# Principles of Least Action and of Least Constraint

Ekkehard Ramm

► To cite this version:

Ekkehard Ramm. Principles of Least Action and of Least Constraint. GAMM-Mitteilungen, 2011, 34 (2), pp.164 - 182. 10.1002/gamm.201110026. hal-01390460

HAL Id: hal-01390460

https://hal.science/hal-01390460

Submitted on 2 Nov 2016

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Principles of Least Action and of Least Constraint

**Ekkehard Ramm**

Institut für Baustatik und Baudynamik, Universität Stuttgart,
Pfaffenwaldring 7, 70550 Stuttgart, Germany

E-mail: ramm@ibb.uni-stuttgart.de

Phone: +49 711 685 66124

Fax: +49 711 685 66130

**Key words** Principle of Least Action, Principle of Least Constraint, Method of Least Squares, C.F. Gauss, P.-L. M. de Maupertuis

*Dedicated to Professor Erwin Stein on the occasion of his 80th birthday with special gratitude as my
mentor and a leading scholar in computational mechanics.*

The present contribution describes the evolution of two major extremum principles in me chanics
proposed in the 18th and the first half of the 19th century, namely the Principle of Least Action associated
with the name Maupertuis and Gauss's Principle of Least Constraint. We will briefly mention the
d'Alembert Principle strongly related to the Principle of Gauss.

## Preliminary Remark

Like the other contributions in this special issue the present paper is also based on a lecture
given in a session on the history of mechanics at the GAMM annual meeting in 2010. The
author not being a historian apologizes for a rather short and partly superficial essay on both
principles, based mainly on secondary sources. It can by far not replace any of the excellent
rigorous and detailed descriptions on the subject, some of them mentioned in the list of refe-
rences. This holds in particular for the description of the scientific controversy on the Principle
of Least Action, probably one of the most documented affairs in the history of sciences.

## 1 Introduction

The search for the existence of extremum principles in nature and technology can be traced
back to the ancient times. The scholars were not only driven by scientific observations; often
their objective was based on metaphysical arguments. In Figure 1 major figures involved
in the development of extremum principles in mechanics are arranged in the respective time
course. In the following essay the era of the mid 18th and the early 19th century is selected; in
particular we will concentrate on Pierre-Louis Moreau de Maupertuis related to the *Principle
of Least Action* in 1744/46 and on Carl Friedrich Gauss who stated the *Principle of Least
Constraint* in 1829.
<figure><img src="image_2.png" /><figcaption>Fig. 1 Evolution of Extremum Principles.</figcaption></figure>

## 2 Principle of Least Action

### 2.1 Maupertuis and his Principle

The discussion on the Principle of Least Action is one of the most extensive examinations of a scientific controversy. The protagonists (Figure 2) are Maupertuis and Euler on one side and

<figure><img src="image_3.png" /><figcaption>Fig. 2 The protagonists.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 3 Pierre-Louis Moreau de Maupertuis.</figcaption></figure>

König and Voltaire on the other side. Leibniz' work also played a role in the fight along with Frederick the Great, King of Prussia, as a leading authority at that time in Berlin.

The French mathematician Pierre-Louis Moreau de Maupertuis (Figure 3) was very much influenced by the work of Newton. Already in papers to the French Academy of Sciences in 1741 and 1744 he mentioned a principle minimizing a quantity which he called action. After he became the President of the Prussian Academy of Sciences in Berlin, on invitation of King Frederick the Great in 1746, he presented the book "Les Loix du Movement et du Repos" (The Laws of Movement and of Rest derived from a Metaphysical Principle) [1], see Figure 4, left. In the introduction he points to his previous work at the Paris Academy in 1744

<figure><img src="image_5.png" /><figcaption>Fig. 4 Maupertuis' Les Loix (1746) [1] and the Principe de la Moindre Action (Principle of Least Action).</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 5 Euler’s Methodus Inveniendi Lineas Curvas (1744) [2] and Supplement II.</figcaption></figure>

and adds the remark "At the end of the same year, Professor Euler published his excellent book *Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes*" (A method for finding curved lines enjoying properties of maximum or minimum) [2], Figure 5. He then continues "In a Supplement to his book, this illustrious Geometer showed that, in the trajectory of a particle acted on by a central force, the velocity multiplied by the line element of the trajectory is always a minimum". In other words Maupertuis was aware of maximum and minimum properties however he reduced his further considerations to a minimum principle.

In his "Les Loix..." Maupertuis first criticized the usual proofs of the existence of God and refers to the fundamental laws of nature. His self-confident statements are remarkable when he writes "After so many great men have worked on this subject, I almost do not dare to say that I have discovered the universal principle upon which all these laws are based. This is the principle of least action, a principle so wise and so worthy of the Supreme Being, and intrinsic to all natural phenomena; one observes it at work not only in every change, but also in every constancy that Nature exhibits".

Finally he states the general principle as follows (Figure 4, right): "When a change occurs in Nature, the Quantity of Action necessary for that change is as small as possible", and defines "The Quantity of Action is the product of the Mass of Bodies times their velocity and the distance they travel. When a Body is transported from one place to another, the Action is proportional to the Mass of the Body, to its velocity and to the distance over which it is transported". Expressing the Action A in a formula yields

$$A \sim M \cdot v \cdot ds \quad (1)$$

With the mass of a particle M, the velocity v and the transported distance ds. The relation of the Action to the kinetic energy is apparent if the distance ds is replaced by velocity times the time increment dt.

In the sequel of this definition Maupertuis adds three examples as a proof for the generality of his principle, namely on the laws of motion for inelastic as well as elastic bodies and the law of mechanical equilibrium. In the mentioned Supplement II "De motu projectorum..." (On a motion of particles in a non-resistant medium, determined by a Method of maxima and minima), Figure 5, right, Leonard Euler (Figure 6) says "1. Since all natural phenomena obey a certain maximum or minimum law, there is no doubt that some property must be maximized
<figure><img src="image_7.png" /><figcaption>Leonhard Euler
(1707-1783)</figcaption></figure>

* 1707 born in Basel

* 1720-1723 studies mathematics at University of Basel under Johann I Bernoulli and Jakob Hermann
* 1723/24 study of theology

* 1727 Member of Academy in St. Petersburg, Professors of Physics (1731) and Mathematics (1733)

* 1741 call by Frederick II to Berlin,

* 1741 works on "Methodus Inveniendi Lineas Curvas..." published in 1744

* 1744 Director of class of mathematics of new Prussian Academy of Sciences, treatise on variational methods

* 1753 takes Maupertuis' part although he must have known the weaknesses and does not claim any priority of the Principle; see "Dissertatio de Principio Minimae Actiones...."

* 1766 after disagreement with Frederick II returned to St. Petersburg;

* in following years publishes several fundamental treatises

* 1771 looses his sight

* 1783 died in St. Petersburg

<caption>Fig. 6 Leonard Euler.</caption>

or minimized in the trajectories of particles acted upon by external forces. However, it does not seem easy to determine which property is minimized from metaphysical principles known a priori. Yet if the trajectories can be determined by a direct method, the property being minimized or maximized by these trajectories can be determined, provided that sufficient care is taken. After considering the effects of external forces and the movements they generate, it seems most consistent with experience to assert that the integrated momentum, i.e. the sum of all momenta contained in the particle's movement, is the minimized quantity".

In section 2 of the Additamentum Euler defines mass *M*, velocity $\sqrt{v}$ (he used $\sqrt{v}$ instead of *v*), infinitesimal distance *ds*, the momentum integrated over the distance *ds* as $M \cdot ds \cdot \sqrt{v}$ and writes: "Now I assert that the true trajectory of the moving particle is the trajectory to be described, from among all possible trajectories connecting the same end point, that minimizes $\int M ds \sqrt{v}$ or since *M* is constant $\int ds \sqrt{v}$. Since the velocity $\sqrt{v}$ resulting from the external forces can be calculated a posteriori from the trajectory itself, a method of maxima and minima should suffice to determine the trajectory a priori. The minimized integral can be expressed in terms of the momentum (as above), but also in terms of the living forces (kinetic energies). For, given an infinitesimal time $dt$ during which the element $ds$ is traversed, we have $ds = dt\sqrt{v}$. Hence $\int ds\sqrt{v} = \int vdt$, i.e. the true trajectory of a moving particle minimizes the integral over time of its instantaneous living forces (kinetic energies). Thus, this minimum principle should appeal both to those who favor momentum for mechanics calculations and to those who favor living forces".

Euler realized that for the entire path the action has to be a sum along all segments $ds$ and essentially defined as quantity (velocity defined as *v* as usual)

$$A = \int M v ds = \int M v^2 dt \quad (2)$$

which is equal to the kinetic energy up to the factor $\frac{1}{2}$.

Thus it turned out that Maupertuis' pretension on universality of his principle was wrong, confer [3, 4]; the examples were not well chosen and partially not correct. His claim that the
<figure><img src="image_8.png" /><figcaption>Fig. 7 Johann Samuel König.</figcaption></figure>

minimization is an economy principle did not hold because some problems lead to a maxi-
mum. Chevalier P. D'Arcy reproached Maupertuis for this error; for example the refraction
of a concave mirror is based on a maximum property, [3]. Consequently his interpretation of
Euler's statements was not correct.

## 2.2 The critics

It was his fellow student and friend Johann Samuel König (Figure 7) who became Maupertuis’ strongest critic. Two years after he was appointed member of the Berlin Academy, strongly supported by the president, the Swiss lawyer and later mathematician König (1712–1757) passed an essay to Maupertuis who approved it for publication without reading. In his article König pled for a principle of extremal energy from which an extremum for the action can be derived. The essay appeared in Nova Acta Eruditorum 1751 [5]. When Maupertuis read it he became upset because not only the limitation of his principle was questioned but it also claimed that G.W. Leibniz (1646–1716) has given a more precise formulation already in a letter to the Swiss mathematician and theologian Jakob Hermann (1678–1733) in 1707. At the end of the paper [5] he quotes from Leibniz’ letter to Hermann [6] (see Figure 8) “*But action is in no way what you think, there the consideration of the time enters; it is the product of the mass and the time*<fnref n="1" />; or of the time by the living force. I have pointed out that in the modifications of movements, it is usually derived as a *Maximum*, or a *Minimum*. From this can be deduced several important propositions; it can be used to determine the curves describing the bodies that are attached to one or several centres”.

According to König the "force vive" (vis viva) has been already introduced by Leibniz expressing $M \cdot v^2$, i.e. it is two times the kinetic energy. It was not his intention though to accuse Maupertuis being a plagiarist or to disparage his merits; he simply wanted to point out the more general formulation of Leibniz (minimum or maximum). Nevertheless it was

<footnote n="1">¹ The quote contains a misprint; in the first definition of the action time must be replaced by distance and speed.</footnote>
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 8** König's reference letter of Leibniz to Hermann (1708) in Nova Acta Eruditorum (1751) [5].</figcaption></figure>

the beginning of intensive mutual disparagements ending in one of the ugliest of all scientific
disputes [7].

The letter König presented was a copy, a fact which would play a key role in the subsequent
vehement quarrel. All attempts to find the original letter ended only in further copies; König
claimed that he got his copy from the Swiss poet and politician Samuel Henzi in Bern who
collected letters of Leibniz among other things. Henzi was decapitated in 1749 because of
conspiration and all his original papers were supposed to be burned. The Berlin Academy
met in spring 1752 charging König of forgery. It is noteworthy that the meeting was headed
by Euler who from the very beginning was a strong supporter of Maupertuis despite his own
more rigorous work on the same principle in 1744 (Supplement II, Figure 5); the reason for
Euler's reaction is still a matter of conjectures; see e.g. [3, 4, 8]. König's appeal was rejected;
he resigned as member of the Academy in summer 1752.

King Frederick used all his authority to support Maupertuis, the President of "his" Academy.
It was the time when another figure entered the scene, namely Voltaire (Figure 9), the famous
French writer. Voltaire liked satires and polemics and was known for his sharp, malicious,
sometimes also witty remarks. Although he was invited by Frederick to come to Berlin he
took part of König; he published an anonymous letter "A reply from an Academician of Berlin
to an Academician of Paris" in fall 1752 defending König (Figure 13, right); he refers to Mon-
sieur Moreau de Maupertuis and says: "He asserts that in all possible cases, Action is always
a Minimum, which has been demonstrated false; and he says he discovered this law of Mini-
mum, what is not less false. Mr. Koenig, as well as other Mathematicians, wrote against this
strange assertion, & he cited among other things, a fragment of Leibnitz in which the great
man has remarked, that in modifications of movement, the action usually becomes either a
Maximum, or a Minimum". König once more referred to copies of Leibniz' letter in a further
pamphlet. Frederick was provoked and defended Maupertuis in a "Letter of an Academician
of Berlin to an Academician in Paris" (Figure 13, right).

Voltaire published further polemics, the most famous being the "Diatribe due Docteur Akakia, Médicin du Pape" (Figure 10, left), which was a defamatory piece of writing. The Greek word Akakia means "without guile". The *Diatribe* printed in Potsdam was burned by
<figure><img src="image_11.png" /><figcaption>**Fig. 9** François-Marie Arouet, called himself Voltaire.</figcaption></figure>

order of King Frederick. However since it was also published in Holland, some extra copies appeared in Berlin and were again burned in front of Voltaire's apartment. This was the final break-up with the king; Voltaire left for Leipzig in spring 1753 where he continued attacking Maupertuis.

Several letters of Voltaire followed compiled in "Histoire du Docteur Akakia et du Natif de St. Malo" in April 1753 (Figure 10, right) again making snide remarks because St. Malo is the birthplace of Maupertuis. Like the *Diatribe* this pamphlet was a sham presented as a defence but in reality being a backhanded compliment. In the *Histoire* a fictitious peace contract between the President (Maupertuis) and the Professor (König) *Traité de Paix conclu entre M.*

<figure><img src="image_12.png" /><figcaption>**Fig. 10** Voltaire's *Diatribe* [9] and *Histoire du Docteur Akakia* [10] (1753).</figcaption></figure>
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 11** Euler's Sur le Principe de la Moindre Action 1751/53 [12].</figcaption></figure>

Le Président et M. Le Professeur, 1st January 1753 [10] was included saying among other things: "In future we promise, not to put the Germans down and admit that the Copernicus's, the Kepler's, the Leibnitz's..., are something, and that we have studied under the Bernoulli's, and shall study again; and that, finally, Professor Euler, who was very anxious to serve us as a lieutenant, is a very great geometer who has supported our principle with formulae which we have been quite unable to understand, but which those who do understand have assured us they are full of genius, like the published works of the professor referred to, our lieutenant", [11].

Thus Voltaire also included Euler in this dispute and holds him up to ridicule. Euler was indeed a vehement advocate of the Academy President despite his own more rigorous exposition of the subject. Dugas refers in [11] to an essay of Euler and quotes: "This great geometer has not only established the principle more firmly than I had done but his method, more ubiquitous and penetrating than mine, has discovered consequences that I had not obtained. After so many vested interests in the principle itself, he has shown, with the same evidence, that I was the only one to whom the discovery could be attributed".

Euler discussed his view of the development of the principle in [12], presented already in 1751 but not printed in the Histoires of the Prussian Academy until February 1753 (Figure 11). He referred to König's role and made a strong statement: "But there is no one with whom we would be less likely to have dispute than Professor Koenig, who boldly denies that there is in nature such a universal law and pushes this absurdity to the point of mocking the Principle of Conservation, which constitutes the Minimum that nature appoints. In addition, he introduces the great Leibnitz as claiming and explaining that he was far from knowing such a principle. From this we see that Mr. Koenig cannot deny our President the discovery of the principle that he himself considers to be false". Finally he said: "The principle that Mr. de Maupertuis discovered is therefore worthy of the greatest of praise; and without a doubt it is far superior to all discoveries that have been made in dynamics up until now" (Figure 11, right). This essay was included in the "Dissertatio de Principio Minimae Actionis...", a bilingual edition in
<figure><img src="image_15.png" /><figcaption>Fig. 12 Bilingual edition of Euler’s *Dissertatio* [13].</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13 Maupertuisiana (1753), List of Essays (figure from [3]).</figcaption></figure>

Latin and French [13] together with an examination of the objections of M. Professor Koenig made against the principle (Figure 12), see also [14].

Shortly thereafter in April 1753 sixteen polemics and letters in dispute were put together in a publication initiated by Voltaire [14]; it was entitled "Maupertuisiana" (Figure 13) and printed in a fictitious location Hambourg; among them are the aforementioned "Reply" and "Letter". The figure on the title page shows Don Quixote (Maupertuis) fighting the windmills with a broken lance shouting "Tremblez" (tremble); behind him Sancha Panza (Euler) is riding a donkey. On the right side a satyr is depicted saying "sic itur ad astra" (thus one reaches the
stars), meaning "thus one is blamed forever". Above the picture a line of Virgil's "Aeneis" is
added: *"Discite Justitiam, moniti"* (be warned and learn justice).

The "Traité de Paix" was also included in the *Maupertuisiana*; in the title page of that
volume the first part of a quote from Horace's Satires was added: "ridiculum acri forties ac
melius (plerumque secat res)" or "ridicule often settles matters of importance better (and with
more effect than severity)".

With the publication of the *Maupertuisiana* the quarrel came more or less to an end. Mau-
pertuis was ill, spent a year in Paris and St. Malo, returned to Berlin which he left in 1756.
After another year in St. Malo he was accommodated by Johann II Bernoulli in Basel where
he died in 1759. S. König being the moral victor in this battle died from a stroke in Holland
two years earlier in 1757. Voltaire had still some active and eventful years; he left Berlin in
1753 for Paris, but was banned by Louis X. He then lived in Geneva and in Ferney across the
French border. In 1778 he returned for the first time in 20 years to Paris where he died after
three months. Euler as the fourth figure in this dispute stayed a couple of years in Berlin, but
returned to St. Petersburg in 1766. He was extremely active and published several fundamen-
tal treatises. He passed away in 1783.

2.3 On Leibniz' Alleged Letter to Hermann

The claim for forgery was taken up again 140 years later when C.I. Gerhardt, editor of Leibniz'
mathematical œuvre at the Prussian Academy, reinvestigated the case. The Physicist H. von
Helmholtz, who gave a speech on the history of the Principle of Least Action in the Academy
in 1887 [15], refers to Gerhardt's remark, that the letter does not fit to the correspondence with
Hermann. Leibniz may have postponed the publication because he planned a later application.
In a detailed report [16] in 1898 (see minutes for a meeting of the Academy) Gerhardt claims
that three of the four letters presented by König are genuine and concludes at the end of his
paper: "From the above explanations it should undoubtedly follow that the letter fragment,
which König published, is not made up, and the entire letter has not been foisted. The letter
is written by Leibniz. Also with a probability close to certainty it has been proven that it
had been directed to Varignon (Pierre de Varignon (1654–1722), French mathematician and
physicist). The coincidental letters of the correspondence with Varignon in the Royal Library
of Hannover have disappeared as the letters of Leibniz in the respective correspondence with
Hermann did". The assumption for the authenticity was underlined by another argument in
1913, when W. Kabitz [17] found a further copy of the letter in Gotha among a collection of
other genuine letters of Leibniz. This was accepted as a "proof" for the originality of the letter
at that time.

Recently Breger has re-examined the case in a remarkably thorough discussion [18] in
which he elaborated on eight arguments against the authenticity of the letter. Among those is
the argument that Leibniz never applied the notion "limites" in the sense as it is used in the
letter. Leibniz also often referred to minimum and maximum properties and also mentioned
the term action, however combining both these concepts would be a singular event; for further
arguments see [18], a paper worth reading. Breger recomments that now it is time to inversely
analyse which arguments are speaking in favor of the genuineness. He also mentions that in
case of forgery the rather strange reaction of Euler is easier to comprehend; he understood
the letter as an action against his own person. A further question namely with respect to the
identity of the falsifier still remains open.
<figure><img src="image_17.png" /><figcaption>Fig. 14 Carl Friedrich Gauss.</figcaption></figure>

## 2.4 Application of Principle of Least Action

As said above the Principle of Least Action has been defined in more rigorous form already by Euler in 1744 as an energy principle. It turned out to be rather a *Principle of Stationary Action*. Lagrange derived the equation of motion based on his newly developed calculus of variations (1760). In 1834/35 Hamilton introduced the Lagrangian function into the variational principle deriving what has been called later the Euler-Lagrange equations. Whereas Euler and Maupertuis concentrated in the Principle of Least Action on the kinetic energy (vis viva, living forces) tacitly assuming constant potential energy the difference between the kinetic and the potential energy entered the Lagrangian in Hamilton's principle. Despite this close relationship the Principle of Least Action has not found its way into the engineering community which mostly refers to the works of Lagrange, d'Alembert and Hamilton. In the 19th century the well known Principles of Minimum of Total Potential Energy (Dirichlet-Green) and Minimum of Total Complementary Energy (Menabrea-Castigliano) became basic theorems in mechanics. The Principles of Virtual Work and Complementary Work entered the scene in particular because of their generality being applicable also for problems where no potentials exist. They are nowadays together with mixed Variational Principles (Hellinger-Reissner, Fraeijs deVeubeke-Hu-Washizu, etc.) the foundations for the derivation of discretization methods.

Opposite to the engineering community the notion of Principle of Least (Stationary) Action is still present in modern physics. Entire chapters of treatises are devoted to the principle, e.g. [19, 20]; see also the famous Feynman Lectures on Physics [21]. Occasionally they are looked upon as generalization of Hamilton's principle rather than as its predecessor. The Principle has been applied for example in the theory of relativity or in quantum mechanics, see [22].
<figure><img src="image_18.png" /><figcaption>Fig. 15 C.F. Gauss’s Method of Least Square (1809), extract from [23].</figcaption></figure>

## 3 Principle of Least Constraint

### 3.1 Method of Least Squares

Carl Friedrich Gauss (1777–1855) (Figure 14) applied his method of least squares in 1801 when he determined the elliptical orbit of the astroid Ceres; however he had developed the basis of this method already in 1795 when he was 18 years old. It was not until 1809 that he published the method in the second volume of his book on the Theory of Celestial Bodies [23] (Figure 15). He said "Our principle which we have made use of since the year 1795 has lately been published by Legendre in the work *Nouvelles methodes pour la determination des orbites des cometes*, Paris 1806 prolatum est, vbi plures aliae proprietates huius principii expositae sunt, quas hic breuitatis caussa supprimimus.". It was a further priority argument in the history of mechanics; however Gauss could present his correspondence with colleagues concerning its use much earlier.

### 3.2 Principle of Least Constraint

Twenty years later in 1829 Gauss wrote an essay on "Über ein neues allgemeines Grundgesetz der Mechanik" (On a new Fundamental Law of Mechanics) [24]. The paper was published in the 4th issue of *Journal für die reine und angewandte Mathematik* (ed. A.L. Crelle), a journal still existing today. He describes his new law on four pages that became known as the Principle of Least Constraint (Figure 16), by the way with very little algebra. It says "The motion of a system of material points... takes place in every moment in maximum accordance with the free movement or under least constraint;...". He continues "the measure of constraint, ..., is considered as the sum of products of mass and the square of the deviation to the free motion".

Applying the usual mathematical notation the acceleration $a_i^{\text{free}}$ of the free unconstrained motion is defined by force $F_i$ divided by mass $m_i$; $r$ is the position vector.

free motion

$$a_i^{\text{free}} = \frac{F_i}{m_i} \qquad a_i = \ddot{r}_i \quad (3)$$

constrained motion

$$a_i \neq a_i^{\text{free}} \qquad (4)$$

The acceleration of the constraint motion is called $a_i$ where kinematic conditions constrain the corresponding material point $i$. For example a kinematic constraint may be given by a
<figure><img src="image_19.png" /><figcaption>Fig. 16 C.F. Gauss in 1828 and his paper on the Principle of Least Constraint (1829) [24].</figcaption></figure>

<figure><img src="image_20.png" /></figure>

prescribed trajectory. The measure Z for the "constraint" (Zwang) is proportional to the sum of squares of the differences between free and constrained accelerations. Each term in the sum is weighted by mass $m_i$.

$$Z \sim \sum_{i=1}^{N} (a_i - a_i^{\text{free}})^2 \qquad (5)$$

$$\text{or } Z = \sum_{i=1}^{N} m_i \left( a_i - \frac{F_i}{m_i} \right)^2 = \text{MIN} \quad + \text{kinematical conditions,} \\ \qquad \qquad \qquad \qquad \qquad \text{e.g. prescribed displacements} \qquad (6)$$

According to the principle Z is supposed to be a Minimum. Thus from all possible motions (accelerations) the actual motion leads under given conditions to the least constraint.

Figure 17 applies the principle to the motion of a pendulum, the circle being the constrained trajectory. As expected the principle yields the equation of motion. This example elucidates that there is a strong relation with d'Alembert's Principle (Jean d'Alembert, 1717-1783). We start from the Principle of Least Constraint and its variation with respect to its free parameters $a_i$

$$\text{constraint} \qquad Z = \sum_{i=1}^{N} m_i \left( a_i - \underbrace{\frac{F_i}{m_i}}_{\text{fix}} \right)^2 = \text{MIN} \qquad (7)$$

$$\text{variation} \qquad \delta Z = \sum_{i=1}^{N} 2m_i \left( a_i - \frac{F_i}{m_i} \right) \underbrace{\delta a_i}_{\hat{=} \delta r_i \hat{=} \delta d_i} = 0 \qquad (8)$$
<figure><img src="image_21.png" /><figcaption>**Fig. 17** Application of the Principle of Least Constraint for Pendulum.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. 18** Principle of d'Alembert.</figcaption></figure>

where the position vector and its variation are

$$ \text{position vector} \qquad r(t + dt) = r(t) + v(t) dt + 1/2 a(t) dt^2 \qquad (9) $$

$$ \text{variation} \qquad \delta r(t + dt) = 1/2 \delta a(t) dt^2 \qquad (10) $$

The variation or virtual displacements satisfy the essential boundary conditions. $\delta Z = 0$ directly yields the Principle of d'Alembert (1743) as a variational principle of the equation of motion (Figure 18).

Gauss finishes his paper with the remark (Figure 19) "It is strange that the free movements, when they cannot withstand the necessary conditions, are modified in the same way as the analyzing mathematician, applying the method of least squares, balances experiences which are based on parameters depending on necessary interactions", see the comparison in Figure 20.
> Es ist sehr merkwürdig, dass die freien Bewegungen, wenn sie mit nothwendigen Bedingungen nicht bestehen können, von der Natur gerade auf dieselbe Art modificirt werden, wie der rechnende Mathematiker, nach der Methode der kleinsten Quadrate, Erfahrungen ausgleicht, die sich auf unter einander durch nothwendige Abhängigkeit verknüpfte Gröfsen beziehen. Diese Analogie liefse sich noch weiter verfolgen, was jedoch gegenwärtig nicht zu meiner Absicht gehört.

<figure><img src="image_23.png" /><figcaption>Fig. 20 Principle of Least Constraint and Least Square Method.</figcaption></figure>

He adds a final sentence to his paper which says: "*This analogy could be further followed up, but this is currently not my intention*". However he never picked up this matter again.

## 3.3 Application of Principle of Least Constraint

Gauss's Principle is not very well known although it is mentioned as a fundamental principle in many treatises, e. g. [3, 25-27], see also [28]; correspondingly it has not been applied too often. Evans and Morriss [26] discuss in detail the application of the Principle for holonomic (constraints depend only on co-ordinates) and nonholonomic constraints (non-integrable constraints on velocity) and conclude "*The correct application of Gauss's principle is limited to arbitrary holonomic constraints and apparently, to nonholonomic constraint functions which are homogeneous functions of the momenta*". Evans et al. [29] applied the principle in the context of statistical mechanics, see also [30]. As another example the work of Glocker [31, 32] is mentioned where accelerations in rigid multibody systems under set-valued forces are evaluated applying Gauss's principle. Recently Udwadia and co-workers extended the Principle to underdetermined systems [33].

As a final remark the author would like to point out to the possibility of using Gauss's Principle in optimization and design of structures under dynamic excitation where the system ought to be tuned for selected accelerations.
References

[1] P.-L. M. de Maupertuis, "Les Loix du Mouvement et du Repos déduites d'un Principe Metaphysique." in: *Histoire de l'Académie Royale des Science et des Belles Lettres*, 1746, pp. 267-294. English Translation: http://en.wikisource.org/wiki/Derivation_of_the_laws_of_motion_and_equilibrium_from_a_metaphysical_principle

[2] L. Euler, *Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes*. Marcum-Michaelem Bousquet & Socios, Lausannæ & Genevæ, 1744. Citation from the Euler Archive (Mathematical Association of America) E65, http://www.math.dartmouth.edu/~euler/pages/E065.html. English Translation of Additamentum II: http://en.wikisource.org/wiki/Methodus_inveniendi/Additamentum-II

[3] I. Szabo, *Geschichte der mechanischen Prinzipien und ihrer wichtigsten Anwendungen*. Birkhäuser, Basel, 1979.

[4] E. Knobloch, "Das große Spargesetz der Natur: Zur Tragikkomödie zwischen Euler, Voltaire und Maupertuis." in: *Leonhard Euler 1707–1783, Mathematiker - Mechaniker - Physiker*. edited by G. Biegel, A. Klein and T. Sonar. Braunschweigisches Landesmuseum, Braunschweig, 2008, pp. 79–89.

[5] S. Koenigio, *De universali principio aequilibrii et motus, in Vi viva reperto, deque nexu inter Vim vivam et Actionem, utriusque minimo, Dissertatio*. Nova Acta Eruditorum, 1751, pp. 125–135, 162–176. http://www.izwt.uniwuppertal.de/repertorium/MS/NAE_1751.html

[6] G. Leibniz, "Lettre de Mr. de Leibnitz, dout Mr. Koenig a cité le Fragment (16. Octobre 1707)" in: *Appel au Public, du Ingement de L'Académie Royal de Berlin, Sur un Fragment de Lettre de Mr. de Leibnitz, cité par Mr. Koenig*. Seconde Edition. A Leide de L'Imp. d'Elie Luzac, Jun, 1753, pp. 166–171.

[7] A. Harnack, *Geschichte der Königlich Preussischen Akademie der Wissenschaften, Band 1.1: Von der Grundung bis zum Tod Friedrich des Großen*. Reichsdruckerei, Berlin, 1900. pp. 331–345.

[8] N. Ariga, *Science and its Public Perception: The Principle of Least Action in Eighteenth-Century Europe*. Department of Philosophy and History of Science, Graduate School of Letters, Kyoto University, Japan, 2008. http://www7b.biglobe.ne.jp/~ariga/strage/eastssw2008_article.pdf

[9] Voltaire, *Diatribe du Docteur Akakia, Médecin du Pape*. Rome, 1753.

[10] Voltaire, *Histoire du Docteur Akakia et du Natif de St. Malo*. 1753. (containes Diatribe, Traité de Paix among other documents) http://www.voltaire-integral.com/Html/23/08DIAL.htm. See also: http://www.slubdresden.de/index.php?id=5363&tx_dlf[id]=9030

[11] R. Dugas, *A History of Mechanics*. Dover Classics of Science and Mathematics. Dover Publications Mineola, New York, 1988.

[12] L. Euler, "Sur la principe de la moindre action" ("On the Principle of Least Action") in: *Histoire de L'Académie Royale des Science et des Belles Lettres*. Haude et Spencer, Berlin, 1753. Akademiebibliothek Berlin-Brandenburger Akademie der Wissenschaften. http://bibliothek.bbw.de/bbw/bibliothek-digital/digitalequellen/schriften/anzeige?band=02-hist/1751 English Translation: Euler Archive (Mathematical Association of America), E 198: http://www.math.dartmouth.edu/~euler/tour/tour_07.html

[13] L. Eulero, *Dissertatio de principio minimae actionis una cum examine obiectionum CL. Prof. Koenigii contra hoc principium factarum*. Berolini, Ex Officina Michaelis, 1753. Bayrische Staatsbibliothek digital. http://www.bsb-muenchen-digital.de/~web/web1008/bsb10080685/
images/index.html?digID=bsb10080685&pimage=6&v=100&nav=0&l=de see
also Euler Archive (Mathematical Association of America), E 186:
http://www.math.dartmouth.edu/~euler/pages/E186.html

[14] Voltaire, *Maupertuisiana, Collection of pamphlets related to controversy on Principle of Least Actions*. Hambourg, 1753.

[15] H. Helmholtz, "Rede über die Entstehungsgeschichte des Principes der kleinsten Action." (1887), in: *Geschichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. Band 2* Urkunden und Actenstücke, edited by A. Harnack. Reichsdruckerei, Berlin, 1900, pp. 282–296.
http://bibliothek.bbw.de/bbw/bibliothek-digital/digitalequellen/schriften/anzeige/index.html?
band=ak-gesch/harn-2&seite:int=295

[16] C.I. Gerhardt, *Über die vier Briefe von Leibniz, die Samuel König in dem Appel au public, Leide MDCCLIII, veroffentlicht hat*. Sitzungsberichte der Königlich Preusschen Akademie der Wissenschaften, Vol. I, 1898, pp. 419–427.

[17] W. Kabitz, "*Über eine in Gotha aufgefundene Abschrift des von S. König in seinem Streite mit Maupertuis und der Akademie veroffentlichten, seinerzeit fur unecht erklarten Leibnizbriefes*. Sitzungsberichte der Königlich Preusschen Akademie der Wissenschaften, Vol. II, 1913, pp. 632–638.

[18] H. Breger, "Über den von Samuel König veröffentlichten Brief zum Prinzip der kleinsten Wirkung." in: *Pierre Louis Moreau de Maupertuis. Eine Bilanz nach 300 Jahren*. edited by H. Hecht. Schriftenreihe des Frankreich-Zentrums der Technischen Universität Berlin. Verlag Arno Spitz GmbH/Nomos Verlagsgesellschaft, Berlin, 1999, pp. 363–381.

[19] L. Pars, *A Treatise on Analytical Dynamics*. Heinemann, London, 1965; reprinted by Ox Bow Press, Woodbridge, CT, USA, 1979.

[20] V. Berdichevsky, *Variational Principles of Continuum Mechanics. Part I Fundamentals*. Springer, Berlin, 2009.

[21] R. Feynman, R. Leighton and M. Sands, *Lectures on Physics, Volume 2, Chapter 19. 1964*. extended 2nd edition. Addison-Wesley, 2005.

[22] E. Taylor, *Principles of Least Action*. MIT, 2008.
http://www.eftaylor.com/leastaction.html

[23] C.F. Gauss, "Theoria Motus Corporum Coelestium in sectionibus conicus solum ambientium." Volume 2, Section 3, pp 205-224, Göttingen 1809, in: *Carl Friedrich Gauss Werke. Königl. Gesellschaft der Wissenschaften zu Göttingen in Commision bei B.G. Teubner, Leipzig, 1906, pp. 236-257*. Georg August Universität Göttingen Niedersächsische Staats- und Universitätsbibliothek Göttingen, http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN236008730&DMDID=dmdlog17

[24] C.F. Gauss, "Über ein neues allgemeines Grundgesetz der Mechanik", *Journal fur die reine und angewandte Mathematik*, herausg. v. Crelle IV:232–235 (1829).

[25] C. Lanczos, *The Variational Principles of Mechanics*. 3rd edition. University of Toronto Press, Toronto, 1966, pp. 106–110.

[26] D. Evans and G. Morriss, *Statistical Mechanics of Nonequilibrium Liquids*. 2nd edition. ANU EPress, The Australian National University, Canberra, 2007, pp. 33–44.

[27] M. Päsler, *Prinzipe der Mechanik*. Walter de Gruyter & Co., Berlin, 1968.

[28] K. Linkwitz, "Numerical Computation and Principles of Mechanics: C.F. Gauss, the Great Mathematician, Muses Comprehensively on Figures and Formulas" in: *Proc. IASS-IACM 2000 Colloquium on Computational Methods for Shell and Spatial Structures, Chania, Crete*. Edited by M. Papadrakakis, A. Samartin and E. Oñate. ISASR, Athens, Crete, 2000.

[29] D. Evans, W. Hoover, B. Failore, B. Moran and A. Ladd "Nonequilibrium molecular dynamics via Gauss's principle of least contraint", *Physical Review A*. 28 (1983).
[30] J. Bright, D. Evans and D. Searles "New observations regarding deterministic, time reversible thermostats and Gauss's principles of least constraint", *J. Chem Phys.* 122(19):1-8 (2005).

[31] C. Glocker "The Principles of d'Alembert, Jourdain and Gauss in Nonsmooth Dynamics. Part I: Scleronomic Multibody Systems", *ZAMM Z. Angew. Math. Mech.* 78:21-37 (1998).

[32] C. Glocker, *Set-Value Force Laws: Dynamics of Non-Smooth Systems*. Lecture Notes in Applied and Computational Mechanics. Springer, Berlin, Heidelberg, 2001.

[33] Y. Fan, R. Kabala, H. Natsuyama and F. Udwadia "Reflections on the Gauss Principle of Least Constraint", *Journal of Optimization Theory and Applications* 127:475-484 (2005).