Geodesic
Modified Hamilton formalism for fields
Teoretičeskaâ i matematičeskaâ fizika, Tome 176 (2013) no. 2, pp. 281-305 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In Hamiltonian mechanics, the equations of motion can be regarded as a condition on the vectors tangent to the solution: they should be null-vectors of the symplectic structure. The passage to the field theory is usually done by replacing the finite-dimensional configuration space with an infinite-dimensional one. We apply an alternative formalism in which the space–time is considered one worldsheet and its maps are studied. Instead of null-vectors of the symplectic $2$-form, null-polyvectors of a higher-rank form on a finite-dimensional manifold are introduced. The action in this case is an integral of a differential form over a surface in the phase space. Such a method for obtaining the Hamiltonian mechanics from the Lagrange function is a generalization of the Legendre transformation. The condition that the value of the action and its extremals are preserved naturally determines this procedure.
Keywords: Hamiltonian mechanics, field theory. 
@article{TMF_2013_176_2_a5,
     author = {I. Danilenko},
     title = {Modified {Hamilton} formalism for fields},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {281--305},
     year = {2013},
     volume = {176},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2013_176_2_a5/}
}
TY  - JOUR
AU  - I. Danilenko
TI  - Modified Hamilton formalism for fields
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2013
SP  - 281
EP  - 305
VL  - 176
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2013_176_2_a5/
LA  - ru
ID  - TMF_2013_176_2_a5
ER  - 
%0 Journal Article
%A I. Danilenko
%T Modified Hamilton formalism for fields
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2013
%P 281-305
%V 176
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2013_176_2_a5/
%G ru
%F TMF_2013_176_2_a5
I. Danilenko. Modified Hamilton formalism for fields. Teoretičeskaâ i matematičeskaâ fizika, Tome 176 (2013) no. 2, pp. 281-305. http://geodesic.mathdoc.fr/item/TMF_2013_176_2_a5/

[1] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, Editorial URSS, M., 2003 | MR

[2] B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, Sovremennaya geometriya. Metody i prilozheniya, v. 2, Geometriya i topologiya mnogoobrazii, Editorial URSS, M., 2001 | MR

[3] V. I. Arnold, A. B. Givental, Simplekticheskaya geometriya, RKhD, Izhevsk, 2000

[4] Dzh. Kharris, Algebraicheskaya geometriya. Nachalnyi kurs, MTsNMO, M., 2006

[5] A. Morozov, Hamiltonian formalism in the presence of higher derivatives, arXiv: 0712.0946

[6] D. M. Gitman, I. V. Tyutin, Kanonicheskoe kvantovanie polei so svyazyami, Nauka, M., 1986 | MR | Zbl

[7] P. I. Dunin-Barkovskii, A. V. Sleptsov, TMF, 158:1 (2009), 72–97 | DOI | MR

[8] P. A. M. Dirak, Lektsii po kvantovoi mekhanike, RKhD, Izhevsk, 1998

[9] L. Takhtajan, Commun. Math. Phys., 160:2 (1994), 295–315, arXiv: hep-th/9301111 | DOI | MR | Zbl

[10] T. Curtright, C. Zachos, Phys. Rev. D, 68:8 (2003), 085001, 29 pp., arXiv: hep-th/0212267 | DOI