Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Communications in Mathematics

Editor-in-Chief: Rossi, Olga

2 Issues per year


Mathematical Citation Quotient (MCQ) 2016: 0.28

Open Access
Online
ISSN
2336-1298
See all formats and pricing
More options …

Variational principles and symmetries on fibered multisymplectic manifolds

Jordi Gaset
  • Corresponding author
  • Department of Mathematics. Ed. C-3, Campus Norte UPC C/ Jordi Girona 1. 08034 Barcelona, Spain
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Pedro D. Prieto-Martínez / Narciso Román-Roy
Published Online: 2016-12-22 | DOI: https://doi.org/10.1515/cm-2016-0010

Abstract

The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multi-symplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is equivalent, conservation laws), symmetries, Cartan (Noether) symmetries, gauge symmetries and different versions of Noether's theorem are studied in this ambient. In this way, this constitutes a general geometric framework for all these topics that includes, as special cases, first and higher order field theories and (non-autonomous) mechanics.

Keywords: Variational principles; Symmetries; Conserved quantities; Noether theorem; Fiber bundles; Multisymplectic manifolds

References

  • [1] V. Aldaya, J. A. de Azcarraga: Variational Principles on r - th order jets of fibre bundles in Field Theory. J. Math. Phys. 19 (9) (1978) 1869-1875.CrossrefGoogle Scholar

  • [2] V. Aldaya, J.A. de Azcarraga: Higher order Hamiltonian formalism in Field Theory. J. Phys. A 13 (8) (1980) 2545-2551.Google Scholar

  • [3] V. I. Arnold: Mathematical methods of classical mechanics. Springer-Verlag, New York (1989).Google Scholar

  • [4] P. Dedecker: On the generalization of symplectic geometry to multiple integrals in the calculus of variations. In: Differential Geometrical Methods in Mathematical Physics. Springer, Berlin (1977) 395-456.Google Scholar

  • [5] M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz-Lecanda, N. Román-Roy: Pre-multisymplectic constraint algorithm for field theories. Int. J. Geom. Meth. Mod. Phys. 2 (2005) 839-871.CrossrefGoogle Scholar

  • [6] M. de León, D. Martín de Diego: Symmetries and Constant of the Motion for Singular Lagrangian Systems. Int. J. Theor. Phys. 35 (5) (1996) 975-1011.CrossrefGoogle Scholar

  • [7] M. de León, D. Martín de Diego, A. Santamaría-Merino: Symmetries in classical field theory. Int. J. Geom. Meths. Mod. Phys. 1 (5) (2004) 651-710.CrossrefGoogle Scholar

  • [8] A. Echeverría-Enríquez, M. De León, M. C. Muñoz-Lecanda, N. Román-Roy: Extended Hamiltonian systems in multisymplectic field theories. J. Math. Phys. 48 (11) (2007). 112901Web of ScienceGoogle Scholar

  • [9] A. Echeverría-Enríquez, M.C. Muñoz-Lecanda, N. Román-Roy: Geometry of Lagrangian first-order classical field theories. Forts. Phys. 44 (1996) 235-280.CrossrefGoogle Scholar

  • [10] A. Echeverría-Enríquez, M.C. Muñoz-Lecanda, N. Román-Roy: Multivector fields and connections: Setting Lagrangian equations in field theories. J. Math. Phys. 39 (9) (1998) 4578-4603.CrossrefGoogle Scholar

  • [11] A. Echeverría-Enríquez, M.C. Muñoz-Lecanda, N. Román-Roy: Multivector Field Formulation of Hamiltonian Field Theories: Equations and Symmetries. J. Phys. A: Math. Gen. 32 (1999) 8461-8484.CrossrefGoogle Scholar

  • [12] M. Ferraris, M. Francaviglia: Applications of the Poincaré-Cartan form in higher order field theories. Differential Geometry and Its Applications (Brno, 1986), Math. Appl.(East European Ser.) 27 (1987) 31-52.Google Scholar

  • [13] P. L. García: The Poincaré-Cartan invariant in the calculus of variations. Symp. Math. 14 (1973) 219-246.Google Scholar

  • [14] P. L. García, J. Muñoz: On the geometrical structure of higher order variational calculus. Atti. Accad. Sci. Torino Cl. Sci. Fis. Math. Natur. 117 (1983) 127-147.Google Scholar

  • [15] G. Giachetta, L. Mangiarotti, G. Sardanashvily: New Lagrangian and Hamiltonian methods in field theory. World Scientific Publishing Co., Inc., River Edge, NJ (1997).Google Scholar

  • [16] H. Goldschmidt, S. Sternberg: The Hamilton-Cartan formalism in the calculus of variations. Ann. Inst. Fourier Grenoble 23 (1) (1973) 203-267.CrossrefGoogle Scholar

  • [17] F. Hélein, J. Kouneiher J: Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl. Adv. Theor. Math. Phys. 8 (2004) 565-601.CrossrefGoogle Scholar

  • [18] S. Kouranbaeva, S. Shkoller: A variational approach to second-order multisymplectic field theory. J. Geom. Phys. 4 (2000) 333-366.CrossrefGoogle Scholar

  • [19] D. Krupka: Introduction to Global Variational Geometry. Atlantis Studies in Variational Geometry, Atlantis Press (2015).Google Scholar

  • [20] D. Krupka, O. Štěpánková: On the Hamilton form in second order calculus of variations. In: Procs. Int. Meeting on Geometry and Physics. (1982) 85-101.Google Scholar

  • [21] L. Mangiarotti, G. Sardanashvily: Gauge Mechanics. World Scientific, Singapore (1998).Google Scholar

  • [22] P. D. Prieto-Martínez, N. Román-Roy: Higher-order mechanics: variational principles and other topics. J. Geom. Mech. 5 (4) (2013) 493-510.CrossrefWeb of ScienceGoogle Scholar

  • [23] P.D. Prieto-Martínez, N. Román-Roy: Variational principles for multisymplectic second-order classical field theories. Int. J. Geom. Meth. Mod. Phys 12 (8) (2015). 1560019Google Scholar

  • [24] W. Sarlet, F. Cantrijn: Higher-order Noether symmetries and constants of the motion. J. Phys. A: Math. Gen. 14 (1981) 479-492.CrossrefGoogle Scholar

  • [25] D.J. Saunders: The geometry of jet bundles. Cambridge University Press, Cambridge, New York (1989). Google Scholar

About the article

Received: 2016-10-17

Accepted: 2016-12-06

Published Online: 2016-12-22

Published in Print: 2016-12-01


Citation Information: Communications in Mathematics, ISSN (Online) 2336-1298, DOI: https://doi.org/10.1515/cm-2016-0010.

Export Citation

© 2016. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Alberto Ibort and Amelia Spivak
Journal of Geometric Mechanics, 2017, Volume 9, Number 1, Page 47

Comments (0)

Please log in or register to comment.
Log in