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Uspekhi Mat. Nauk, 1975, Volume 30, Issue 1(181), Pages 173–198 (Mi umn4140)  

This article is cited in 17 scientific papers (total in 18 papers)

What is the hamiltonian formalism?

A. M. Vinogradov, I. S. Krasil'shchik


Abstract: In this paper the basic concepts of the classical Hamiltonian formalism are translated into algebraic language. We treat the Hamiltonian formalism as a constituent part of the general theory of linear differential operators on commutative rings with identity. We take particular care in motivating the concepts we introduce. As an illustration of the theory presented here, we examine the Hamiltonian formalism in Lie algebras. We conclude by presenting a version of the “orbit method” in the theory of representations of Lie groups, which is a natural corollary of our view of the Hamiltonian formalism.

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English version:
Russian Mathematical Surveys, 1975, 30:1, 177–202

Bibliographic databases:

UDC: 517.4
Received: 19.06.1974

Citation: A. M. Vinogradov, I. S. Krasil'shchik, “What is the hamiltonian formalism?”, Uspekhi Mat. Nauk, 30:1(181) (1975), 173–198; Russian Math. Surveys, 30:1 (1975), 177–202

Citation in format AMSBIB
\Bibitem{VinKra75}
\by A.~M.~Vinogradov, I.~S.~Krasil'shchik
\paper What is the hamiltonian formalism?
\jour Uspekhi Mat. Nauk
\yr 1975
\vol 30
\issue 1(181)
\pages 173--198
\mathnet{http://mi.mathnet.ru/umn4140}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=650307}
\zmath{https://zbmath.org/?q=an:0327.70006}
\transl
\jour Russian Math. Surveys
\yr 1975
\vol 30
\issue 1
\pages 177--202
\crossref{https://doi.org/10.1070/RM1975v030n01ABEH001403}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. A. Kirillov, “Local Lie algebras”, Russian Math. Surveys, 31:4 (1976), 55–75  mathnet  crossref  mathscinet  zmath
    2. A. M. Vinogradov, B. A. Kupershmidt, “The structures of Hamiltonian mechanics”, Russian Math. Surveys, 32:4 (1977), 177–243  mathnet  crossref  mathscinet  zmath
    3. V. N. Shander, “Vector fields and differential equations on supermanifolds”, Funct. Anal. Appl., 14:2 (1980), 160–162  mathnet  crossref  mathscinet  zmath
    4. Yvette Kosmann-Schwarzbach, “Hamiltonian systems on fibered manifolds”, Lett Math Phys, 5:3 (1981), 229  crossref  mathscinet  zmath  isi
    5. A. V. Belyaev, “On the motion of a multidimensional body with fixed point in a gravitational field”, Math. USSR-Sb., 42:3 (1982), 413–418  mathnet  crossref  mathscinet  zmath
    6. Jedrzej Śniatycki, Alan Weinstein, “Reduction and quantization for singular momentum mappings”, Lett Math Phys, 7:2 (1983), 155  crossref  mathscinet  zmath  isi
    7. V. V. Trofimov, A. T. Fomenko, “Liouville integrability of Hamiltonian systems on Lie algebras”, Russian Math. Surveys, 39:2 (1984), 1–67  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    8. A.M. Vinogradov, “The -spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory”, Journal of Mathematical Analysis and Applications, 100:1 (1984), 41  crossref
    9. M. V. Karasev, “Analogues of the objects of Lie group theory for nonlinear Poisson brackets”, Math. USSR-Izv., 28:3 (1987), 497–527  mathnet  crossref  mathscinet  zmath
    10. A.M. Astashov, A.M. Vinogradov, “On the structure of Hamiltonian operators in the field theory”, Journal of Geometry and Physics, 3:2 (1986), 263  crossref
    11. S. S. Akbarov, “Smooth structure and differential operators on a locally compact group”, Izv. Math., 59:1 (1995), 1–44  mathnet  crossref  mathscinet  zmath  isi
    12. S. S. Akbarov, “Construction of the cotangent bundle of a locally compact group”, Izv. Math., 59:3 (1995), 445–470  mathnet  crossref  mathscinet  zmath  isi
    13. G. Marmo, G. Vilasi, A.M. Vinogradov, “The local structure of n-Poisson and n-Jacobi manifolds”, Journal of Geometry and Physics, 25:1-2 (1998), 141  crossref
    14. Daniel R. Farkas, Gail Letzter, “Ring theory from symplectic geometry”, Journal of Pure and Applied Algebra, 125:1-3 (1998), 155  crossref
    15. Janusz Grabowski, Giuseppe Marmo, “The graded Jacobi algebras and (co)homology”, J Phys A Math Gen, 36:1 (2003), 161  crossref  mathscinet  zmath  isi
    16. JOSÉ F. CARIÑENA, XAVIER GRÀCIA, GIUSEPPE MARMO, EDUARDO MARTÍNEZ, MIGUEL C. MUÑOZ-LECANDA, NARCISO ROMÁN-ROY, “GEOMETRIC HAMILTON–JACOBI THEORY”, Int. J. Geom. Methods Mod. Phys, 03:07 (2006), 1417  crossref
    17. V. S. Kalnitsky, “Symmetries of a flat cosymbol algebra of the differential operators”, J. Math. Sci. (N. Y.), 222:4 (2017), 429–436  mathnet  crossref  mathscinet
    18. A. M. Astashov, I. V. Astashova, A. V. Bocharov, V. M. Bukhshtaber, V. A. Vasilev, A. M. Verbovetskii, A. M. Vershik, A. P. Veselov, M. M. Vinogradov, L. Vitalyano, R. F. Vitolo, F. F. Voronov, V. G. Kats, I. Kosmann-Shvartsbakh, I. S. Krasilschik, I. M. Krichever, A. P. Krischenko, S. K. Lando, V. V. Lychagin, M. Marvan, V. P. Maslov, A. S. Mischenko, S. P. Novikov, V. N. Rubtsov, A. V. Samokhin, A. B. Sosinskii, Dzh. Stashef, D. B. Fuks, A. Ya. Khelemskii, N. G. Khorkova, V. N. Chetverikov, A. S. Shvarts, “Aleksandr Mikhailovich Vinogradov (nekrolog)”, UMN, 75:2(452) (2020), 185–190  mathnet  crossref
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