Quantum Mechanics, republished as a book in 2005, with L. M. Brown as editor], and the last conference he attended, shortly before his death, was on action/variational principles [Difficulties in Applying the Variational Principle to Quantum Field Theories, in Variational Calculations in Quantum Field Theory, edited by L. P. Polley and D. Pottinger, 1988, p. 28]. Many others have been so captivated, including Coopersmith and me. In fact, in preparing this review, I found myself “making more analyses on the thing.” Jennifer Coopersmith has written a most welcome book, the first historically and philosophically motivated full study since two classics written nearly a half-century ago (both still available in inexpensive Dover editions): C. Lanczos, The Variational Principles of Mechanics, 4th ed. (1970), and W. Yourgrau and S. Mandelstam, Variational Principles in Dynamics and Quantum Theory, 3rd ed. (1968). This follows her earlier acclaimed book Energy, the Subtle Concept, revised edition, 2015, and is written in the same style, a pleasing combination of qualitative and quantitative arguments.

Action principles are usually taught at the senior undergraduate and graduate levels, but some scholars are making the case for their introduction earlier in the curriculum [see, e.g., Edwin Taylor, A Call to Action, AJP guest editorial 71(4), 2003]. The attractions of action principles include the following: (1) they are brief and elegant in expressing the laws of motion, (2) they are covariant (valid for any choice of coordinates), (3) they readily yield conservation laws from the symmetries of the system (Nöther's theorem), (4) they automatically generate covariant equations of motion (Euler-Lagrange equations), and (5) they can also generate true trajectories directly, bypassing equations of motion, via the direct or Rayleigh–Ritz method of the calculus of variations. Action principles transcend classical particle and rigid body dynamics and extend naturally to other branches of physics such as relativistic mechanics, continuum mechanics (e.g., fluid mechanics and elasticity theory), quantum mechanics, and field theory (classical and quantum) and thus play a unifying role. They have occasionally assisted in developing new laws of physics, with general relativity being perhaps the best example. Each of these points can and has been greatly elaborated. Briefly, regarding point (1), for example, the role of the importance and usefulness of elegance in physics and in mathematics, engineering and other sciences, in both theory and experiment, has a huge literature; a recent discussion in this journal is given by Stanley Deser [Truth, Beauty and Supergravity, AJP guest editorial 85(11), 2017]. Many scientists endorse elegance, but there are some distinguished dissenters, e.g., Boltzmann and the applied mathematician, historian of science, and iconoclast, Clifford Truesdell. But even Truesdell reserves his negative opinion on the elegance of variational principles to the application of such principles to continuum mechanics and states explicitly that the variational principles in particle mechanics are beautiful and elegant [C. Truesdell and R. Toupin, The Classical Field Theories, 1960, p. 595]. Regarding point (5), the direct method has been widely applied as an approximation technique, in competition with and complementary to perturbation theory, in classical continuum mechanics and field theory, and in quantum particle mechanics, but not in classical particle mechanics and quantum field theory.

Here is a brief summary of the action principles. There are two major versions of the action, due to Maupertuis (1744) and Hamilton (1834), and two corresponding action principles. The Hamilton principle is nowadays more used. The Hamilton action S for any conceivable (real or imagined) trajectory AB, connecting two given points qA and qB in a given time T, is defined by the integral S=0TL(q,q̇,t)dt along the space-time trajectory q(t) connecting the specified space-time events, initial event A = (qA, tA = 0) and final event B = (qB, tB = T), where the Lagrangian L depends on coordinate q, its rate of change q̇, and in general the time t. For a system with f degrees of freedom, q will stand for the complete set of generalized coordinates q1,q2,…,qf. In the simplest cases, L(q, q̇) has no explicit time-dependence, it is equal to the difference in the kinetic and potential energies KV, the kinetic energy K depends quadratically on q̇, and the potential energy V depends only on q (conservative system). Hamilton's principle states that among all conceivable trajectories q(t) connecting qA to qB in time T, a true one is one for which S is stationary, i.e., S is unchanged to first order when the trajectory is varied a small amount δq(t) from the true one, with the true one and the varied one each satisfying the prescribed constraints (q(t) = qA at t = 0 and q(t) = qB at t = T). There may be more than one true trajectory, or no true trajectory, satisfying these constraints (boundary conditions). A briefer statement is that the extremals of the functional S[q(t)], defined by the first order variation condition δS = 0, are the true trajectories. The terminology “least” action is historical. S is stationary for true trajectories; whether it is actually a minimum for true trajectories, or a saddle point, depends on the sign of the second-order variation, usually not discussed in the textbooks [see Am. J. Phys. 75, 434 (2007)]. The second major version of the action is Maupertuis' action W which can be defined for a conceivable trajectory by the spatial integral W=qAqBpdq, with p being the canonical momentum (which equal to the ordinary momentum is the simplest cases). For f degrees of freedom, p dq stands for p1dq1 + p2dq2 +⋯+ pfdqf. For normal systems (systems with K quadratic in q̇), an equivalent time-integral definition is W=0T2Kdt. Maupertuis' principle states that with prescribed end points qA and qB and prescribed trajectory energy E, W is stationary (δW = 0) for true trajectories. Unlike the Hamilton principle, the Maupertuis principle is restricted to conservative systems but has been generalized to apply to nonconservative systems in recent years. It is very important to note the different constraints which apply to the two principles. With Hamilton, two points and the time are fixed; with Maupertuis, two points and the energy are fixed. Thus, with Hamilton, T is fixed and E is not, with the reverse holding for Maupertuis. For conservative systems, the two principles can be shown to be related by a Legendre transformation (see the last paragraph); for nonconservative systems, the generalized Maupertuis principle is again related to the Hamilton principle by a Legendre transformation. The solution of the Hamilton variational problem δS = 0 yields true space-time trajectories. The solution of the Maupertuis variational problem δW = 0 yields true space-time trajectories if one uses the time-dependent form for W and yields true orbits (spatial paths) if one uses the time-independent form for W.

The usual procedure is to use the action principles to derive Euler-Lagrange equations of motion for the true trajectories and then to solve these equations for the true trajectories. But as noted in point (5) in the second paragraph, we can obtain the true trajectories directly from the action principles without the equations of motion. There has always been speculation about whether something deeper underlies the action principles, with questions like “why does nature act in such an economical way?” Some (including the author of The Lazy Universe, p. 194) see the justification of Hamilton's principle in the Feynman path integral of quantum mechanics, which implies Hamilton's principle in the classical limit. Others feel any justification of classical action principles should be done within the framework of classical mechanics [T. Toffoli, “What is the Lagrangian Counting?” Int. J. Theor. Phys. 42, 363 (2003)]. Many others do not think that the question is relevant and no underlying a priori physical principle exists: the action is not always a minimum, and besides, it is just mathematics since any equation can be reformulated as a variational statement [E. Gerjuoy, A. Rau, and L. Spruch, “A unified formulation of the construction of variational principles,” Rev. Mod. Phys. 55, 725 (1983)].

Jennifer Coopersmith has attempted and succeeded admirably I believe in her aim to write a modern book on the history and philosophy of the action principles, as well as to give the technical details. She accomplishes this difficult task by focusing the main text (nine chapters) on the history, philosophy, and intuitive notions involved, while relegating many of the mathematical details to short appendices (twelve in all). Her intended audience is the general reader. She states explicitly that her book is not a textbook (there are no problem sets), as it is meant to be a more qualitative and discursive treatment, trying to bring the equations to life with a lot of motivating discussion of their meanings. Still I think that students can profit by reading this book in parallel with a textbook, as the technical matters they need are indeed covered, many clearly and succinctly, in the appendices. Getting a qualitative feel for a subject is greatly assisted by seeing different viewpoints. She aims for a “shorter and simplified” version of Lanczos' classic book, mentioned in the first paragraph. An aside: Lanczos was an outstanding scientist. Some of his early work was in general relativity; he spent a year in Berlin as Einstein's assistant, he published a paper on an equation for quantum energy eigenfunctions just before Schrödinger, based on Heisenberg's recently published paper on matrix mechanics, but it had the form of an integral equation with the kernel not determined and so he did not quite discover wave mechanics, and he later published some very useful books on applied mathematics and some interesting popular books, as well as his pedagogical masterpiece on action principles. There is a very nice short biography by Barbara Gellai [The Intrinsic Nature of Things: the Life and Science of Cornelius Lanczos (AMS, 2010)].

In the early chapters, The Lazy Universe discusses the precursors (e.g., Fermat's principle in optics) and early history well, including anecdotes about Maupertuis, his multiple talents, idiosyncrasies (e.g., teleological views), and the priority dispute over the origin of the Maupertuis principle. The work of the greats Euler, Lagrange, Hamilton, and Jacobi is well covered. Jacobi devised a third form for the Maupertuis action; in multiple dimensions, Jacobi's form involves an integral along the arc length in q-space so that the action principle then resembles a geodesic problem of finding the shortest path between two points. The calculus of variations background needed to carry out variations of the action integrals is nicely summarized. Hamilton's principle is developed from the dynamical version (D'Alembert) of the principle of virtual work, and the exposition of the latter principle is thorough and meticulous. Maupertuis' principle is not derived. Lagrangian and Hamiltonian mechanics each get a chapter. The Nöther theorem relating conservation laws to symmetries of the action is discussed well for the special case of energy conservation arising from time translational invariance (pp. 129, 222). She gives Jacobi's simple intuitive argument (p. 127) why the action, although always stationary for true trajectories, can never be a maximum. The argument is that if someone claims to have a trajectory with maximum action, I will modify that trajectory by adding wiggles to it at some place along the trajectory, of high frequency but small amplitude, so that K increases a lot while V is not much changed. Thus, L = KV increases in the region of the wiggles and so does its integral S. (We also need to ensure that the overall travel time T does not change.) Hence, the original trajectory cannot be one of maximum S. Now, that is an elegant argument! And so is this book as a whole.

My criticisms are few and mostly minor.

  • Misprint (p. 24). Maupertuis made his discovery in 1744, not 1644.

  • Notation. She uses the modern notations used above for the actions, S (Hamilton) and W (Maupertuis), on. p. 250, but mostly (e.g., pp. 223, 241) she uses Lanczos's notation A for the Hamilton action and S for the Maupertuis action. This may cause confusion for modern readers.

  • Why kinetic energy K is quadratic in the velocity (pp. 119, 217). I think that she missed the essential point of the Landau-Lifshitz argument, that the assumption of Galilean invariance yields this result.

  • Frictional forces and nonholonomic constraints. These are not handled easily by the standard action principles. But, she overlooks (p. 184) modern work which can include friction directly in the Lagrangian for some systems, including all one-dimensional ones, and thereby can avoid the awkward Rayleigh dissipation function method [see, e.g., V. Chandrasekhar et al., “On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator,” J. Math. Phys. 48, 032701 (2007)]. She briefly mentions that systems with nonholonomic constraints (which depend on velocity, not position, e.g., a disc rolling without slipping on a horizontal plane) do not obey the standard action principles, but it would have been instructive to mention that many researchers, even some distinguished ones, have stumbled over this point.

  • Confusing statements. On p. 126, she states that “L has exactly the form KV and no other,” but on the very next page, she agrees that L need not have the form KV. On pages 187 and 192, she states that the initial and final events A and B used to define the action principles “must be close together,” but on p. 193, she states (correctly) that they can be “a finite ‘distance’ apart.” On p. 185, she gives a relativistic Lagrangian for a free particle. But she forgets her own admonition (p. 127) that one can multiply L by any constant C with no change in the physics. If she chooses C = −1 in this case (as does Lanczos on. p. 324 of his book), her equating the relativistic principle of maximal aging with the principle of least action breaks down; the principle of maximal aging will now be equivalent to a principle of greatest action. Both equivalences are correct, as are both action principle statements; the action allows a great deal of arbitrariness in its definition.

  • Stationary or minimum action. On p. 112 (also see p. 4), she states that in section 6.6, we will see that the action for true trajectories is a minimum, not just stationary. But in Sec. 6.6 (p. 126), she (correctly) states the opposite that the action need not be a minimum: it can be a minimum or a saddle point. (She does not delve into when one gets one or the other but refers readers to the relevant literature.)

  • Einstein's 1917 paper. She has a lovely appendix (p. 256) on Einstein's somewhat under-appreciated paper. Einstein considers the most general quasi-periodic systems with f degrees of freedom, systems which are not “separable” like the hydrogen atom, which has separable radial and angular actions. Using action-angle variables, one can define the f “good actions” J1,…,Jf, the ones for which the Hamiltonian H depends only on the actions and not the angles so that (from Hamilton's equations of motion) they are constants of the motion. Ji are special values of the Maupertius W action. In modern language, the actions define “invariant tori” surfaces in phase space (q, p), which confine the motion of the representative point to these subspaces of the constant energy surface in phase space. If one then quantizes these actions by setting J1 = n1h, J2 = n2h, etc., where n's are integers and h is Planck's constant, one gets an energy quantization using the relation E = E(J1,…,Jf). The n's become the “good quantum numbers.” This is the basis of what is nowadays called the Einstein–Brillouin–Keller or EBK semiclassical quantization scheme for quasiperiodic systems (not pointed out in the book), which generalizes the Bohr-Sommerfeld-Wilson or BSW scheme for separable systems. But there is a second gem in Einstein's paper, not mentioned by Coopersmith. Einstein realized that this scheme would fail for chaotic or ergodic systems where f good actions do not exist, and for an ergodic system energy E is the only constant of the motion and so the representative point in phase space wanders pseudo-randomly over the whole energy surface defined by H(q,p) = E; he raised the question of how one would quantize such systems. The problem was forgotten with the advent of matrix and wave mechanics in 1925–1926 but was resurrected in the 1970s by Martin Gutzwiller and others. Gutzwiller showed that for such systems, one can relate the complete set of quantum energy levels to the complete set of classical periodic orbits. The main result, Gutzwiller's semiclassical trace formula, is derived using path integrals [M. Gutzwiller, Chaos in Classical and Quantum Mechanics, 1990]. I think that it would be interesting to readers of The Lazy Universe to learn about this solution to the problem raised by Einstein.

The Lazy Universe is a fine book, and in my opinion, its author has succeeded in achieving her stated goals. My hope is that in future editions (Lanczos needed four to perfect his story), she will bring her considerable pedagogical skills to bear in discussing additional topics such as (1) The Legendre transform relation between S and W. She states (p. 250) the relation S = WET, which holds for the actions along true trajectories but does not give the general relation S = WE¯T valid for arbitrary (conceivable) trajectories, where E¯ is the mean energy along the trajectory between the endpoints. From this general relation, one can derive the Maupertuis principle from the Hamilton principle and vice-versa. (2) The action principles that have been developed in recent years, including the extension of the Maupertuis principle to nonconservative systems, reciprocal Hamilton and reciprocal Maupertuis principles, and principles which allow end-point variations and relaxation of other constraints. (3) The relations between the classical action principles and the quantum variational principles. One can in fact derive the reciprocal Maupertius principle from the quantum variational principle in the classical limit, and the reciprocal Maupertuis principle provides the basis of a simple semiclassical quantization scheme. (4) The direct variational method. One can employ the Maupertuis and Hamilton principles, and the newer action principles mentioned in point (2), to obtain trajectories directly from an action principle without using equations of motion. This method can be used to get exact analytical expressions for trajectories only for simple systems (just as with the equations of motion) but provides approximate analytical expressions when used as an approximation method (similar to the Rayleigh-Ritz method widely used in quantum mechanics), and if used as a numerical method, it can produce essentially exact trajectories. (5) Continuum mechanics and field theory. Inclusion of these topics might extend the book too much, but a sketch, an example, and some good references could be appropriate. A bit more on the relativistic principles might also be worth considering.

As one expects from OUP, the book is beautifully produced and is reasonably priced.

Chris Gray is a University Professor Emeritus at the University of Guelph, Ontario, Canada, and is a theoretical physicist. He has worked in molecular physics, the statistical mechanical theory of liquids, and most recently computational biophysics/simulations. He is a coauthor of two books on the theory of molecular liquids (1984 and 2011) and has published extensively on classical, semiclassical, and quantum variational principles, including several papers in AJP on this topic, and an elementary review in Scholarpedia which includes discussion of some modern developments.