The Geometrization of Dynamical Systems

  • José F. Cariñena
  • Alberto Ibort
  • Giuseppe Marmo
  • Giuseppe Morandi


In this chapter we would like to move a step forward and discuss the notions discussed in the previous chapter in such a way that they will not depend on being defined on a linear space. Such carrier space, as it was suggested before would be closely related to the notion of the space of ‘states’ of our system, and it is not always true that there is a linear structure on it compatible with the given dynamics.


  1. [Ma66]
    Malgrange, B.: Ideals of differentiable functions. Oxford University Press, Published by the Tata Institute of Fundamental Research, Bombay (1966)Google Scholar
  2. [Wh44]
    Whitney, H.: The self intersections of a smooth \(n\)-manifold in \(2n\)-space. Annals Math. 45, 220–246 (1944)Google Scholar
  3. [NS03]
    Navarro González, J., Sancho de Salas, J.: \(^\infty \)-Differentiable Spaces. Lecture Notes in Maths, 1824. Springer, Berlin (2003)Google Scholar
  4. [BM10]
    Balachandran, A.P., Marmo, G.: Group Theory and Hopf Algebras: Lectures for Physicists. World Scientific, Hackensack (2010)Google Scholar
  5. [AM78]
    Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Benjamin, Reading (1978)Google Scholar
  6. [Ha70]
    Haefliger, A.: Homotopy and Integrability. Lecture Notes in Maths, vol. 197, pp. 133–164, Springer, Berlin (1970)Google Scholar
  7. [LF12]
    Loja Fernandes, R., Frejlich, P.: A \(h\)-principle for symplectic foliations. Int. Math. Res. Not. 2012, 1505–1518 (2012)Google Scholar
  8. [Gr86a]
    Gromov, M.: Soft and hard symplectic geometry. In: Proceedings of the International Congress of Mathematicians at Berkeley, vol. 1, pp. 81–98 (1986)Google Scholar
  9. [Gr86b]
    Gromov, M.: Partial Differential Relations. Springer, Berlin (1986)Google Scholar
  10. [Em84]
    Emch, G.G.: Mathematical and Conceptual Foundations of 20th Century Physics. North-Holland, Amsterdam (1984)Google Scholar
  11. [FF13]
    Falceto, F., Ferro, L., Ibort, A., Marmo, G.: Reduction of Lie-Jordan Banach algebras and quantum states. J. Phys. A Math. Theor. 46, 015201 (2013)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • José F. Cariñena
    • 1
  • Alberto Ibort
    • 2
  • Giuseppe Marmo
    • 3
  • Giuseppe Morandi
    • 4
  1. 1.Departamento de Física TeóricaUniversidad de ZaragozaZaragozaSpain
  2. 2.Departamento de MatemáticasUniversidad Carlos III de MadridMadridSpain
  3. 3.Dipartimento di FisicheUniversitá di Napoli “Federico II”NapoliItaly
  4. 4.INFN Sezione di BolognaUniversitá di BolognaBolognaItaly

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