Physics, specifically, mechanics, is rife with what are ultimately themes from the calculus of variations, trained on either Lagrangian or Hamiltonian formalisms (or both, in a way). Roughly speaking, a Hamiltonian encodes the total energy of a system of particles (whatever that might mean), i.e. the sum of potential and kinetic energies, whereas a Lagrangian sets up a difference between these two types of energy. In the book under review, Lanczos puts it this way:
L, defined as the excess of kinetic energy over potential energy, is the most fundamental quantity in the mathematical analysis of mechanical problems. It is frequently referred to as the ‘Lagrangian function’… (p. 113)
and then, ten pages later (p. 123) we read that
the ‘total energy’ of the mechanical system … is, together with the Lagrangian … L, the most important scalar associated with the mechanical system. In fact, … in the Hamiltonian form of dynamics this function … takes precedence over the Lagrangian … and, expressed in the proper variables, becomes the ‘Hamiltonian function’ H of the mechanical system which completely replaces the original Lagrangian function L.
These selections, in fact, hint at a number of fundamental features of this business of classical mechanics, viz. the importance of assigning proper variables to the players in the game, and then to exploit various fundamental physical principles using the calculus of variations. The aforementioned players include positions and momenta, and the physical principles include, for instance conservation laws. A paradigm for all this is the principle of Hamilton (again, p. 113):
the motion of [a] … system occurs in such a way that the definite integral [of L over time] becomes stationary for arbitrary possible variations of the configurations of the system, provided the initial and final configurations are prescribed.
Thus we immediately discern the historical roots of the whole discipline in the work of Euler and Lagrange, and Lanczos adds the historical note that William Rowan Hamilton “gave an improved mathematical formulation of [what Euler and Lagrange had done] … [and] the name ‘Hamilton’s principle’ [was] coined by Jacobi …” Indeed this is classical mechanics par excellence.
Parenthetically, although Lanczos’s book only touches upon quantum mechanics obliquely (cf. e.g., p. 253), and does so only in the context of Delauney’s method (coming from astronomy) applied to the Bohr model of the hydrogen atom, it is of great historical, philosophical, physical, and ultimately mathematical significance to note how classical mechanics gives rise to quantum mechanics. This is all in accord with the seminal work of Heisenberg, Born, Schrödinger, Dirac, and (conducting the orchestra, so to speak) Bohr, i.e. the evolution of the Copenhagen Interpretation of quantum mechanics, with canonical quantization at the heart of it. The tectonic shift in perspective is to replace the classical mainstays of position and momentum by operators on a Hilbert space, specifically the space of states of the given quantum mechanical system (as opposed to a classical, deterministic mechanical system), and to impart a Fourier-analytic constraint on these operators’ interaction: nothing less than the Uncertainty Principle of Heisenberg. In this formalism, measurables, or observables, are such courtesy of the solvability of an eigenvalue problem associated with a PDE, the wave equation of Schrödinger, which (if you’ll pardon the pun) integrally involves the Hamiltonian (observable) of the system. So, vis à vis classical mechanics, the names are the same, but the players have changed: now they’re operators on a Hilbert space, and the numbers, i.e. measurable values, that we can compare to experimental data see the light of day through the services of, ultimately, very sexy functional analysis. Of course, the prominence of the latter as an organic quasi-axiomatized system for quantum mechanics is really mostly due to one of Hilbert’s most effective assistants at Göttingen during Hilbert’s physics period, the ever-so-redoubtable John von Neumann.
There is a good case to be made that for mathematicians his Mathematical Foundations of Quantum Mechanics is the most satisfying place to learn about quantum mechanics, its age notwithstanding. (And if I may be allowed a plug for another truly fabulous book, where functional analysis softens the blows of what physicists are apt to do, Prugovečki’s Quantum Mechanics in Hilbert Space is absolutely irresistible.)
Well, back to classical mechanics and Lanczos. It’s an old book, originally appearing in 1949; the present (4th) edition goes back to 1970. The later editions of the book differ from the earlier ones by the inclusion of a discussion of Emmy Noether’s “physics theorem” (as opposed to her first isomorphism theorem in algebra) which Lanczos describes in his Third Edition Preface as “not easy reading” in the setting of her original paper; in his book he hits the theorem (cf. p. 401) in the context of “added variables of the variational problem, for which the Euler-Lagrange equations can be found.” And the upshot, then, is the beautiful correspondence between conservation laws and symmetries which is credited to Noether, evidently proven at the behest of Hilbert. Yes, this subject’s pedigree is first-rate.
Beyond the Prefaces, then, we encounter a welcome expansive Introduction, rich in historical analysis and with the calculus of variations featured — it cannot be otherwise, of course. Then it’s eleven beefy chapters, followed by two appendices. It starts with analytical mechanics and the calculus of variations, after which we get “virtual work” (about which one can easily construct any number of cheap puns, but let’s let it go…), and, following that, d’Alembert’s Principle and the Lagrangian and canonical equations of motion. Then, after a discussion of canonical transformations, we get to PDEs of Hamilton-Jacobi type, followed by relativistic mechanics (Einstein’s gravitational theory appears on p. 330), a historical survey, and, finally, a closing chapter on the mechanics of the continua: hydrodynamics, elasticity, electromagnetism (with Maxwell’s equations appearing), Noether’s Principle coming in a prelude to its dissection in the later Appendix, and a section titled, “The ten conservation laws.”
This tantalizingly named section includes a discussion of what is still startling after all these decades: “Newton speaks of the mass of the body instead of its energy. The replacement of the word ‘mass’ by ‘energy’ is in complete harmony with Einstein’s fundamental discovery … that mass and energy are identical (in a time scale c = 1 …). It was Planck … who pointed out that the field theoretical interpretation of Einstein’s principle can only be the symmetry of the energy momentum tensor …” And this symmetry is also needed for Newton’s physics: “Nor could a non-symmetric energy-momentum tensor guarantee that the law of inertia, according to which the centre of mass of an isolated system moves in a straight line with constant velocity.” Stunning. One is reminded of one of Dirac’s aphorisms: “God used beautiful mathematics in creating the world.” Lanczos’s fine book is really a paean to this beautiful mathematics; it is also a wonderful experience to encounter a treatment of this material in a style that is now all but absent, evincing historical scholarship and riddled with discussions of roots and connections, with all of it presented both leisurely and thoroughly.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.