Talk:Principle of least action

Reviewer A

This is a well-written article on the Principle of Least Action by one of the leaders in this field. Here are some comments about the material presented in some of the eleven sections.

In Section 1, it should perhaps be pointed out that, like Fermat's Principle of "Least" Time, Maupertuis' Principle of Least Action is a "geodesic" principle (since it involves the infinitesimal length element ds) while Hamilton's Principle of Least Action is a "dynamic" principle (since it involves the infinitesimal time dt). Lastly, in Eq.(5), it might be more appropriate to place indices on the q variable.

In Section 4, it should be pointed out that the second variation \(\delta^{2}S\) can be expressed in terms of the variation \(\delta x\) and the Jacobian deviation \(u\) (at fixed time t) as \(\delta^{2}S = \int_{0}^{T} \frac{\partial^{2}L}{\partial \dot{x}^{2}} \left( \delta\dot{x} \;-\; \delta x\;\frac{\dot{u}}{u} \right)^{2} \geq 0,\) which vanishes only when \(\delta\dot{x}\;u = \delta x\;\dot{u}\). The latter expression defines the kinetic focus. An explicit reference (such as C. Fox, An Introduction to the Calculus of Variations, Dover, 1987) would be appropriate in addition to Gray and Taylor's.

In Section 7, the exact solution of the quartic-potential problem is given in terms of the Jacobi elliptic function \({\rm cn}(z|m)\) as \(x(t) \;=\; (4 E/C)^{1/4}\;{\rm cn}\left( \frac{4K}{T}\;t \;\left|\;\frac{1}{2}\right. \right),\) where \(K = K(1/2) = 1.85407...\) is the complete elliptic integral of the first kind evaluated at \(m = 1/2\) (the lemniscatic case) and the period is \(T = 4 K (m^{2}/4 EC)^{1/4}.\) When we compare the exact angular frequency \(\omega = 2\pi/T\) with Eq.(15), we indeed find that \(\omega/[{\rm Eq.(15)}] = \pi/(2^{3/4}\,K) = 1.0075.\)

In Section 8, while the choice of sign for the relativistic Lagrangian (or action) given in Eq.(16) might appear to be a matter of choice, the incorrect sign chosen in Eq.(16) is incompatible with the conservation laws derived from it by Noether method.

In Section 9, it might be useful to mention the wonderfully elegant derivation of the Schroedinger equation by Feynman and Hibbs in Chapter 4 of their textbook "Quantum Mechanics and Path Integrals" (McGraw-Hill, 1965). The reader should also be refered to Yourgrau and Mandelstam for additional historical comments.

Reviewer B

Since my expertise is restricted to classical (i.e. non-relativistic and non-quantum) mechanics, my review refers only to the classical part of the article. This is one of the best articles of its kind, i.e. among those found in electronic and traditional encyclopaedias (Wiki, Britannica etc.). It's brief, authoritative, with unusual detail (e.g. study of second variation), and highly readable, i.e. no uneccessary formalisms ("epsilonics"). Its author has published several original papers on this subject. If I could make one suggestion for improvement, that would be to use small Greek delta for isochronous (vertical) variations, i.e. time kept fixed, and UPPER case Greek delta for non-isochronous (non-vertical, or skew) ones; see e.g. E. Whittaker's "Analytical Dynamics" (1937, pp.246 ff.) Therefore I do recommend its publication.