On the Work of Benjamin Olinde Rodrigues (1795-1851) -- in particuler, on Expression of Spatial Motions
aa r X i v : . [ m a t h . HO ] M a y On the Work of Benjamin Olinde Rodrigues (1795-1851)— in particuler, on “ Expression of Spatial Motions ”—
Takeshi HIRAI*
Abstract.
This is a translation of Proceedings of 22 th Symposium on History of Mathematics,Tsuda University 2011, on the work of Benjamin Olinde Rodrigues and on his life. His chef-d’oeuvre isthe work on Euclidean motion group in 1840. He invented
Rodrigues expression of rotation and gaveexplicit calculation formula for product of two rotations, which might be considered as a discovery ofquaternion product rule ahead of Hamilton. We follow a new proof of É. Cartan in his book on spineurs in 1938 for Rodrigues formula, which was called as Euler-Olind-Rodrigues formula mistakenly. Weadd as Appendix important parts of Lecture Note on applications of quaternion. There are givendescription of rotational movements in Rodrigues expression and an interesting compact formula fortime derivative of rotation, applicable in many purposes.
Why I came to pay attention to Benjamin Olinde Rodrigues (1795/10/6 – 1851/12/17) isbecause, in fact, I wanted to know the historical process of the quaternion of Hamilton. Ac-tually, in the workshop "Non-Commutative Harmonic Analysis" at Będlewo, Poland, Augustof 2007, I gave a talk on projective representations of complex reflection groups containingsymmetric groups, and necessarily I discussed about the trilogy [Sch1, 1904] – [Sch3, 1911] ofSchur’s theory on spin representations of finite groups and noticed that, in the third paper[Sch3], there appeared substantially a triplet of matrices which is called later as
Pauli matri-ces , found independently by Physicist Pauli in [Paul, 1927]. About this historical comment,Prof. Marek Bożejko gave me a question “
How about the case of quaternion of Hamilton ? ”,which means that in the theory of quaternion in 19 th century, “ Is there something like suchtriplet of matrices ? ” It might be possible, and mathematicians nowadays know generallythat the unit ball B consisting of quaternions with norm 1 gives a double covering group ofthe 3-dimensional rotation group SO (3) . However, at that time, I lacked totally such kind ofhistorical point of view, and couldn’t answer his question. I felt deeply that I know nothingabout Hamilton to answer this question and that I ought to study this sometime in future. : Primary 20-03, 01A70; Secondary 20C99, 01A85. Key Words and Phrases.
Cartan’s spineur, quaternion of Hamilton, projective or spin representations ofSO(3), Pauli matrices, time derivative of Rodrigues formula of rotation Translation (with small changes) of Proceedings of 22 th Symposium on History of Mathematics in
Reportsof Institute for Mathematics and Computer Sciences, Tsuda University , (2012), 59–79. ote 1. Three matrices appeared in p.198 of Schur’s paper [Sch3, 1911] are F = (cid:18) (cid:19) , A = (cid:18) (cid:19) , B = (cid:18) − (cid:19) , C = (cid:18) − (cid:19) , (1.1) with (cid:26) A = F, B = − F, C = F, CBA = F,AB = − BA = − C, BC = − CB = − A, CA = − AC = B, and they are used, as fundamental ingredients, to write down explicitly doubly-valued pro-jective representations of symmetric groups S n and alternating group A n .On the other hand, three matrices appered in Pauli’s paper [Paul, 1927] and later called asPauli matrices, express rotation moment (spin) of electron, act on ( C -valued) wave function ψ as follows: s x ( ψ ) = (cid:18) (cid:19) ψ , s y ( ψ ) = (cid:18) − ii (cid:19) ψ , s z ( ψ ) = (cid:18) − (cid:19) ψ , (1.2)with commutation relation [ s x , s y ] = 2 is z , [ s y , s z ] = 2 is x , [ s z , s x ] = 2 is y , i = √− . Further, Élie Cartan discussed in his paper [Car1, 1913], without introducing the terminol-ogy such as spineur ( spinor in French), representations of SU (2) double covering of rotationgroup SO (3) , actually containing spin representations together with linear representations,but did not appear there any triplet of matrices.In the meantime, I was asked in 2008 from Prof. Satoshi Kawakami to give a SummerIntensive Course of one week for Mathematics Master Course of Nara Education University,and so I proposed as a subject “ Quaternion, 3-dimensional Rotation Group and Introductionto Representation Theory of Groups ” and begun to prepare a Lecture Note. In that occason,when I searched documents about quaternion in website, I found a paper [Alt2]
Hamilton,Rodrigues, and the Quaternion scandal . What does it mean scandal , in such an academicsituation ? I wondered and read it with a keen interest, then I noticed that the historical factswhich I imagined myself until then, arround universal covering group of 3-dimensional rotationgroup, or parameter expressions of 3-dimensional rotations and so on, differ considerably fromthe true story. Thus I gradually went deeply into Rodrigues, the central figure of the presentpaper. As for Lecture Note above, I attach its important part with deep relation to this paperas an Appendix for more detailed commentary. See also [Hir9]. ∼ , After the 1815 Restoration the Catholic hierarchy took control of educationaland academic institutions, and Jewish people could not obtain any teaching position.1816, The main part of the above thèse was published in Mémoire (*) sur l’attraction des sphéroïdes,
PREMIÈRE PARTIE.
Formulesgénérales pour l’attraction des corps quelquonques, et application de ces formulesà la sphère et aux ellipsoïdes.
SECONDE PARTIE.
Attraction des sphéroïdesinfiniment peu différens d’une sphère, et dévelopement générale de la fonction V. ibid., (1814–1816): pp.361–385, 1816.This paper contains Rodrigues formula for Legendre polynomials : P n ( x ) = 12 n n ! d n dx n ( x − n . —– between 1817–1837, no papers in Mathematics were published —–1838, Three papers were published in Journal de Mathématiques pures et appliquées,vol. (1938), pp.547–548, pp.549–549, pp.550–551. (He was 42 years old)The contents of 3 papers are the following:[R3-1] The number of ways to decompose a convex polygon into triangles by diagonals [R3-2]
The number of ways to make product with n factors [R3-3] Elementary and purely algebraic proof of development of binomial (1 + a ) x intopower series with negative or frational powers Length of each paper is short, but contents are important. The first two gave compactand clear proofs for main results of long papers published very recently in journals.1839, A paper [R4-1] was pulished in ibid. , (1839), 236–240. Its contents are3R4-1] Number of inversions in the order of products of permutations
Contents.
The generating function for this number was given, and this result continuedto have important influences until nowadays. For σ ∈ S n , count the number of pairs i < j such that the inversion “ σ ( i ) > σ ( j ) ” occurs, and let N n ( k ) be the number of σ for whichthe number of inversins is just equal to k . Define the generating function of N n ( k ) as R n ( q ) := X k N n ( k ) q k . Rodriques gave a method of computing R n ( q ) inductively. Note.
For this problem, later there were studies of Netto and MacMahon. But, onlyin 1970, Leonard Carlitz discovered this paper [R4-1].1840, A paper [R5-1] was pulished in ibid. vol. , pp.380–440. Its subjects are[R5-1] Eulidean Motions in the Space, in particular Rotations
This long paper gives very important results and can be said as his Chef d’Oeuvre.The contents, containing substantial discovery of Quaternion, will be explaind in §3.1843, Two papers were published in ibid. vol. ,[R8-1] pp.217–224, [R8-2] pp.225–234.Cf. 1843/10/16, Didscovery of Quaternion by William Rowan Hamilton (1805–1865) :by giving the fundamental formula i = j = k = ijk = − .1843/10/17, Letter from Sir W. R. Hamilton to John T. Graves, Esq. on Quaternions (which is hand written) [Later, in printed form, 1 line about75 letters and total 167 lines, 4 pages, containing footnotes, in Collected Works.] Note.
In these two days, he wrote, with a quill and ink, a letter of this volumeand its fair copy for himself. This is a great concentration of a man of genius.1851/12/17, Benjamin Olinde RodriguesĄi56 years oldĄjdied.
Paper [R5-1] by Olinde Rodrigues:
Des lois géométriques qui régissent les déplacements d’unsystème solide dans l’espace, et la variation des coordonnées provenant de ses déplacementsconsidérés indépendamment des causes qui peuvent les produire.
Translation of Title:
Geometric rules which govern the movements of a system of solidbodies in the space, and changes of coordinates coming from its displacements consideredindependently of the reason which produces them.
The Style of Writing of this paper.
There is no independent Introducion. The paperis separated into parts with numbers from to , which we call here as parts (maybe numeros in French). Each part has only its number and no title. As general style, there are18 Titles in italic under each of which a group of parts are gathered. But there exist severalexceptions, for instance in some parts , there are one or two italic titles (something like as4ubsections in a section). Theorems are not separated from ground sentences as in modernstyle where theorems are numbered and their assertions are written in italic. In the middleof part o Théorème fondamental.
But this might be a title of a subnumero (like subsubsection in a subsection) and I foundseveral of such italic titles. There are neither Propositions, Lemmas nor Diagrams.
Contents.
He discussed very generally on displacements of solid bodies, that is,
Eu-clidean motion group in modern language. Naturally there are two kind of motions, rotationsand parallel translations. He treated the latter as a kind of infinitesimally small rotations with rotation axis situated in the perpendicular direction at the infinite long distances. Thisis the general idea (Idée générale) throughout of this paper. Accordingly, he asserted (inn o and n o ) that “the properties of parallel translations are contained in the properties ofrotations ”. In his original expression,Ainsi donc, toute translation d’un système peut rigoureusement être consid-éré comme une rotation d’une amplitude infiniment petite autour d’un axe fixeinfiniment éloigné et normal à la direction de cette translation.On ne sera donc pas surpris de trouver ultérieurement toutes les propriétés des translations comprises dans celles des rotations, · · · · · · Thus, as methods of discussions, repeatedly he used calculations using infinitely smallpararell displacements ∆ x, ∆ y, ∆ z , of the directions of x -axis and so on, and limit transitionsfrom spheirical triangles to planar triangles. List of Titles and the corresponding numbers of p arts : Idée générale de la translation et de la rotation d’un système solide. n o ∼ Du déplacement d’un système d’un point fixe. n o (Displacement fixing a point, or a rotation arround a fixing point.) Du déplacement quelconque d’un système solide dans l’espace. n o ∼ De la composition des rotations successives d’un solide autour dedeux axes convergents. n o (Product of rotations arround two axis that intersect each other, or product of tworotations arround the intersecting point.)This is one of highlight points of the paper where the composition of two rotations is treated.Here jumping over the usual product structure in the rotation group SO (3) , there appearsthe product structure in its universal covering (double covering) group Spin (3) . Thit is the product formula in Rodrigues expression of rotations. I quote his original sentences at thisimportant critical point :Telle est la différence caractéristique à signaler entre la composition des rotationset celle des translations successives. Il y a d’ailleurs entre des deux sortes decomposition l’analogie qui existe entre les propriétés du triangle rectiligne et celles5u triangle sphérique; et si l’on compare les translations parallèles aux trois côtésd’un triangle rectiligne, aux sinus des demi-rotation accomplies autour des troiscôtés d’un angle trièdre, les valeurs des translations et celles de ces sinus serontégalement proportionnelles aux sinus des angles opposés aux côtés respectifs dansle triangle rectiligne et dans l’angle trièdre. (Explanation as I understand) The first sentence, in response to the previous sen-tences, refers to “difference between rotation composition and translation composition ”. Thefollowing sentence explains how to calculate the compositions of two rotations arround thesame center O . But it is difficult for me to translate this and relating portions of this n o O , take two point A and B and put n A = −→ OA, n B = −−→ OB .Denote by R ( φ A n A ) with φ A ∈ R the rotation arround the unit vector n A (as rotation axis)with the angle φ A to the direction of right-handed screw. Then the assertion is Assertion 3.1. The product R ( φ B n B ) R ( φ A n A ) of two rotations is expressed as R ( φ C n C ) with rotation axis n C = −−→ OC and rotation angle φ C given as follows: rotate the plane OAB arround n A by angle − φ A / , then we get a line (= big circle) on the sphere. Similarly rotatethe plane
OAB arround n B by angle φ B / , then we get another line on the sphere. Twolines intersect at a point C (the nearest intersecting point), and we put inner angle at C as π − φ C / (or outer angle φ C / ).Proof. We draw several auxiliary lines (cf. Figure 3 in [Alt1]). Rotate the plane
OAB arround n A by angle φ A / , then we get a line on the other side of ∆ ABC , and rotate
OAB arround n B by angle − φ B / , we get another line, and they cross each other at a point C ′ . ∆ ABC ′ is a mirror image of ∆ ABC . Rotate
OAB arround n A by angle − φ A , then we get aline on the other side of ∆ ABC , and rotate
OAB arround n B by angle φ B , we get anotherline, and they cross each other at a point A ′ . C is invariant under R ( φ B n B ) R ( φ A n A ) .In fact, under the first rotation R ( φ A n A ) , C is mapped to C ′ , and under the second R ( φ B n B ) , C ′ is mapped back to C . A is mapped to A ′ . In fact, under the first rotation, A is mapped to A , and underthe second, A is mapped to A ′ . ∠ ACA ′ = φ C .In fact, denote by D the crossing point of BC and AA ′ . Then ∆ ACD and ∆ DCA ′ aremutually mirror images of the other. So, ∠ ACA ′ = 2 ∠ ACD = 2( φ C /
2) = φ C . ✷ Thus Assertion 3.1 is proved and so sin( φ C / and cos( φ C / can be calculated usingknown formula in spherical trigonometry (cf. Note A3.1 in Appendix below). This givesa synthetic calculation method, and one can say that the calculation rule of quaternion hasappeared substantially here (cf. [Agn], [Alt1]). It seems that French mathematicians at thetime had rather considerable background in spherical trigonometry. The title
Assertion 3.1 is temporarily given here by me for convenience of quotation. Note that if we rotate the plane
OAB by angle π (a half of 2 π ), then it comes back to itself. omposition des rotations infiniment petites. n o Here the translations are treated as infinitesimally small rotations arround an axisat infinity, and discuss their compositions (= products).
De la composition des rotations autour de deux axes parallèles. n o ∼ Composition of rotations arround two axis parallel to each other.
De la composition des rotations autour d’axes fixes nonconvergents en nombre quelconques. n o Composition of rotations arround non-intersecting two axis, very complicated.
Examen du cas particulier des axes non convergents. n o Examination in the special case of two axis non-intersecting.
De la composition des déplacements successifs d’un systèmecombinés de rotations et de translations. n o ∼ On the composition of rotations and translations.
Équation de l’axe central. n o Equations which determines the axis of the composed rotation.
Examen du cas des variations infiniment petites. n o ∼ Examination of the case of infinitely small displacements.
De la composition analytique des rotations autour d’axes nonconvergents. n o Calculation formula for composition of rotations around two axis non-intersecting.
Composition des rotations successives autour de trois axesrectangulaires. n o Composition of rotations arround three orthogonal axis (this corresponds so-calledEuler product of rotation). De la composition des déplacements infiniment petits successifsd’un système solide. n o ∼ Conditions d’ équilibre de plusieurs déplacements successifs infinimentpetits. n o ( missing)Discussions on the realization of the state of Equilibrium , that is, the condition forthat, after successive displacements, the solid body comes back to the originalposition.
Analogie de ces lois de composition et d’équilibre avec celles de la composition et del’équilibre des forces appliquées à un système invariable. n o Discussions on the striking analogy (l’analogie frappante) between the above stateof
Equilibrium and
Equilibrium state when force is applied . De la détermination des variations des coordonées d’un système solid dues à un dé-placement quelconque de ce système, analytiquement déduites des conditions By the way, Euler discussed the existence of axis for a rotation, but he didn’t discuss Euler expression ofa rotation, as I understand. e l’invariabilité de ce système. n o ∼ Conclusion. — Loi générale de la Statistique. n o Élie Cartan quoted in n o
59. Représentation d’une rotation , p.57, in his book [Car2],1938, one of main results of Rodrigues as follows and gave a proof of his own. I quote thecentral part of the proof of Cartan:La formule (3) permet de retrouver les formules d’Euler-Olinde-Rodrigues.Soit L le vecteur unitaire porté sur l’axe de rotation et θ l’angle de rotation ;les deux vecteurs unitaires A, B ont pour produit scalaire cos θ et leur produitvectoriel ( AB − BA ) est égal à iL sin θ . On en déduit BA = cos θ − iL sin θ , AB = cos θ + iL sin θ , d’où (5) X ′ = (cid:16) cos θ − iL sin θ (cid:17) X (cid:16) cos θ + iL sin θ (cid:17) . Si l’on désigne par l , l , l les cosinus directeurs de L , les paramètres d’Euler-Olinde-Rodrigues sont les quatre quantités ρ = cos θ , λ = l sin θ , µ = l sin θ , ν = l sin θ , dont la somme des carrés est égal à 1.Here a matrix X is associated to a vector −→ x (as defined in n o ) as X = (cid:18) x x − ix x + ix − x (cid:19) ←→ −→ x = x x x ∈ R , and λE = diag ( λ, λ ) is identified with a scalar λ , where E is the identity matrix of order2. Take a rotation R ( θ l ) arround a unit rotation axis l = t ( l , l , l ) and of rotation angle θ .the matrix L = (cid:18) l l − il l + il − l (cid:19) is associated to the axis l , and X ′ is the image of X under R ( θ l ) . The matrices A and B are associated to unit vectors −→ a and −→ b respectively, chosenin such a way that ( AB + BA ) = h−→ a , −→ b i E = cos θ E , ( AB − BA ) = iL sin θ .Thus the formula (5) gives correctly Rodrigues expression of the rotation R ( θ l ) given inthe paper [R5-1]. One of the main results of the paper [R5-1], 1840, of Benjamin Olinde Rodrigues. The author’s name ofthis paper is written as Olinde Rodrigues. By definition, of the length 1. It is very much regrettable that Cartan did not give an exact reference to this paper, and he misunderstoodthe name of its author Olinde Rodrigues as names of two persons called Olinde and Rodrigues respectively(cf.
Comment 5.1 below). Explanation of this quotation)
Let a matrix X be associated to a vector −→ x as above.Then we have det X = − ( x + x + x ) E = −k−→ x k E , where k−→ x k denotes Euclidian normof −→ x . Moreover X = k−→ x k E , and A = E = 1 , A − = A for unit vector −→ a . Let Y beassociated to −→ y , then ( XY + Y X ) = (cid:10) −→ x , −→ y (cid:11) , ( XY − Y X ) = i −→ x ∧−→ y , where −→ x ∧−→ y denotes the vector product of −→ x and −→ y . Two vectors −→ x and −→ y are perpen-dicular to each other if and only if XY = − Y X , and in such a case −→ x ∧−→ y is called bivecteur (by Cartan) and is represented by − i XY = − i ( XY − Y X ) .In n o , a triplet H , H , H of × matrices is introduced as H = (cid:18) (cid:19) , H = (cid:18) − ii (cid:19) , H = (cid:18) − (cid:19) , and in n o , another triplet I j := − iH j ( j = 1 , , is introduced. Each of them has thefollowing relations respectively H j = 1 ( j = 1 , , , H j H k = − H k H j ( j = k ) , H H H = i,I j = − j = 1 , , , I j I k = − I k I j ( j = k ) , I I I = − . The matrix X associated to −→ x is X = x H + x H + x H , and { I , I , I } is a tripletsatisfying the fundamental formula of Quaternion (this is remarked in n o
57, Relation avecla théorie des quaternions ). Moreover { , H , H , H , i, I , I , I } gives a basis over R of M (2 , C ) of full matrix algebra of order 2 over C .Now we come to explain the meaning of the above quotation. Consider a unit vector −→ a and the reflection ( symétrie in [Car2]) Ref( −→ a ) with respect to the hyperplain orthogonal toit, which is given as −→ x ′ = −→ x − −→ a (cid:10) −→ x , −→ a (cid:11) . Translating this into the matrix form, we have Ref( −→ a ) X = X ′ = − AXA (4.1)In fact, X ′ = X − A
12 ( XA + AX ) = X − AXA − A X = − AXA ( ∵ A = 1) . Take another unit vector −→ b , then Ref( −→ b )Ref( −→ a ) X = BAXAB . Also take a vector −→ y perpendicular to −→ x and take bivecteur −→ u = −→ x ∧ −→ y which is represented by U := − i XY .Then, under Ref( −→ a ) , XY is transformed to X ′ Y ′ = A ( XY ) A = A ( XY ) A − , and so Ref( −→ b )Ref( −→ a ) U = ( BA ) U ( AB ) , AB = ( BA ) − .Thus stated, we should come back to the fundamental principle of Cartan’s idea for theproof of so-called les paramètres d’Euler-Olinde-Rodrigues above. In n o
10. Décompositiond’une rotation en un produit de symétries of [Car2], it is proved that, in the space ofdimension n over R or C , The fomula below is exacly ‘ La formule (3) ’ at the top of the above quotation from [Car2], p.57. oute rotation est le produit d’un nomble pair n de symétries. Hence, in the case of n = 3 over R , every rotation is a product of two reflections ( symétries ),that is, for any non-trivial rotation R , there exists two unit vectors −→ a and −→ b , with the anglefrom −→ a to −→ b smaller than π , such that R = Ref( −→ b )Ref( −→ a ) .The rotation axis −→ l ( = l in our notation) of R is a positive multiple of −→ a ∧ −→ b andlet the angle from −→ a to −→ b be θ . Then (cid:10) −→ a , −→ b (cid:11) = cos θ or ( AB + BA ) = cos θ , and ( AB − BA ) = i −→ a ∧−→ b = sin θ · iL .If we check the movement of Ref( −→ b )Ref( −→ a ) on the 2-dimensional plane spanned by −→ a and −→ b , then it is exactly the movement of R ( θ l ) , as we can see easily.Thus the assertion in the quotation above is newly proved by Cartan. Comment 5.1.
For the middle name Olinde of Rodrigues, I have checked the lists ofChristian Saints and the lists of traditional French boys’ names, downlorded from website.Curiously enough, I couldn’t find Olinde in these lists. I understood that this name is addedby his father under the order of Christian Church arround 1808, but in some literature it isexplained that this name Olinde, along with the second names of his brother and sisters, wastaken by his father from literary works and the like. Anyhow he signed to his mathematicalpapers as Olinde Rodrigues, not using Benjamin. The reason why, I cannot imagine, butthis seems to work against him. For instance, Élie Cartan misunderstood it as two person’snames, and in his book [Car2] used the terminology as les formules (and les paramètres ) d’Euler-Olinde-Rodrigues . My friend Prof. emeritus Michel Duflo helped me very much tosearch Rodrigues’ papers, difficult to take copies. He wrote me that he didn’t know thereexists French name Olinde in that time.
Comment 5.2.
About confusions on the first names of mathematicians, also there isthe case of J. Schur (1875–1941) and I. Schur (Issai Schur). About 90 years later of the caseof Benjamin Olind Rodrigues, it was the times when hidden (?) Jewish misanthropy turnedinto persecution by Nazi who held power. I cannot but shed tears in the latest years whenSchur was becoming very unhappy because of the persecution (Cf. [Hir6]).
Comment 5.3.
Some years ago, at about 2008, when I was checking systematicallywebsite files under the questioning title ‘mathematician Benjamin Rodreagues’ or somethingsimilar, the considerable date and time were necessary while throwing away useless files tolook for possibly valuable documents. In addition, there were many files which put wronginformations. For example, there appeared a file saying ‘the mathematician who wrote onlyone article during life’ (it means [R5-1]), and after many files passing, ‘the mathematician whowrote only two articles during life’, and so on. Also about his birth place and nationality, thereare estimations ‘Spain or Portgal from the spelling of family name’, and someone reported as‘I found a family documents in a Portuguese ancient document house, so it’s done’ and so on.Finally I found a file reporting a third paper of Rodrigues, and I felt somethig really curious10nd so decided to study seriously the situation, in partricular, search how many papers arethere of him etc. and wanted to collect copies of all of them.Once I asked Prof. M. Duflo to find out the above 3rd paper and so on in
Correspondencesur l’École Impériale Polytechnique, . Then, together with copies of all papers of his inJ. Math. Pures Appl., I could make a report on Rodrigues and sent him my draft. Afterthat I again asked him to find another document [R4*] in Bulletin Scientifique de la SociétéPhilomatique de Paris, as shown below in a part-copy of his e-mail :Cher Takeshi,Merci beaucoup pour ton intéréssant et amusant texte sur Olinde. Je suiscontent d’avoir pu t’aider à obtenir de la documentation. > Je te demandrai cette fois-ci encore de trouver l’article> \bibitem[R04]{Rodr04} Olinde Rodrigues,> Sur quelques propri\’et\’es des int\’egrales doubles et des rayons> de courbure des surfaces,> Bulletin Scientifique de la Soci\’et\’e Philomatique de Paris,> pp.34-36, 1815. [Signed \lq P.’ by Poisson.]
Cela a été difficile ! Un excellent exercice d’internet !Je ne l’ai pas trouvé dans les bibliothèques parisiennes, y compris la biblio-thèque nationale, jussieu, ihp, ens... J’ai peut-être mal cherché, je l’ai découvertaprès, il s’appelle souvent Rodrigue (sans s) ce qui rend les recherches difficiles.Je ne l’ai pas trouvé sur le site web de la société philomathique de paris. Enfait c’est amusant de consulter ce site web; je ne connaissais pas l’éxistence decette société; · · · · · · · · ·
Puis j’ai pensé a books.google.com. C’est genial : j’ai mis "philomathique paris1815" dans la boite de recherches, et j’ai pu lire le livre — et donc tu peux aussi lefaire. J’ignore dans quelle bibliothèque Google a scanné ce volume, probablementune bibliothèque d’une université americaine. · · · · · ·
Comment 5.4.
In this occasion (at the beginning of 2012), when I tryed again to lookarround website putting ‘ mathematician Benjamin Olinde Rodrigues ’ in the box of research,I was quite surprised that the situation has been completely changed from that of severalyears ago so that many files containing wrong data disappeared and the files, which shouldbe appeared even then, such as [AlOr] etc. appear ranked high enough.
Quoted from review [Dav] on the book [AlOr], AMS-LMS, 2005.Rodrigues produced only 17 mathematical papers but wrote extensively aboutsocial, economic, and political matters, on banking and on alleviating problems oflabor. From 1816 to 1837, Rodrigues produced no mathematical papers. Between1838 and 1845, he wrote eight, including one on transformation groups that someconsider his chef-d’oeuvre. Taken at face value, this is a remarkable achievement.11ow many of us could get back into mathematical shape after doing something else(writing reviews or becoming a provost, say) for two decades ? Perhaps Rodrigueswas theorematizing all along but didn’t have the time to write up his findingsproperly. He left no personal papers, so we can’t tell. We can safely conjecture,though, that he kept abreast of the contents of the mathematical journals of theday.Quoted from [URL1] on Rodrigues. · · · · · ·
Rather in 1807 Jews living in France were required to modify their familynames and in the following year they were required to add a name of French origin.At this point Olinde was added to Rodrigues name. · · · · · ·
Appendix. Quaternion, 3-dimensional Rotation Group andIntroduction to Theory of Reprentations of Groups
Section 1 is omitted. Section 2, Introduction and §2.1, are omitted. Below begins with§2.2. We add the character A to the top of section numbers and subsection numbers etc.
A2. Represent Complex Numbers and Quaternions bymeans of × Matrices
A2.2. Complex × matrices representing quaternions The total of quaternion number q = α + β i + γ j + δ k ( α, β, γ, δ ∈ R ) gives a non-commutative number field, and we denote it by H . Quaternion q with α = 0 is called purequaternion and we denote by H − their totality. Here i , j , k are imaginary units invented byW.R. Hamilton (1805–’65) satisfyng i = j = k = ijk = − , (2.1) ij = − ji = k , jk = − kj = i , ki = − ik = j , (2.2)and, together with 1, they form a basis of H over R . Hamilton discovered on the way ofmorning walk on the 16th Oct. 1843 that, with three imaginary units i , j , k satisfying theso-called fundamental formula (2.1), in the totality of q = α + β i + γ j + δ k , four arithmeticoperations are possible. His paper [Ham1] appeared in 1843. Besides that, he wrote in thenext day a detailed report about this dicovery to his friend J.T. Graves as a long letter, writtenwith a quill and ink, which is reprinted in 1844. In the printed form of Collected Works ofHamilton, 4 pages, total 167 lines [Ham2].Quaternion H is a linear algebra over R and can be immersed into M (2 , C ) . The immer-sion Ψ is given as an linear extension of the correspondence, with i = √− ∈ C , i → I = (cid:18) −
11 0 (cid:19) , j → J = (cid:18) ii (cid:19) , k → K = (cid:18) − i i (cid:19) Lecture Note for Summer Concentrated Course for Mathematics Master Course (one week) of NaraEducational University, 2008
12o that for α, α ′ ∈ R and q, q ′ ∈ H , (cid:26) Ψ( α q + α ′ q ′ ) = α Ψ( q ) + α ′ Ψ( q ′ ) , Ψ( qq ′ ) = Ψ( q )Ψ( q ′ ) . (2.3)The conjugate of q = α + β i + γ j + δ k ∈ H is defined as q = α − β i − γ j − δ k and thenorm of q by k q k = p qq = ( α + β + γ + δ ) / . Then we have q − = k q k − q . Problem 2.2.1.
Prove that the system of relations (2.2) is equivalent with the system ofrelations (2.1).
Problem 2.2.2.
Prove that the triplet of matrices { I, J, K } satisfies the similar relationsas the triplet { i, j, k } . Problem 2.2.3.
Prove the formula (2.3). Also prove det Ψ( q ) = k q k , and qq ′ = q ′ q (the order of the product is inverted). Note A2.2.4.
When Hamilton discovered quaternion on the way of morning walk, hewas near to a bridge, and at that time he curved the so-called fundamental formula on a stoneof the bridge. It is the formula (2.1). I read that even today some peoples of Departmentof Mathematics are used to take a morning walk to the bridge on the same date as the 16thOctober, the date of Grate Discovery.
A3. Quaternion and rotation group SO (3) , Hamilton’sdiscrepancies A3.1. Expression of a 3-dimensional rotation
After the discovery, Hamilton was pursuing applications of quaternion. One of the themeswas the problem of describing a rotation in 3D Euclidean space E . As is explained in §A2.1(omitted), a rotation in 2D Euclidean plane E ∼ = C can be expressed in a simple way bymultiplication of complex number with modulu 1, and so Hamilton was aiming for somethingsimilar to that with respect to quaternion.Let us take a bijective correspondence between the space H − of pure quaternions and 3DEuclidean space E given as H − ∋ x = x i + x j + x k x = t ( x , x , x ) = x x x ∈ E . (3.1)Here x is a vertical vector but we express it by a transposed horizontal vector to save space.The length of x is given by k x k = q x + x + x , and so we put k x k := k x k . We can express the length preserving isomorphism (3.1) by asymbol as H − ∼ = E (this modern symbol expression is powerful but didn’t exist in the timesof Hamilton !). 13ecall that a rotation in E fixing the origin is expressed by an orthogonal matrix U =( u ij ) i,j =1 ∈ SO (3) as x → x ′ = U x , where, with E the unit matrix of order 3, SO (3) = { U ∈ M (3 , R ); U t U = t U U = E , det U = 1 } . (3.2)On the other hand, denote by B the unit ball of H given as B := { a ∈ H ; k a k = 1 } ,then it is a group under multiplication. From the beginning, Hamilton estimated that, as inthe case of 2D rotation group SO (2) and the torus group T := { z ∈ C ; | z | = 1 } , The group B should be isomorphic to the group SO (3) . He took as above the space of pure quaternions H − as Euclidean space E and look forways of action of B on it. As we see, the simplest way of action is the left multiplication as L ( a ) : H − ∋ x ax ∈ H − ( a ∈ B ) . Alas ! for Hamilton. The image L ( a ) x = ax belongsto H − only when a is orthogonal to x , that is, h a , x i = ( ax + xa ) = 0 (Cf. Problem 3.1below). So the big and difficult problem for Hamilton was What kind of action of a ∈ B on H − ∼ = E gives an isomorphism from B onto SO (3) ? However, judging from the result, the above estimation of Hamilton was a misleadingwrong estimate or a wrong button, from the beginning. Before 40 years old, He discoveredquaterionĄC and after that he wrote several huge books e.g. [Ham3] and [Ham4], and triedto spread the theory of quaternion in the world, but it seems that the wrong buttons werestuck for his whole later life. In the essay [Alt2], Altmann wrote this situation in detail usingsome emotional terms such as
The sad truth or entirely unacceptable or Optical illusion or causing endless damage etc. Still more expressed as · · · · · · , and that Hamilton committed a serious error of judgement in basinghis parametrization on the special case of the rectangular transformation. (this is the transformation appeared in Problem 3.1 below). Problem 3.1.
Put B − := B ∩ H − and express a ∈ B as a = cos θ + sin θ w with w ∈ B − , θ ∈ R . Prove that, in case sin θ = 0 , we have ax ∈ H − for an x ∈ H − ∼ = E if and only if h x , w i = 0 , that is, x ⊥ w (perpendicular to each other).Also prove that, in that case, the left multication L ( a ) = L (cos θ + sin θ w ) induces onthe hyperplane w ⊥ := { x ∈ H − ; x ⊥ w } a rotation of angle θ arround the origin.Well now, what is the correct expression of the rotation group by means of quaternion ? An answer has been given substantially in the paper [R5-1] of Rodrigues in 1840, but it isignored historically until very recently, except an early comment by É. Cartan in [Car2].
Lemma A3.2.
For a ∈ B , we have a − = a , and the group B acts on H − ∼ = E through T ( a ) : H − ∋ x −→ x ′ = axa − = axa ∈ H − , (3.3) 14 hat is, T ( a ) T ( b ) = T ( ab ) ( a , b ∈ B ) .Proof. Since qq ′ = q ′ q , we have x ′ = a x a = a ( − x ) a = − x ′ , whence x ′ ∈ H − .Moreover T ( a ) T ( b ) x = a ( bxb ) a = ( ab ) x ( ab ) = T (cid:0) ab (cid:1) x . ✷ Lemma A3.3.
For a w ∈ B − = H − ∩ B , put g w ( θ ) = cos θ + sin θ w ( θ ∈ R ) . Then θ g w ( θ ) is a one-parameter subgroup of B , and dg w ( θ ) dθ (cid:12)(cid:12) θ =0 = w .Proof. For θ, θ ′ ∈ R , we have, from w = − , g w ( θ ) g w ( θ ′ ) = (cid:0) cos( θ ) cos( θ ′ ) − sin( θ ) sin( θ ′ ) (cid:1) + (cid:0) sin( θ ) cos( θ ′ ) + cos( θ ) sin( θ ′ ) (cid:1) w = g w ( θ + θ ′ ) . ✷ Problem 3.4.
For a w ∈ B − , take u , v ∈ B − in such a way that u , v } , { w gives aright-handed orthonormal coordinate system. Then uv = w , vw = u , wu = v , and for g w ( θ ) , we have T ( g w ( θ )) u = cos(2 θ ) u + sin(2 θ ) v ,T ( g w ( θ )) v = − sin(2 θ ) u + cos(2 θ ) v ,T ( g w ( θ )) w = w . (3.4)The matrix expressin of T ( g w ( θ )) with respect to the basis { u , v , w } is cos(2 θ ) − sin(2 θ ) 0sin(2 θ ) cos(2 θ ) 00 0 1 . (3.5) Theorem A3.5.
The group B is a double covering and universal covering of rotationgroup SO (3) and a covering map is given by T ( a ) ( a ∈ B ) .Proof. The map T is surjective. In fact, it is known that any rotation g ∈ SO (3) hasa non-zero invariant vector w ∈ E and so it is a rotation of of some angle φ arround w . Takethe vector w ∈ H − ∼ = E corresponding to it and put θ = φ/ . Then, as seen from (3.5) inProblem 3.4, we have T (cos θ + sin θ w ) = g . The kernel of T is {± } ⊂ B . In fact, as seen from (3.4), T ( g w ( θ )) = E (unitmatrix) if and only if θ ≡ mod π ) , whence θ ≡ mod π ) and so g w ( θ ) = ± . The unit ball B is topologically homeomorphic to 3-dimensional sphere S , and issimply connected. ✷ Problem 3.6.
Prove the following. For q ∈ H , define exp q by an abosolutely convergentinfinite series as exp q = ∞ X n =0 q n n ! = 1 + q + q
2! + · · · . (3.6) 15hen, for w ∈ B − , exp( θ w ) = cos θ + sin θ w = g w ( θ ) ( θ ∈ R ) . The most important point of above dicussions is that, under the correspondence H − ∋ θ w → exp( θ w ) ∈ B → T (cid:0) exp( θ w ) (cid:1) ∈ SO (3) (3.7)with θ ∈ R , w ∈ B − , the angle of rotation is doubled as θ → θ as is shown in (3.5). Thismeans that a = exp( θ w ) and − a = exp (cid:0) ( θ + π ) w (cid:1) have the same image T ( a ) = T ( − a ) ,and the map T : B → SO (3) is a 2:1 correspondence. Hamilton seems to have been insistingparticularly, with the great pioneer’s stubbornness, to obtain 1:1 correspondence. Accordingto some biographies, Hamilton became eventually to suffer from excessive alcohol intake [Bell].Altmann points out, as one of the reasons, a serious psychological distress in this rotationexpression problem. Looking at the cause of his worries, from the present age of mathematicalstandard, I can suspect that, in this problem there are two different objects such as1) object which operate on something (operators),2) object to be operated (operands),however they both are the same quaternion and might be confused mutually or might not beclearly distinguished.Dear readers ! You may not feel much sympathy to Hamilton’s serious anxiety, whenreading this explanation. However it is because firstly you have been taught already undera modern mathematical basic training, and secondly here the author (Hirai) have chosenadequate notation in such a way thatfor the object 1), the characters such as a , b , g w etc.,for the object 2), the characters such as x , y etc.Thus, because of the hints drawn carefully for the reader, your understanding is unconsciouslyguided in the right direction.I have already mentioned the misunderstandings that Hamilton had. For his life and alsoabout quaternion, somewhat ironic story telling can be found in websitehttp://members.fortunecity.com/jonhays/clifhistory.htmand the author of this website MR. jonhays noted that at age 17 he read about Hamilton in Men of Mathematics by Eric Temple Bell (1883–1960). Problem 3.7 (formula for calculation).
For u, v ∈ H − , the corresponding elementsin E are denoted by u = t ( u , u , u ) , v = t ( v , v , v ) , and their inner product is defined as h u, v i := u v + u v + u v and denoted by u · v = h u, v i . Then, prove the following formula: uv = − u · v + u × v , (3.8) where u × v ∈ H − , u × v := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i u v j u v k u v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Problem 3.8.
Prove the following formula: u × v = − v × u , ( u × v ) ⊥ u , ( u × v ) ⊥ v . (3.9) Chapter 19,
An Irish Tragedy , pp.340–361. A3.2.1. Rodrighues parameter θ w ∈ H − ( θ ∈ R , w ∈ B − ) It was proved by Euler that any rotation ρ in 3D Euclidean space E arround the originhas necessarily a rotation axis. Let w ∈ E , k w k = 1 , be the axis and θ the angle of ρ arroundthe axis w in right-handed screw rotation. Then, as is shown above, ρ is expressed in (3.7) as ρ = R ( θ w ) := T (cid:0) (exp( θ w ) (cid:1) ∈ SO (3) , (3.10)where w ∈ H − ∼ = E corresponds to w . We call this expression as Rodrigues expression ,and ( θ, w ) ∈ R × B − or θ w ∈ H − as Rodrigues parameter of rotation ρ .This expression is, unlike the expression by means of Euler angles (cf. §A3.3 below), theparameters are seamless, and locally univalent but globally multivalent. If one wishes to makecompletely univalent and put some restriction on θ w , there appears inevitably some breaksin the parameter. So that, as parameter space, it is natural to take the whole space H − and enjoy the advantage of capability of describing smoothly multi-rotations of machines orairplanes etc. A3.2.2. Product formula for two rotations
The main contribution of Rodrigues is the description of the product of two rotations R ( θ w ) R ( θ ′ w ′ ) = R ( θ ′′ w ′′ ) . To describe Rodrigues parameters θ ′′ w ′′ from θ w and θ ′ w ′ , wecalculate it according to the quaternion product rule in H as (cid:0) cos( θ ) + sin( θ ) w (cid:1)(cid:0) cos( θ ′ ) + sin( θ ′ ) w ′ (cid:1) = cos( θ ′′ ) + sin( θ ′′ ) w ′′ . It gives us the so-called
Rodrigues formula in [R5-1] in our notations as cos( θ ′′ ) = cos( θ ) cos( θ ′ ) − sin( θ ) sin( θ ′ ) w · w ′ , sin( θ ′′ ) w ′′ = cos( θ ) sin( θ ′ ) w ′ +sin( θ ) cos( θ ′ ) w ++ sin( θ ) sin( θ ′ ) w × w ′ . (3.11)Originally he induced his product formula (equivalent to the above) from some formulas forspherical trigonometric functions, and thus we may say that Rodrigues substantially gave theproduct rule for quaternion, in advance of Hamilton. Note A3.9.
When θ, θ ′ are both small, we can obtain the first approximation from thesecond equation above as θ ′′ w ′′ + θ ′ w ′ + θ w . Furthermore if w , w ′ are near to each other,then the first approximation is θ ′′ + θ + θ ′ , w ′′ + ( w + w ′ ) . (3.12)Note that, in the case of Euler angle expression, there does not exist such an approximation. Note A3.10.
Basic formulas for spherical trigonometry are given as follows. For aspherical triangle
ABC , let the interior angles be α, β, γ , the opposite side lengths be a, b, c ,the area be S , and the radius of the sphere be ρ . Then a formula Spherical Excess is α + β + γ − π = S/ρ > , (3.13) 17elow, put radius ρ = 1 , then sin formula sin a sin α = sin b sin β = sin c sin γ , (3.14) cos formula cos a = cos b cos c + sin b cos c cos α, cos b = cos c cos a + sin c cos a cos β, cos c = cos a cos b + sin a cos b cos γ, (3.15) cos formula cos α = − cos β cos γ + sin β sin γ cos a, cos β = − cos γ cos α + sin γ sin α cos b, cos γ = − cos α cos β + sin α sin β cos c, (3.16) sin cos formula sin a cos β = cos b sin c − sin b cos c cos α, sin b cos γ = cos c sin a − sin c cos a cos β, sin c cos α = cos a sin b − sin a cos b cos γ. (3.17)In (3.14)–(3.17), the radius ρ = 1 , and so the side length is equal to the angle measured inradians. For example, the length c of AB is equal to ∠ AOB ( O is the center of sphere). Thefirst formula in (3.11) comes from the 3rd formula in (3.16). In fact, α = θ, β = θ ′ , γ = π − θ ′′ , c = ∠ AOB, cos c = w · w ′ . A3.3. Expression of 3D rotation by means of Euler angles
Let { e , e , e } be an orthonormal system giving coordinate in E as x e + x e + x e ↔ x = t ( x , x , x ) . Denote by g ( θ ) a rotation arround e at right-handed screw angle θ , andsimilarly g ( θ ) , g ( θ ) for e , e respectively, then g ( θ ) = θ − sin θ θ cos θ , g ( θ ) = cos θ θ − sin θ θ , g ( θ ) = cos θ − sin θ θ cos θ
00 0 1 . A rotation ρ of E or ρ ∈ SO (3) can be expressed as ρ = g ( ϕ ) g ( θ ) g ( ψ ) ( − π < ϕ, ψ ≤ π, ≤ θ ≤ π )= cos ϕ − sin ϕ ϕ cos ϕ
00 0 1 cos θ θ − sin θ θ cos ψ − sin ψ ψ cos ψ
00 0 1 = cos ϕ cos θ cos ψ − sin ϕ sin ψ − sin ϕ cos θ cos ψ − cos ϕ sin ψ sin θ cos ψ cos ϕ cos θ sin ψ + sin ϕ cos ψ − sin ϕ cos θ sin ψ + cos ϕ cos ψ sin θ sin ψ − cos ϕ sin θ sin ϕ sin θ cos θ . When we apply Euler angle expression to calculations such as product of two rotations,we encounter immediately some dificulties, for instances: The computational load is heavy. In fact, to calculate the product of two rotations,first calculate the product of 6 matrices of Euler angles, then decompose the product into 3Euler angle components again. 18 ) It is not possible to evaluate calculation errors. Even if the change of rotations aresmall enough, the deviations between their Euler angles can be very big, sometimes therewould be jumps.For more details, you can read Altmann’s text book [Alt1].
A4. Applications of Rodrigues expression, Time derivative
Recently, in many directions, Rodrigues expression of rotation is applied adequately. Itsparameter is given by θ w ∈ H − ( θ ∈ R , w ∈ B − ) and the rotation R ( θ w ) = T (cid:0) (exp( θ w ) (cid:1) acts on x ∈ H − ∼ = E as H − ∋ x → exp( θ w ) x exp( θ w ) − (4.1) = (cid:0) cos( θ ) + sin( θ ) w (cid:1) x (cid:0) cos( θ ) − sin( θ ) w (cid:1) ∈ H − ∼ = E , A4.1. Examples of application in various fields
In geophysics, it is important to describe rotations for problems such as : • In plate tectonics, describe plates movement according to geological time, by means ofa rotation which leaves the earth center invariant. • In geodesy, describe the relationship of the inertial coordinate system of the universe,which is the basis of Newton’s equation of motion, with the Earth coordinate system. Thisis used for satellite orbit calculation. • In seismology, it is necessary to quantify the two rotations “ difference ” to quantify howmuch the fault plane deviates from the reference plane.For more purposes such as • Computer graphics, • Aircraft design, spacecraft attitude control, dynamics such as aviation.For more detailed comments, see e.g. [Agn], 2006.
A4.2. Time derivative of rotation in Rodriques expression
When we study rotations in a dynamical system, such as in aeronautical engineering, therotation that depends on the time t is treated. We give here an interesting compact formulafor time derivative of rotation, which can be easily applied in many purposes.In case θ and w depend on the time t , put Q ( t ) := R ( θ w ) , and let us study its timederivative ˙ Q ( t ) = dQ ( t ) dt . Theorem A4.1.
For w ∈ B − ( ⊂ H − ) depending on t , we have ˙ ww = − w ˙ w and so ˙ w ⊥ w in H − and ˙ ww = ˙ w × w , ⊥ ˙ w , ⊥ w (mutually orthogonal). The time derivative ˙ Q ( t ) of rotation Q ( t ) is given as ˙ Q ( t ) x = Q ( t ) (cid:16)(cid:2) ˙ θ w + sin θ ˙ w + (1 − cos θ ) ˙ ww (cid:3) × x (cid:17) , (4.2) 19 r Q ( t ) − ˙ Q ( t ) x = (cid:2) ˙ θ w + sin θ ˙ w + (1 − cos θ ) ˙ ww (cid:3) × x , where x ∈ H − ( ∼ = E ) is a fixed vector, and for a , b ∈ H − , a × b denotes the vector productin the 3-dimensional vector space H − .Proof. Differentiate the both side of w = ww = − with respect t , then ˙ w w + w ˙ w = ∴ w ⊥ ˙ w (in H − ) ∴ ˙ w w = ˙ w × w . (4.3) ddt exp( θ w ) = ddt (cid:0) cos( θ ) + sin( θ ) w (cid:1) == (cid:0) − sin( θ ) + cos( θ ) w (cid:1) ˙ θ + sin( θ ) ˙ w = exp( θ w ) ˙ θ w + sin( θ ) ˙ w . Therefore ddt Q ( t ) x = ddt (cid:8) exp( θ w ) x exp( − θ w ) (cid:9) = ddt (cid:0) exp( θ w ) (cid:1) x exp( − θ w ) + exp( θ w ) x ddt (cid:0) exp( − θ w ) (cid:1) = exp( θ w ) (cid:0) ˙ θ (cid:1)(cid:0) wx − xw (cid:1) exp( − θ w ) ++ exp( θ w ) (cid:8)(cid:0) cos( θ ) − sin( θ ) w (cid:1) sin( θ ) ˙ wx ++ x (cid:0) − sin( θ ) ˙ w (cid:1)(cid:0) cos( θ ) + sin( θ ) w (cid:1)(cid:9) exp( − θ w )= exp( θ w ) (cid:8) ˙ θ w × x + cos (cid:0) θ (cid:1) sin (cid:0) θ (cid:1) ( ˙ wx − x ˙ w (cid:1) − sin (cid:0) θ (cid:1) (cid:0) w ˙ w x + x ˙ w w ) (cid:9) exp( − θ w ) (cid:0) use (4.3) (cid:1) = Q ( t ) (cid:8)(cid:2) ˙ θ w + sin θ ˙ w + (1 − cos θ ) ˙ ww (cid:3) × x (cid:9) . ✷ (Omitted Below) Acknowledgements.
The author would like to express heartful thanks to Prof. M. Duflofor documents of Rodrigues and to Prof. Kyo Nisiyama for documents of Hamilton.
References [**]
Papers of Benjamin Olinde Rodrigues (1795-1851): [R01]
Sur le mouvement de Rotation des corps libres, par M.
Rodrigues, licencié ès-sciences ,Correspondence sur l’École Impériale Polytechnique, (1814–1816): pp.32–36, 1814.[R02] De l’angle de contingence d’une courbe à double courbure, par M.
Rodrigues, licencié ès-sciences , ibid., (1814–1816): pp.36–37, 1814.[R03] Sur la résistance qu’éprouve un point matériel assujetti à se mouvoir sur une courbe donnée,par M.
Rodrigues, licencié ès-sciences , ibid., (1814–1816): pp.37–39, 1814.[R04*] Sur quelques propriétés des intégrales doubles et des rayons de courbure des surfaces; par
M. Rodrigue [nom mal écrit, sans s, à la place de Rodrigues, avec s], Bulletin Scientifique dela Société Philomatique de Paris, pp.34-36, 1815. [Extraits écrit par S.D. Poisson, et signéP. à la fin de l’article.]
Note.
This is not written by Rodrigues but written by Poisson, on the paper [R08]. R05]
De la manière d’employer le principe de la moindre action, pour obtenir les équations dumouvement, rapportées aux variables indépendantes, par M.
Rodrigues, licencié ès-sciences ,Correspondence sur l’École Impériale Polytechnique, (1814–1816): pp.159–162, 1815.[R06] Recherches sur la théorie analytique des lignes et des rayons de courbure des surfaces, etsur la transformation d’une classe d’intégrales doubles, qui ont un rapport direct avec lesformules de cette théorie, par M.
Rodrigues, ibid., (1814–1816): pp.162–179, 1815.[R07] Addition aux recherches précédentes, par M.
Rodrigues, ibid., (1814–1816): pp.180–182,1815.[R08] Mémoire (*) sur l’attraction des sphéroïdes, par M.
Rodrigues,
Docteur ès-siences.
PRE-MIÈRE PARTIE.
Formules générales pour l’attraction des corps quelquonques, et applica-tion de ces formules à la sphère et aux ellipsoïdes.
SECONDE PARTIE.
Attraction desSphéroïdes infiniment peu différens d’une sphère, et dévelopement générale de la fonction V. ibid., (1814–1816): pp.361–385, 1816. (*) Ce Mémoire a été le sujet d’une thèse soutenue pour le doctorat, devant la Faculté des Siencesde Paris, le 23 juin 1815, sous la présidence de M. Lacrois, Doyen de la Faculté. [ Note ] Correspondence sur l’École Polytechnique was founded in 1804 by J.N.P. Hachette, Professorof the school at that time, and ended when he was forced to resign the post on 1816 becauseof the
Restauration . Three volumes were published up to 1816. •• No papers of Rodrigues were published between 1817–1837 •• [R3-1] Olinde Rodrigues, Sur le nombre de manière de décomposer un Polygone en triangles aumoyen de diagonales,
J. Math. Pures et Appl., (1838), 547–548.[R3-2] Olinde Rodrigues, Sur le nombre de manière de d’effectuer un produit de n facteurs, ibid., (1838), 549–549.[R3-3] Olinde Rodrigues, Démonstration élémentaire et purement algébrique du développement d’unbinome élevé à une puissance négative ou fractionnaire, ibid., (1838), 550–551.[R4-1] Olinde Rodrigues, Note sur les inversions, ou dérangements produits dans les permutations, ibid., (1839), 236–240.[R5-1] Olinde Rodrigues, Des lois géométriques qui régissent les déplacements d’un système solidedans l’espace, et la variation des coordonnées provenant de ses déplacements considérésindépendamment des causes qui peuvent les produire, ibid., (1840), 380–440. (The part ofthis paper on the composition of rotations may be thought of as a discovery – predatingW.R. Hamilton – of the quaternions)[R8-1] Benjamin (Olinde) Rodrigues, Du développement des fonctions trigonométriques en produitsde facteurs binômes, ibid., (1843), 217–224.[R8-2] Benjamin (Olinde) Rodrigues, Note sur l’évaluation des arcs de cercle, en fonction linéairedes sinus ou des tangentes de fractions de ces arcs, décroissant en progression géométrique, ibid., (1843), 225–234.[**] Related Papers and Literatures [Agn] D. Agnew,
Finite Rotations , 2006 (in website).[Alt1] S. Altmann,
Rotations, Quaternins, and Double Groups , Clarendon Press, 1986. Alt2] S. Altmann,
Hamilton, Rodrigues, and the Quaternion scandal,
What went wrong with one ofthe major mathematical discoveries of the nineteenth century,
Mathematics Magazine , (1989),291-308.[AlOr] S. Altmann and E. Ortiz edit., The Rehabilitation of Olinde Rodrigues, Mathematics andSocial Utopias in France: Olinde Rodrigues and His Times , Amer. Math. Soc. and LondonMath. Soc., 2005, 168 pages.[Bell] E. Bell,
Men of Mathematics , Simon & Schuster (1st v. 1937), 1986 (paperback; ISBN 0-671-62818-6).[Car1] É. Cartan, Les groupes projectifs qui ne laissent invariante aucune multiplicité plane, Bull.Soc. Math. France, (1913), 53-96.[Car2] É. Cartan, Leçons sur la théorie des spineurs
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