Book suggestions on Lagrangian and Hamiltonian mechanics?

Question

What are some books that talk about the origins of Lagrangian and Hamiltonian mechanics? I would like to dig deeper into the fundamental approach and the fundamentals of action and its principles.

Answer 1

In my opinion the existing books are not good sources of information.

In particular: what is presented as 'historical information' is often incorrect.

I believe only the following two categories of information source are reliable:
-the original works; the primary sources
-articles/books by authors who are dedicated historians of science

David R Wilkins has created many transcripts of Hamilton's articles, including the ones on mechanics.

It would appear that Wilkins regards the articles that were published in the Proceedings of the Royal Society as the core of Hamilton's work on dynamics.

The page titled On a general method in dynamics offers links links to those articles.

However, while the articles published in the Proceedings of the Royal Society do not contain any interpretation, there is another article that does:

On a General Method of expressing the Paths of Light and of the Planets by the Coefficients of a Characteristic Function

This was published in: Dublin University Review and Quarterly Magazine

Presumably that magazine had only a small circulation.

Quoting from the above paper:

the quantity pretended to be economised is in fact often lavishly expended. In optics, [...] in reflexion at a curved mirror this economy is often violated. [...]

In mathematical language, the integral called action, instead of being always a minimum, is often a maximum;

Larger quote, to provide the context of the above snippets.

But although the law of least action has thus attained a rank among the highest theorems of physics, yet its pretensions to a cosmological necessity, on the ground of economy in the universe, are now generally rejected. And the rejection appears just, for this, among other reasons, that the quantity pretended to be economised is in fact often lavishly expended. In optics, for example, though the sum of the incident and reflected portions of the path of light, in a single ordinary reflexion at a plane, is always the shortest of any, yet in reflexion at a curved mirror this economy is often violated. If an eye be placed in the interior but not at the centre of a reflecting hollow sphere, it may see itself reflected in two opposite points, of which one indeed is the nearest to it, but the other on the contrary is the furthest; so that of the two different paths of light, corresponding to these two opposite points, the one indeed is the shortest, but the other is the longest of any. In mathematical language, the integral called action, instead of being always a minimum, is often a maximum; and often it is neither the one nor the other: though it has always a certain stationary property, of a kind which has been already alluded to, and which will soon be more fully explained. We cannot, therefore, suppose the economy of this quantity to have been designed in the divine idea of the universe: though a simplicity of some high kind may be believed to be included in that idea. And though we may retain the name of action to denote the stationary integral to which it has become appropriated--which we may do without adopting either the metaphysical or (in optics) the physical opinions that first suggested the name--yet we ought not (I think) to retain the epithet least: but rather to adopt the alteration proposed above, and to speak, in mechanics and in optics, of the Law of Stationary Action.

About Joseph Louis Lagrange:

An english translation of 'Mecanique Analytique' is available on archive.org.

The first part covers Statics
The second part covers Dynamics

Section 1 of the Second Part (Dynamics) is titled 'the various principles of Dynamics'

Lagrange gives an overview of the knowledge of that time, including historical information. (Some of Lagrange's historical information may be incorrect; I have not examined that.)

On page 183 Lagrange arrives at the action concept that was available at the time: Maupertuis' action

Lagrange's opinion was that Maupertuis' action is not particularly relevant.

I finally arrive at the fourth principle which I call Least Action in analogy to the one to which Maupertuis has applied this designation and which the works of several illustrious authors have subsequently made so famous. This principle, viewed analytically, consists of the following: in the motion of bodies which act on one another, the sum of the products of the masses with the velocities and the spaces traversed is a minimum. The author deduced from it the laws of reflection and refraction of light as well as the laws governing the percussion of bodies in two memoirs read to the Academie des Sciences of Paris in 1744 and two years later at the Academie de Berlin, respectively.

However, these applications are too restrictive to be used to establish the truth of a general principle. They have also something vague and arbitrary about them which can only make the consequence which one could draw for the accuracy of the principle itself uncertain. Thus it would be wrong, it seems to me, to put this principle as it is presented on the same level with the ones we just discussed. However, there is another way to view it, more general and more rigorous and which alone deserves the attention of geometers. Euler has given the first interpretation at the end of his treatise on isoperimetric problems published at Lausanne in 1744, 17 by showing that for the trajectories described by bodies following central forces the integral of the product of the velocity by the element of the curve is always a maximum or a minimum.

The property which Euler found in the motion of isolated bodies and which seems to be confined to these bodies, I later expanded using the Conservation des Forces Vives, to the motion of any system of bodies acting upon one another in an arbitrary fashion. A new general principle derives from this effort, to wit, that the sum of the products of the masses with the integrals of the velocities multiplied by the element of the spaces traversed is always a maximum or a minimum.

This is the principle to which I improperly gave the name of Least Action and which I view not as a metaphysical principle but as a simple and general result of the laws of mechanics. [...]

About William Rowan Hamilton and Carl Gustav Jacob Jacobi

In his 1837 series of lectures 'Vorlesungen über Dynamik' Jacobi devotes a lot of attention to Hamilton's work.

(As far as I know there is no english translation of that series of lectures.)

Available for download (pdf file): the book Vorlesungen über Dynamik

First Lecture, page 3

Hamilton hat den schönen Zusammenhang, den er gefunden hat, etwas unzugänglich gemacht und verdunkelt, und zwar dadurch, dass er seine charakteristische Function noch zugleich von einer zweiten partiellen Differentialgleichung abhängen lässt. Die Hinzufügung dieser Bestimmung macht die ganze Entdeckung unnöthig complicirt, da eine genauere Untersuchung zeigt, dass die zweite partielleDifferentialgleichung vollkommen überflüssig ist.

[My translation]
Hamilton has made the beautiful interconnection he found somewhat inaccessible and obscure because he has made his characteristic function dependent on a second partial differential equation. This addition makes the whole discovery unnecessarily complicated because a closer examination shows that the second partial differential equation is completely superfluous.

19th lecture, page 153

Hamilton [...] steht zugleich noch eine zweite partielle Differentialgleichung auf, welcher V ebenfalls genügen soll. Diese kann man aber fortlassen, weil sie sich aus der schon aufgestellten herleiten lässt [...]

[My translation]
Hamilton sets up a second partial differential equation that V must satisfy. This can however be omitted because it can be derived from the already stated equation.

[Jacobi proceeds to present that derivation]

My experience has been that authors of physics books aren't particularly interested in understanding the stationary action concept. Authors seem to be content with being vague and superficial.

Really, the most probing thoughts I have encountered were those in the quote from William Rowan Hamilton. It's not about minimum or maximum; both can occur. Therefore the name 'stationary action' is intrinsically superior.

But the above quote is all; it would appear that after 1835 (Hamilton died in 1865) Hamilton did not publish about stationary action again.

It appears there is no record of Hamilton responding to Jacobi.

On the attitude of the physics community as a whole:

It appears that in the physics community there is a preference for the name 'least action'. I think that is because people are eager to believe that "something" is minimized. As pointed out by Hamilton, that belief system is a dead end.

On my stackexchange profile page there is a link to a page on my own website, with discussion of Hamilton's stationary action. (Discussion of the fundamentals.)