Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
Variational principles; Symmetries; Conserved quantities; Noether theorem; Fiber bundles; Multisymplectic manifolds.
Variational principles; Symmetries; Conserved quantities; Noether theorem; Fiber bundles; Multisymplectic manifolds.
Summary:
The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multisymplectic 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.
References:
[1] Aldaya, V., Azcarraga, J. A. de: Variational Principles on $r-th$ order jets of fibre bundles in Field Theory. J. Math. Phys., 19, 9, 1978, 1869-1875, DOI 10.1063/1.523904 | MR 0496116
[2] Aldaya, V., Azcarraga, J.A. de: Higher order Hamiltonian formalism in Field Theory. J. Phys. A, 13, 8, 1980, 2545-2551, MR 0582906 | Zbl 0467.58013
[3] Arnold, V. I.: Mathematical methods of classical mechanics. 60, 1989, Springer-Verlag, New York, MR 0997295 | Zbl 0692.70003
[4] Dedecker, P.: On the generalization of symplectic geometry to multiple integrals in the calculus of variations. Differential Geometrical Methods in Mathematical Physics, 570, 1977, 395-456, Springer, Berlin, MR 0458478 | Zbl 0352.49018
[5] León, M. de, Marín-Solano, J., Marrero, J. C., Muñoz-Lecanda, M. C., Román-Roy, N.: Pre-multisymplectic constraint algorithm for field theories. Int. J. Geom. Meth. Mod. Phys., 2, 2005, 839-871, DOI 10.1142/S0219887805000880 | MR 2177288 | Zbl 1156.70317
[6] León, M. de, Diego, D. Martín de: Symmetries and Constant of the Motion for Singular Lagrangian Systems. Int. J. Theor. Phys., 35, 5, 1996, 975-1011, DOI 10.1007/BF02302383 | MR 1386775
[7] León, M. de, Diego, D. Martín de, Santamaría-Merino, A.: Symmetries in classical field theory. Int. J. Geom. Meths. Mod. Phys., 1, 5, 2004, 651-710, DOI 10.1142/S0219887804000290 | MR 2095443
[8] Echeverría-Enríquez, A., León, M. De, Muñoz-Lecanda, M. C., Román-Roy, N.: Extended Hamiltonian systems in multisymplectic field theories. J. Math. Phys., 48, 11, 2007, 112901. DOI 10.1063/1.2801875 | MR 2370237 | Zbl 1152.81420
[9] Echeverría-Enríquez, A., Muñoz-Lecanda, M.C., Román-Roy, N.: Geometry of Lagrangian first-order classical field theories. Forts. Phys., 44, 1996, 235-280, DOI 10.1002/prop.2190440304 | MR 1400307 | Zbl 0964.58015
[10] Echeverría-Enríquez, A., Muñoz-Lecanda, M.C., Román-Roy, N.: Multivector fields and connections: Setting Lagrangian equations in field theories. J. Math. Phys., 39, 9, 1998, 4578-4603, DOI 10.1063/1.532525 | MR 1643297 | Zbl 0927.37054
[11] Echeverría-Enríquez, A., Muñoz-Lecanda, M.C., Román-Roy, N.: Multivector Field Formulation of Hamiltonian Field Theories: Equations and Symmetries. J. Phys. A: Math. Gen., 32, 1999, 8461-8484, DOI 10.1088/0305-4470/32/48/309 | MR 1733703 | Zbl 0982.70019
[12] Ferraris, M., Francaviglia, M.: 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, MR 0923342 | Zbl 0659.58010
[13] García, P. L.: The Poincaré-Cartan invariant in the calculus of variations. Symp. Math., 14, 1973, 219-246, MR 0406246
[14] García, P. L., Muñoz, J.: On the geometrical structure of higher order variational calculus. Atti. Accad. Sci. Torino Cl. Sci. Fis. Math. Natur., 117, 1983, 127-147, MR 0773483 | Zbl 0569.58008
[15] Giachetta, G., Mangiarotti, L., Sardanashvily, G.: New Lagrangian and Hamiltonian methods in field theory. 1997, World Scientific Publishing Co., Inc., River Edge, NJ, MR 2001723 | Zbl 0913.58001
[16] Goldschmidt, H., Sternberg, S.: The Hamilton-Cartan formalism in the calculus of variations. Ann. Inst. Fourier Grenoble, 23, 1, 1973, 203-267, DOI 10.5802/aif.451 | MR 0341531 | Zbl 0243.49011
[17] Hélein, F., J, J. Kouneiher: 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, MR 2105190 | Zbl 1115.70017
[18] Kouranbaeva, S., Shkoller, S.: A variational approach to second-order multisymplectic field theory. J. Geom. Phys., 4, 2000, 333-366, DOI 10.1016/S0393-0440(00)00012-7 | MR 1780759 | Zbl 0987.70020
[19] Krupka, D.: Introduction to Global Variational Geometry. 2015, Atlantis Studies in Variational Geometry, Atlantis Press, MR 3290001 | Zbl 1310.49001
[20] Krupka, D., Štěpánková, O.: On the Hamilton form in second order calculus of variations. Procs. Int. Meeting on Geometry and Physics, 1982, 85-101, MR 0760838
[21] Mangiarotti, L., Sardanashvily, G.: Gauge Mechanics. 1998, World Scientific, Singapore, MR 1689375
[22] Prieto-Martínez, P. D., Román-Roy, N.: Higher-order mechanics: variational principles and other topics. J. Geom. Mech., 5, 4, 2013, 493-510, DOI 10.3934/jgm.2013.5.493 | MR 3180709 | Zbl 1284.35014
[23] Prieto-Martínez, P.D., Román-Roy, N.: Variational principles for multisymplectic second-order classical field theories. Int. J. Geom. Meth. Mod. Phys, 12, 8, 2015, 1560019. MR 3400659
[24] Sarlet, W., Cantrijn, F.: Higher-order Noether symmetries and constants of the motion. J. Phys. A: Math. Gen., 14, 1981, 479-492, DOI 10.1088/0305-4470/14/2/023 | MR 0601885 | Zbl 0464.58010
[25] Saunders, D.J.: The geometry of jet bundles. London Mathematical Society, Lecture notes series, 142, 1989, Cambridge University Press, Cambridge, New York, MR 0989588 | Zbl 0665.58002
Search
Partner of
