Ukrainian Mathematical Journal

, Volume 53, Issue 8, pp 1220–1236 | Cite as

Ostrohrads'kyi Formalism for Singular Lagrangians with Higher Derivatives

  • V. V. Nesterenko


We generalize the Ostrohrads'kyi method for the construction of the Hamiltonian description of a nondegenerate (regular) variational problem of arbitrary order to the case of degenerate (singular) Lagrangians. These Lagrangians are of major interest in the contemporary theory of elementary particles. For simplicity, we consider the Hamiltonization of a variational problem defined by a singular second-order Lagrangian. Generalizing the Ostrohrads'kyi method, we derive equations of motion in the phase space. We determine a complete collection of constraints of the theory.


Phase Space Elementary Particle Variational Problem Major Interest Arbitrary Order 
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  1. 1.
    A. A. Slavnov and L. D. Faddeev, Introduction to the Quantum Theory of Gauge Fields [in Russian], Nauka, Moscow (1978).Google Scholar
  2. 2.
    M. Kaku, “Superconformal gravity in Hamiltonian form: Another approach to the renormalization of gravitation,” Phys. Rev. D, 27, No. 12, 2809-2818 (1983).Google Scholar
  3. 3.
    B. Podolsky and P. Schwed, “Review of generalized electrodynamics,” Rev. Mod. Phys., 20, No. 1, 40-50 (1948).Google Scholar
  4. 4.
    R. Utiyama and B. S. De Witt, “Renormalization of a classical gravitational field interacting with quantized matter fields,” J. Math. Phys., 3, No. 4, 608-618 (1962).Google Scholar
  5. 5.
    K. S. Stelle, “Renormalization of higher-derivative quantum gravity,” Phys. Rev. D, 16, No. 4, 953-969 (1997).Google Scholar
  6. 6.
    E. S. Fradkin and A. A. Tseytlin, “Renormalizable asymptotically free quantum theory of gravity,” Nucl. Phys. B, 201, No. 3, 469-491 (1982).Google Scholar
  7. 7.
    D. G. Boulware and S. Deser, “String generated gravity models,” Phys. Rev. Lett., 55, No. 24, 2656-2660 (1985).Google Scholar
  8. 8.
    B. Zwiebach, “Curvature squared terms and string theories,” Phys. Lett. B, 156, No. 5, 6, 315-317 (1985).Google Scholar
  9. 9.
    D. G. Boulware, “Quantization of higher derivative theories of gravity,” in: Quantum Theory of Gravity, Adam Hilger, Bristol (1984), pp. 267-294.Google Scholar
  10. 10.
    S. W. Hawking, “Who's afraid of (higher derivative) ghosts?,” in: Quantum Field Theory and Quantum Statistics, Vol. 2, Adam Hilger, Bristol (1987), pp. 129-139.Google Scholar
  11. 11.
    V. V. Nesterenko, “Curvature and torsion of the world curve in the action of the relativistic particle,” J. Math. Phys., 32, No. 12, 3315-3320 (1992).Google Scholar
  12. 12.
    V. V. Nesterenko, “Torsion in the action of the relativistic particle,” Class. Quant. Grav., 9, 1101-1114 (1992).Google Scholar
  13. 13.
    V. V. Nesterenko, “Canonical quantization of a relativistic particle with torsion,” Mod. Phys. Lett. A, 6, 719-726 (1991).Google Scholar
  14. 14.
    V. V. Nesterenko, “Relativistic particle with curvature in an external electromagnetic field,” Int. J. Mod. Phys. A, 6, 3989-3996 (1991).Google Scholar
  15. 15.
    V. V. Nesterenko, “On a model of a relativistic particle with curvature and torsion,” J. Math. Phys., 34, 5589-5595 (1993).Google Scholar
  16. 16.
    V. V. Nesterenko, “On squaring the primary constraints in a generalized Hamiltonian dynamics,” Phys. Lett. B, 327, 50-55 (1994).Google Scholar
  17. 17.
    M. S. Plyushchay, “Massive relativistic point particle with rigidity,” Int. J. Mod. Phys., 4, No. 15, 3851-3865 (1989).Google Scholar
  18. 18.
    M. S. Plyushchay, “Massless particle with rigidity as a model for the description of bosons and fermions,” Phys. Lett. B, 243, No. 4, 383-388 (1990).Google Scholar
  19. 19.
    M. S. Plyushchay, “The model of the relativistic particle with torsion,” Nucl. Phys. B, 362, No. 1, 2, 54-72 (1991).Google Scholar
  20. 20.
    Yu. A. Kuznetsov and M. S. Plyushchay, “The model of the relativistic particle with curvature and torsion,” Nucl. Phys., 389, No. 1, 181-205 (1993).Google Scholar
  21. 21.
    V. V. Nesterenko and Nguyen Swan Han, “The Hamiltonian formalism in the model of the relativistic string with rigidity,” Int. J. Mod. Phys., 3, No. 10, 2315-2329 (1988).Google Scholar
  22. 22.
    M. V. Ostrohrads'kyi, “A memoir on the differential equations related to an isoperimetric problem,” in: Complete Collection of Works [in Russian], Vol. 2, Ukrainian Academy of Sciences (1961), pp. 139-233. [see also Variational Principles of Mechanics [in Russian], Fizmatgiz, Moscow (1959), pp. 315–387.]Google Scholar
  23. 23.
    E. T. Whittaker, A Treatise of the Analytical Dynamics of Particles and Rigid Bodies, University Press, Cambridge (1927).Google Scholar
  24. 24.
    V. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1979).Google Scholar
  25. 25.
    P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, New York (1964).Google Scholar
  26. 26.
    A. J. Hanson, T. Regge, and C. Teitelboim, Constrained Hamiltonian Systems, Acad. Naz. dei Lincei, Roma (1976).Google Scholar
  27. 27.
    V. V. Nesterenko and A. M. Chervyakov, Singular Lagrangians: Classical Dynamics and Quantization [in Russian], Preprint No. P2-86-323, Joint Institute for Nuclear Research, Dubna (1986).Google Scholar
  28. 28.
    D. M. Gitman and I. V. Tyutin, Canonical Quantization of Fields with Constraints [in Russian], Nauka, Moscow (1986).Google Scholar
  29. 29.
    B. M. Barbashov and V. V. Nesterenko, “Continuous symmetries in field theory,” Fortsch. Phys., 31, No. 10, 535-567 (1983).Google Scholar
  30. 30.
    V. V. Nesterenko and A. M. Chervyakov, “On some properties of constraints in theories with degenerate Lagrangians,” Teor. Mat. Fiz., 64, No. 1, 82-91 (1985).Google Scholar
  31. 31.
    C. Battle, J. Gomis, J. M. Pons, and N. Roman, “Lagrangian and Hamiltonian constraints,” Lett. Math. Phys., 13, No. 1, 17-23 (1987).Google Scholar
  32. 32.
    K. Kamimura, “Singular Lagrangian and constrained Hamiltonian systems, generalized canonical formalism,” Nuovo Cim. B, 68, No. 1, 33-54 (1982).Google Scholar
  33. 33.
    R. Pisarski, “Field theory of paths with a curvature-dependent term,” Phys. Rev. D, 34, No. 2, 670-673 (1986).Google Scholar
  34. 34.
    F. Alonso and D. Espriu, “On the fine structure of strings,” Nucl. Phys. B, 283, No. 3, 4 393-412 (1987).Google Scholar
  35. 35.
    P. A. Griffiths, Exterior Differential Systems and the Calculus of Variations, Birkhäuser, Berlin (1983).Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • V. V. Nesterenko
    • 1
  1. 1.Joint Institute for Nuclear ResearchDubnaRussia

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