| Title:
|
Variational principles and symmetries on fibered multisymplectic manifolds (English) |
| Author:
|
Gaset, Jordi |
| Author:
|
Prieto-Martínez, Pedro D. |
| Author:
|
Román-Roy, Narciso |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
24 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
137-152 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Variational principles |
| Keyword:
|
Symmetries |
| Keyword:
|
Conserved quantities |
| Keyword:
|
Noether theorem |
| Keyword:
|
Fiber bundles |
| Keyword:
|
Multisymplectic manifolds. |
| MSC:
|
49S05 |
| MSC:
|
53D42 |
| MSC:
|
55R10 |
| MSC:
|
70H50 |
| MSC:
|
70S05 |
| MSC:
|
70S10 |
| idZBL:
|
Zbl 06697287 |
| idMR:
|
MR3590211 |
| . |
| Date available:
|
2017-02-28T16:45:09Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146017 |
| . |
| Reference:
|
[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, MR 0496116, 10.1063/1.523904 |
| Reference:
|
[2] Aldaya, V., Azcarraga, J.A. de: Higher order Hamiltonian formalism in Field Theory.J. Phys. A, 13, 8, 1980, 2545-2551, Zbl 0467.58013, MR 0582906 |
| Reference:
|
[3] Arnold, V. I.: Mathematical methods of classical mechanics.60, 1989, Springer-Verlag, New York, Zbl 0692.70003, MR 0997295 |
| Reference:
|
[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, Zbl 0352.49018, MR 0458478 |
| Reference:
|
[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, Zbl 1156.70317, MR 2177288, 10.1142/S0219887805000880 |
| Reference:
|
[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, MR 1386775, 10.1007/BF02302383 |
| Reference:
|
[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, MR 2095443, 10.1142/S0219887804000290 |
| Reference:
|
[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. Zbl 1152.81420, MR 2370237, 10.1063/1.2801875 |
| Reference:
|
[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, Zbl 0964.58015, MR 1400307, 10.1002/prop.2190440304 |
| Reference:
|
[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, Zbl 0927.37054, MR 1643297, 10.1063/1.532525 |
| Reference:
|
[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, Zbl 0982.70019, MR 1733703, 10.1088/0305-4470/32/48/309 |
| Reference:
|
[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, Zbl 0659.58010, MR 0923342 |
| Reference:
|
[13] García, P. L.: The Poincaré-Cartan invariant in the calculus of variations.Symp. Math., 14, 1973, 219-246, MR 0406246 |
| Reference:
|
[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, Zbl 0569.58008, MR 0773483 |
| Reference:
|
[15] Giachetta, G., Mangiarotti, L., Sardanashvily, G.: New Lagrangian and Hamiltonian methods in field theory.1997, World Scientific Publishing Co., Inc., River Edge, NJ, Zbl 0913.58001, MR 2001723 |
| Reference:
|
[16] Goldschmidt, H., Sternberg, S.: The Hamilton-Cartan formalism in the calculus of variations.Ann. Inst. Fourier Grenoble, 23, 1, 1973, 203-267, Zbl 0243.49011, MR 0341531, 10.5802/aif.451 |
| Reference:
|
[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, Zbl 1115.70017, MR 2105190 |
| Reference:
|
[18] Kouranbaeva, S., Shkoller, S.: A variational approach to second-order multisymplectic field theory.J. Geom. Phys., 4, 2000, 333-366, Zbl 0987.70020, MR 1780759, 10.1016/S0393-0440(00)00012-7 |
| Reference:
|
[19] Krupka, D.: Introduction to Global Variational Geometry.2015, Atlantis Studies in Variational Geometry, Atlantis Press, Zbl 1310.49001, MR 3290001 |
| Reference:
|
[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 |
| Reference:
|
[21] Mangiarotti, L., Sardanashvily, G.: Gauge Mechanics.1998, World Scientific, Singapore, MR 1689375 |
| Reference:
|
[22] Prieto-Martínez, P. D., Román-Roy, N.: Higher-order mechanics: variational principles and other topics.J. Geom. Mech., 5, 4, 2013, 493-510, Zbl 1284.35014, MR 3180709, 10.3934/jgm.2013.5.493 |
| Reference:
|
[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 |
| Reference:
|
[24] Sarlet, W., Cantrijn, F.: Higher-order Noether symmetries and constants of the motion.J. Phys. A: Math. Gen., 14, 1981, 479-492, Zbl 0464.58010, MR 0601885, 10.1088/0305-4470/14/2/023 |
| Reference:
|
[25] Saunders, D.J.: The geometry of jet bundles.London Mathematical Society, Lecture notes series, 142, 1989, Cambridge University Press, Cambridge, New York, Zbl 0665.58002, MR 0989588 |
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