Abstract
AbstractThis work reviews the classical Darboux theorem for symplectic, presymplectic, and cosymplectic manifolds (which are used to describe mechanical systems), as well as certain cases of multisymplectic manifolds, while extends the Darboux theorem in new ways to k-symplectic and k-cosymplectic manifolds (all these structures appear in the geometric formulation of first-order classical field theories). Moreover, we discuss the existence of Darboux theorems for classes of precosymplectic, k-presymplectic, k-precosymplectic, and premultisymplectic manifolds, which are the geometrical structures underlying some kinds of singular field theories, i.e. with locally non-invertible Legendre maps. Approaches to Darboux theorems based on flat connections associated with geometric structures are given, while new results on polarisations for (k-)(pre)(co)symplectic structures arise.
Funder
Ministerstwo Edukacji i Nauki
Ministerio de Ciencia, Innovación y Universidades
Ministerio de Ciencia e Innovación
Departament d’Innovació, Universitats i Empresa, Generalitat de Catalunya
Publisher
Springer Science and Business Media LLC
Reference61 articles.
1. Abraham, R., Marsden, J.E.: Foundations of Mechanics, vol. 364, 2nd edn. AMS Chelsea Publishing. Benjamin/Cummings Pub. Co., New York (1978). https://doi.org/10.1090/chel/364
2. Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis and Applications, Applied Mathematical Sciences, vol. 75, 2nd edn. Springer, New York (1988). https://doi.org/10.1007/978-1-4612-1029-0
3. Albert, C.: Le théorème de réduction de Marsden-Weinstein en géométrie cosymplectique et de contact. J. Geom. Phys. 6(4), 627–649 (1989). https://doi.org/10.1016/0393-0440(89)90029-6
4. Awane, A.: $$k$$-symplectic structures. J. Math. Phys. 33(12), 4046 (1992). https://doi.org/10.1063/1.529855
5. Awane, A.: Polarized symplectic structures. Balk. J. Geom. Appl. 25(1):1–18 (2020). http://www.mathem.pub.ro/bjga/v25n1/B25-1aw-ZBG99.pdf