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The Variational Principles of Mechanics

Cornelius Lanczos
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Michael Berg
, on

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.

1. The variational approach to mechanics
2. The procedure of Euler and Lagrange
3. Hamilton's procedure
4. The calculus of variations
5. Comparison between the vectorial and the variational treatments of mechanics
6. Mathematical evaluation of the variational principles
7. Philosophical evaluation of the variational approach to mechanics
I. The Basic Concepts of Analytical Mechanics
1. The Principal viewpoints of analytical mechanics
2. Generalized coordinates
3. The configuration space
4. Mapping of the space on itself
5. Kinetic energy and Riemannian geometry
6. Holonomic and non-holonomic mechanical systems
7. Work function and generalized force
8. Scleronomic and rheonomic systems. The law of the conservation of energy
II. The Calculus of Variations
1. The general nature of extremum problems
2. The stationary value of a function
3. The second variation
4. Stationary value versus extremum value
5. Auxiliary conditions. The Lagrangian lambda-method
6. Non-holonomic auxiliary conditions
7. The stationary value of a definite integral
8. The fundamental processes of the calculus of variations
9. The commutative properties of the delta-process
10. The stationary value of a definite integral treated by the calculus of variations
11. The Euler-Lagrange differential equations for n degrees of freedom
12. Variation with auxiliary conditions
13. Non-holonomic conditions
14. Isoperimetric conditions
15. The calculus of variations and boundary conditions. The problem of the elastic bar
III. The principle of virtual work
1. The principle of virtual work for reversible displacements
2. The equilibrium of a rigid body
3. Equivalence of two systems of forces
4. Equilibrium problems with auxiliary conditions
5. Physical interpretation of the Lagrangian multiplier method
6. Fourier's inequality
IV. D'Alembert's principle
1. The force of inertia
2. The place of d'Alembert's principle in mechanics
3. The conservation of energy as a consequence of d'Alembert's principle
4. Apparent forces in an accelerated reference system. Einstein's equivalence hypothesis
5. Apparent forces in a rotating reference system
6. Dynamics of a rigid body. The motion of the centre of mass
7. Dynamics of a rigid body. Euler's equations
8. Gauss' principle of least restraint
V. The Lagrangian equations of motion
1. Hamilton's principle
2. The Lagrangian equations of motion and their invariance relative to point transformations
3. The energy theorem as a consequence of Hamilton's prin
4. Kinosthenic or ignorable variables and their elimination
5. The forceless mechanics of Hertz
6. The time as kinosthenic variable; Jacobi's principle; the principle of least action
7. Jacobi's principle and Riemannian geometry
8. Auxiliary conditions; the physical significance of the Lagrangian lambda-factor
9. Non-holonomic auxiliary conditions and polygenic forces
10. Small vibrations about a state of equilibrium
VI. The Canonical Equations of motion
1. Legendre's dual transformation
2. Legendre's transformation applied to the Lagrangian function
3. Transformation of the Lagrangian equations of motion
4. The canonical integral
5. The phase space and the space fluid
6. The energy theorem as a consequence of the canonical equations
7. Liouville's theorem
8. Integral invariants, Helmholtz' circulation theorem
9. The elimination of ignorable variables
10. The parametric form of the canonical equations
VII. Canonical Transformations
1. Coordinate transformations as a method of solving mechanical problems
2. The Lagrangian point transformations
3. Mathieu's and Lie's transformations
4. The general canonical transformation
5. The bilinear differential form
6. The bracket expressions of Lagrange and Poisson
7. Infinitesimal canonical transformations
8. The motion of the phase fluid as a continuous succession of canonical transformations
9. Hamilton's principal function and the motion of the phase fluid
VIII. The Partial differential equation of Hamilton-Jacobi
1. The importance of the generating function for the problem of motion
2. Jacobi's transformation theory
3. Solution of the partial differential equation by separation
4. Delaunay's treatment of separable periodic systems
5. The role of the partial differential equation in the theories of Hamilton and Jacobi
6. Construction of Hamilton's principal function with the help of Jacobi's complete solution
7. Geometrical solution of the partial differential equation. Hamilton's optico-mechanical analogy
8. The significance of Hamilton's partial differential equation in the theory of wave motion
9. The geometrization of dynamics. Non-Riemannian geometrics. The metrical significance of Hamilton's partial differential equation
IX. Relativistic Mechanics
1. Historical Introduction
2. Relativistic kinematics
3. Minkowski's four-dimensional world
4. The Lorentz transformations
5. Mechanics of a particle
6. The Hamiltonian formulation of particle dynamics
7. The potential energy V
8. Relativistic formulation of Newton's scalar theory of gravitation
9. Motion of a charged particle
10. Geodesics of a four-dimensional
11. The planetary orbits in Einstein's gravitational theory
12. The gravitational bending of light rays
13. The gravitational red-shirt of the spectral lines
X. Historical Survey
XI. Mechanics of the Continua
1. The variation of volume integrals
2. Vector-analytic tools
3. Integral theorems
4. The conservation of mass
5. Hydrodynamics of ideal fluids
6. The hydrodynamic equations in Lagrangian formulation
7. Hydrostatics
8. The circulation theorem
9. Euler's form of the hydrodynamic equations
10. The conservation of energy
11. Elasticity. Mathematical tools
12. The strain tensor
13. The stress tensor
14. Small elastic vibrations
15. The Hamiltonization of variational problems
16. Young's modulus. Poisson's ratio
17. Elastic stability
18. Electromagnetism. Mathematical tools
19. The Maxwell equations
20. Noether's principle
21. Transformation of the coordinates
22. The symmetric energy-momentum tensor
23. The ten conservation laws
24. The dynamic law in field theoretical derivation
Appendix I; Appendix II; Bibliography; Index