Abstract
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.
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References
V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian], Editorial URSS, Moscow (2003); English transl. prev. ed., Springer, Berlin (1978).
B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry: Methods and Applications [in Russian], Vol. 2, Geometry and Topology of Manifolds, Editorial URSS, Moscow (2001); English transl. prev. ed.: B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov Modern Geometry — Methods and Applications: Part II. The Geometry and Topology of Manifolds (Grad. Texts Math., Vol. 104), Springer, New York (1985).
V. I. Arnold and A. B. Givental, Symplectic Geometry [in Russian], RKhD, Izhevsk (2000); English transl. prev. ed.: “Symplectic geometry,” in: Dynamical Systems IV (Encycl. Math. Sci., Vol. 4), Springer, Berlin (2001).
J. Harris, Algebraic Geometry, Springer, Berlin (1992).
D. M. Gitman and I. V. Tyutin, Canonical Quantum Fields with Constraints [in Russian], Nauka, Moscow (1986).
A. Morozov, “Hamiltonian formalism in the presence of higher derivatives,” arXiv:0712.0946v3 [hep-th] (2007).
P. I. Dunin-Barkovskii and A. V. Sleptsov, Theor. Math. Phys., 158, 61–81 (2009).
P. A. M. Dirac, Lectures on Quantum Mechanics, Acad. Press, New York (1964).
L. Takhtajan, Commun. Math. Phys., 160, 295–315 (1994); arXiv:hep-th/9301111v1 (1993).
T. Curtright and C. Zachos, Phys. Rev. D, 68, 085001 (2003); arXiv:hep-th/0212267v3 (2002).
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 176, No. 2, pp. 281–305, August 2013.
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Danilenko, I.A. Modified Hamilton formalism for fields. Theor Math Phys 176, 1067–1086 (2013). https://doi.org/10.1007/s11232-013-0089-y
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DOI: https://doi.org/10.1007/s11232-013-0089-y