# What did Feynman mean when he said "Action" historically had a different meaning?

In his lecture on the principle of least action, Feynman says

Also, I should say that S is not really called the ‘action’ by the most precise and pedantic people. It is called ‘Hamilton’s first principal function.’ Now I hate to give a lecture on ‘the-principle-of-least-Hamilton’s-first-principal-function.’ So I call it ‘the action.’ Also, more and more people are calling it the action. You see, historically something else which is not quite as useful was called the action, but I think it’s more sensible to change to a newer definition. So now you too will call the new function the action, and pretty soon everybody will call it by that simple name.

What is it that he is referring to when he says "historically something else which is not quite as useful was called the action"? Also, does anyone know where the term "action" in its modern sense came from, when it started being used, and why its denoted S?

• The original "least action principle" was proposed by Maupertuis in 1745 and differed from Hamilton's later version. Wikipedia explains the differences between them. In particular, Hamilton integrates the Lagrangian over time, varied between two fixed end times and endpoints, while Maupertuis integrates $p\,dq$ over the generalized coordinates, varied along all constant energy paths with fixed endpoints. Commented Aug 23 at 11:13

The earlier action concept is attributed to Maupertuis.

Pierre Louis Maupertuis attempted to formulate a unification of light propagation theory and mechanics theory.

In Maupertuis' time the interpretation of index of refraction was not yet settled. Index of refraction is about a ratio, and that allows an interpretation where light propagating through a medium such as glass goes faster than in air (or interplanetary space).

(Huygens' principle offers the view that propagation of light is a wave phenomenon. But that raises the question of how light traverses interplanetary space. The supposition that light is corpuscular doesn't have that problem; if light consists of particles then it can traverse interplanetary space without need for a medium.)

During Maupertuis' time the interpretation of the index of refraction was an open question.

Maupertuis noticed a way to obtain Snell's law with light that is faster in a denser medium. Maupertuis' action is summed over the entire trajectory. For each section Maupertuis' action is the product of the velocity with the length of that section.

As we know, it was later proved that in a denser medium light propagates slower. So that part of Maupertuis' idea didn't pan out.

On the subject of the Maupertuis action: on Wikisource the following three resources are available:

Maupertuis 1744:
Translation:
Accord between different laws of Nature that seemed incompatible

Maupertuis 1746:
Translation:
Derivation of the laws of motion and equilibrium from a metaphysical principle

Leonhard Euler 1744
Translation of the second appendix to the book in which Euler introduced Calculus of Variations:

These translations were contributed by the same person, who on Wikisource uses the nick 'WillowW'.

Important to note: in particular in the case of Euler: in the translation the notation of the mathematics is modified to make it more accessible to a modern audience.

For comparison: the translation by Ian Bruce reproduces the original notation of the mathematics:
Euler 1744, Calculus of Variations: Appendix II

Maupertuis discussed only cases of collision, in which case the transfer of momentum can be treated as instantaneous.

Euler investigated how to treat cases where a force is acting continuously, so that there is continuous change of momentum.

In every case that Euler discusses he makes use of the fact that there is a relation that describes interconversion of Kinetic energy and Potential energy. For instance, in section 4, in which the case of uniform gravitational force is discussed, Euler uses the relation $$v^2 = v_0^2 + 2gy$$, where $$y$$ is the vertical coordinate, and $$g$$ is gravitational acceleration. (Euler can omit the factor $$m$$ for mass there because inertial mass and gravitational mass are equivalent.)

(The concept used at the time was 'Living Force' (latin 'Vis Viva'), defined as $$mv^2$$. It was only around 1850 that the physics community shifted to the definition $$\tfrac{1}{2}mv^2$$, and the name 'kinetic energy')

Euler's method results in an expression of one position coordinate as a function of another position coordinate. For example: if polar coordinates are used to evaluated orbital motion then the expression obtained will give radial distance $$r$$ as a function of angular coordinate $$\theta$$.

Of course, what is needed is an expression for the trajectory as a function of time. So the expression for one position coordinate as a function of another position coordinate has to be processed further. That is what Feynman refers to when he mentions 'not quite as useful'.

In the late 1750's Euler and Lagrange collaborated on further development of Calculus of Variations.

In 1788 Lagrange's large work 'Mécanique Analytique' was published. Mécanique Analytique consists of two parts, the first dedicated to Statics and the second dedicated to Dynamics.

Lagrange used Calculus of Variations for cases in statics, but not for cases in Dynamics.

Lagrange explained why he didn't use Maupertuis' action, I give an extensive quote from 'Mécanique Analytique' in an appendix.

As we know: the key elements of Lagrangian mechanics are:

• Systematic use of generalized coordinates in cases where is beneficial to used generalized coordinates
• Expressing the physics taking place in terms of interconversion of Kinetic energy and Potential energy

Hamilton's stationary action, the subject of that chapter in the Feynman Lectures, was introduced by William Rowan Hamilton in 1834.

Nowadays the usual presentation is to use Hamilton's stationary action as motivation to derive the Euler-Lagrange equation, and often it is suggested that Lagrangian mechanics is based on Hamiltons stationary action.

However: the development of Lagrangian mechanics predated Hamilton's statioary action by more than half a century.

Returning to Maupertuis' action:

I would appear that Maupertuis was the only one to believe that Maupertuis' action is a principle.

Still, Lagrange took the time to explain to his readers why he dismissed Maupertuis' action. That indicates that at the time there was some interest in Maupertuis' action, enough so that Lagrange felt he had to explain his point of view.

About the contributions of William Rowan Hamilton

Hamilton introduced the stationary action concept, and he was also the one to point out that the name 'stationary action' is the appropriate name.

Hamilton points out that there are also cases where the true trajectory corresponds to a maximum of the action. The criterion that is always valid is: the true trajectory corresponds to a point in variation space such that the derivative of the action is zero.

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

A quote from that article:

And though we may retain the name of action to denote the stationary integral to which it has become appropriated [...] 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.

The name 'least action' has the following disadvantage: it suggests something that isn't actually the case.

Appendix: Lagrange on Maupertuis' action

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. [...]