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La Mathématisation De La Mécanique Relativiste Mémoire De M2 HPDS Sous La Direction De M. Christophe Eckes Et M. Amaury Thuil

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La Mathématisation De La Mécanique Relativiste Mémoire De M2 HPDS Sous La Direction De M. Christophe Eckes Et M. Amaury Thuil La mathématisation de la mécanique relativiste Johann Colombano-Rut [email protected] Mémoire de M2 HPDS Sous la direction de M. Christophe Eckes et M. Amaury Thuillier Université Claude Bernard Lyon 1 2009 Réalisé en LATEX Remerciements J’adresse tous mes remerciements aux directeurs de ce mémoire M. Christophe Eckes et M. Amaury Thuillier pour leur disponibilité et leur support, et pour l’aide précieuse qu’ils m’ont apporté dans l’élaboration de ce travail. 2 Sommaire Introduction 4 I. La relativité restreinte et sa géométrisation 6 1. Cinématique classique et relativiste . .6 a) Relativité galiléenne . .6 Transformations de Galilée b) Relativité poincaréenne (1902-1905) . .9 c) Relativité einsteinienne (1905) . 12 Transformations de Lorentz d) Histoire et rôle secondaire de la théorie des groupes chez Einstein . 17 2. La géométrisation de la relativité restreinte (1907-1914) . 19 a) Göttingen, centre mondial des mathématiques de la fin du XIXe siècle . 19 b) La conférence de Cologne (1908) . 21 c) L’espace-temps de Minkowski . 22 Groupe de Galilée Groupe de Lorentz Interprétation hyperbolique d) Apports de cette géométrisation . 31 e) Nécessité de cette géométrisation . 32 Plus qu’un outil, un univers Une chrono-géométrie objective 3. L’axiomatisation . 36 a) Une théorie à principes . 37 b) L’axiomatisation de la physique (1900 -) . 37 c) L’axiomatisation de la relativité restreinte (1907 -) . 39 II. Relativité restreinte et programme d’Erlangen 43 1. Présentation du programme d’Erlangen (1871-1872) . 43 a) Unification des géométries . 44 b) La géométrie par les groupes de transformations . 46 c) Sous-groupes et géométries subordonnées . 47 2. Application de la théorie des groupes à la géométrie . 48 a) La géométrie projective . 49 b) La géométrie affine . 52 c) Les géométries affines euclidiennes et non-euclidiennes . 53 3. Application de la théorie des groupes à la relativité restreinte (1910-1914) . 56 a) L’espace de Minkowski et son groupe principal . 57 b) Schéma récapitulatif : Transformations affines en dimension 4 ................ 58 III. Conclusion 60 Bibliographie 61 Annexe 1 : Histoire des transformations de Lorentz 65 Annexe 2 : Discussion de priorité entre Einstein et Poincaré 69 Annexe 3 : L’ellipse de Steiner 72 3 Introduction « L’œuvre qui appartient en propre au génie consiste ici dans l’observation de la dépendance de certaines déterminantes partielles des phénomènes méca- niques. L’énonciation formelle et précise de cette dépendance fut au contraire l’ouvrage du travail circonspect qui élabora les différents concepts et les diffé- rents principes de la mécanique. Ce n’est que par la recherche de leurs sources historiques que l’on peut déterminer la véritable valeur et le sens de ces prin- cipes et de ces notions. » [Mach1], La mécanique, 1883, p243. Le développement de la mécanique est indissociable de celui des mathématiques. Le terme mécanique désigne à l’époque de Galilée à la fois la science du mouvement et la construction de machines (les arts mécaniques du moyen-âge). Son ouvrage Le meccaniche (1593) est même centré sur la statique, science de l’équilibre. La polysémie du terme mécanique est encore manifeste par le pluriel du titre de sa traduction française Les méchaniques de Galilée publiée en 1674 par le père Marin Mersenne (1588-1648). Dans l’Encyclopédie, ou dictionnaire raisonné des sciences, des arts et des métiers (1772) de Denis Diderot (1713-1784) et Jean le Rond D’Alembert (1717-1783), la Méchanique est définie comme « partie des mathéma- tiques mixtes, qui considère le mouvement et les forces motrices, leur nature, leurs lois et leurs effets dans les machines »(p10 :222). Nous ne nous intéresserons bien sûr pas à la mécanique des machines, mais à la mécanique dite "classique", qui s’exprime depuis Isaac Newton (1643-1727), c’est-à-dire après la science du mouvement du XVIIème siècle, dans un cadre rigide et immuable, fondé sur des notions de temps et d’espace indépendantes et absolues, et en particulier sur l’existence d’un système de référence au repos absolu. Cet espace et ce temps universels sont identiques pour tous les observateurs, et c’est en leur sein que furent décrites les lois de la statique concernant les corps en équilibre, plus généralement de la cinématique pour les corps en mouvement et de toute la physique contemporaine de Newton. Après trois siècles d’évolution cumulée 1, la mécanique voit l’élargissement du domaine de validité du principe de relativité, qui passe de la mécanique seule à toutes les lois de la physique, dont l’électromagnétisme, et lui attribue un rôle unificateur. Le cadre de la physique évolue alors de façon significative, perdant tout caractère absolu : l’espace unique, fondamentalement immobile, sous le nom d’éther n’est plus nécessaire et ce nouveau cadre modifie considérablement la structure des lois physiques, c’est la cinématique relativiste. Il est accompagné d’une géométrisation et d’une mathématisation remarquables auxquelles participent plusieurs mathématiciens comme Henri Poincaré (1854-1912). De la notion de fonction formalisant les relations entre grandeurs physiques, aux structures algébriques et géo- métriques caractérisant des régularités dans les lois de la nature, en passant par les équations différentielles formalisant des lois de comportement des grandeurs physiques, l’historien montre un lien étroit entre le déve- loppement de la physique et des mathématiques. Tout comme les théorèmes mathématiques, les lois physiques peuvent-elles être soumises à un ordre logique, et reposer sur une axiomatique ? Après plusieurs tentatives d’axiomatisation de la mécanique, celle de la physique est un programme de recherche souligné par le sixième problème de David Hilbert (1862-1943), partiellement établi avec l’axiomatisation de la relativité restreinte et générale. Nous nous intéresserons à la première de ces deux théories physiques, et nous analyserons, depuis l’article fon- dateur d’Albert Einstein Sur l’électrodynamique des corps en mouvement (1905), les apports mathématiques d’Hermann Minkowski (1864-1909) en géométrie, de Félix Klein (1849-1925) en théorie des groupes, ainsi que l’héritage de ces différents aspects de la "mathématisation" de la théorie de la relativité restreinte. Nous désignerons par le terme de mathématisation le développement formel de la théorie physique de la relativité, centré sur l’expression mathématique des transformations (changements de variables entre repères mathéma- tiques) qu’elle met en jeu, des propriétés géométriques et algébriques (structurelles) de ces transformations et de l’espace dans lequel elles sont exprimées, ainsi que de l’axiomatisation de cette théorie. Cette mathématisation est-elle un processus pratique mais non essentiel à l’expression des lois physiques, ou bien est-elle révélatrice d’un lien profond avec les lois de la nature ? Nous observerons les points de vue des mathématiciens et physiciens de l’époque sur la nature de ce lien, tantôt purement fictif, comme dans le conven- tionnalisme de Poincaré, ou objectif, comme dans l’harmonie préetablie de Minkowski. Nous n’avons pas pour ambition de présenter une philosophie générale du lien entre physique et mathéma- 1. Nous nous autorisons un saut historique important car il serait chimérique de vouloir résumer de façon à la fois concise et exhaustive l’histoire de la mécanique classique, et ses relations avec les mathématiques durant ces trois siècles, qui sortent largement de notre périodisation. 4 tiques, mais nous nous contenterons d’exposer, au prix d’une présentation technique d’outils mathématiques indispensables, des éléments de connaissance représentant, selon nous, l’un des meilleurs exemples des interac- tions complexes entre ces deux disciplines. Notre étude historique, questionnant le rôle des mathématiques dans la théorie de la relativité restreinte, com- mencera autour de la formulation du principe de relativité par Poincaré en 1902, en passant par l’article d’Ein- stein de 1905, le formalisme de Minkowski dans son article Espace et temps (1909), jusqu’à la reformulation de Klein en 1910 2. Cette dernière reformulation, dans le cadre de la théorie des groupes, nécessitera de préciser rapidement l’histoire interne du concept de groupe, prenant place tout au long du XIXème siècle 3. Nous tenterons d’éclairer l’appari- tion de ce concept mathématique dans la physique relativiste, en établissant les raisons de son rôle secondaire dans l’article d’Einstein de 1905. Ces réflexions spécifiques sur l’histoire du concept de groupe introduiront tou- tefois notre étude du programme d’Erlangen de Klein, qui repose essentiellement sur ses articles Sur la géométrie dite non euclidienne (1871) et Considérations comparatives sur les recherches géométriques modernes (1872), programme de recherche central dans la reformulation mathématique de la relativité restreinte. Cette étude historique se doublera d’une analyse épistémologique du principe de relativité chez Poincaré et Einstein, puis des concepts d’espace (d’espace-temps chez Minkowski), et sur la nature de la géométrie, ces points éclairant particulièrement les relations complexes et dynamiques entre la physique et les mathématiques. En particulier, nous verrons que le statut épistémologique du principe de relativité est fortement lié à sa for- malisation en un cadre géométrique structurel de la théorie physique. De plus, des considérations telles que la quadridimensionnalité de l’espace-temps jouèrent un rôle important dans le déplacement de l’intérêt des physiciens, de l’étude des mouvements
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