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entropy Article FromTools inSymplecticandPoissonGeometry to J.-M.Souriau’sTheoriesofStatisticalMechanics andThermodynamics † Charles-MichelMarle InstitutdeMathématiquesdeJussieu,UniversitéPierreetMarieCurie,4,Place Jussieu,75252ParisCedex05, France; charles-michel.marle@math.cnrs.fr † Inmemoryof Jean-MarieSouriau(1922–2012). AcademicEditors: FrédéricBarbarescoandFrankNielsen Received: 28 July2016;Accepted: 5October2016;Published: 19October2016 Abstract: I present in thispaper some tools in symplectic andPoissongeometry inviewof their applications in geometricmechanics andmathematical physics. After a short discussion of the LagrangiananHamiltonianformalisms, includingtheuseofsymmetrygroups,andapresentation of the Tulczyjew’s isomorphisms (which explain some aspects of the relations between these formalisms), I explain the concept of manifold of motions of a mechanical system and its use, due to J.-M.Souriau, instatisticalmechanicsandthermodynamics. Thegeneralizationof thenotion of thermodynamicequilibriuminwhichtheone-dimensionalgroupof timetranslations is replaced by amulti-dimensional,maybenon-commutativeLie group, is fully discussed and examples of applications inphysicsaregiven. Keywords:Lagrangianformalism;Hamiltonianformalism;symplecticmanifolds;Poissonstructures; symmetrygroups;momentummaps; thermodynamicequilibria;generalizedGibbsstates 1. Introduction 1.1. Contentsof thePaper,SourcesandFurtherReading Thispaperpresents tools in symplectic andPoissongeometry inviewof their application in geometric mechanics and mathematical physics. The Lagrangian formalism and symmetries of Lagrangiansystemsarediscussed inSections2and3, theHamiltonian formalismandsymmetries of Hamiltonian systems in Sections 4 and 5. Section 6 introduces the concepts of Gibbs state and of thermodynamic equilibrium of a mechanical system, and presents several examples. For a monoatomic classical ideal gas, eventually in a gravity field, or a monoatomic relativistic gas theMaxwell–BoltzmannandMaxwell–Jüttnerprobabilitydistributionsarederived. TheDulong andPetit lawwhichgoverns the specificheat of solids is obtained. Finally Section 7presents the generalizationof theconceptofGibbsstate,due to Jean-MarieSouriau, inwhich thegroupof time translations is replacedbya(multi-dimensionalandeventuallynon-Abelian)Liegroup. Several books [1–11] discuss, much more fully than in the present paper, the contents of Sections2–5. The interested reader is referred to these books for detailedproofs of resultswhose proofsareonlybrieflysketchedhere. Therecentpaper [12]containsdetailedproofsofmost results presentedhere inSections4and5. ThemainsourcesusedforSections6and7are thebookandpapersby Jean-MarieSouriau[13–17] andthebeautiful smallbookbyMackey[18]. The Euler–Poincaré equation,which is presentedwith Lagrangian symmetries at the end of Section3, isnot reallyrelatedtosymmetriesofaLagrangiansystem,since theLiealgebrawhichacts Entropy2016,18, 370 3 www.mdpi.com/journal/entropy