On Dirac's generalized Hamiltonian dynamics

V. V. Kozlov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 79 (2024) no. 4, pp. 649-681

Résumé

Various aspects of Dirac's generalized Hamiltonian dynamics are under consideration. The starting point is a Hamiltonian system on a symplectic manifold on which, in addition, a distribution of multidimensional tangent planes is defined. The Hamiltonian vector field must be modified so that this distribution becomes invariant under the phase flow of the modified dynamical system. This problem can be solved in various ways. The simplest approach is to project the Hamiltonian vector field onto the planes of the distribution by using the symplectic structure, the closed nondegenerate 2-form on the symplectic manifold (which defines the symplectic geometry on tangent planes to the phase space). If the distribution in question is integrable, then this approach leads to generalized Hamiltonian dynamics developed by Dirac (and other authors) to quantize systems whose Lagrangian is degenerate in velocities. In application to the mechanics of Lagrange's systems with non-integrable constraints, this approach yields classical non-holonomic systems. Another approach is based on the definition of the motion of Hamiltonian systems as an extremal of a variational problem with constraints. Then in the case of non-holonomic constraints we obtain dynamical systems of a quite different type. In application to Lagrange's systems with non-integrable constraints this approach yields the equations of motion of so-called vaconomic dynamics. For example geometric optic is considered, which is based on Fermat's variatonal principle with Largangian which is homogeneous of degree one with respect to the velocities. Bibliography: 34 titles.

Détail

Keywords: Hamiltonian system, symplectic structure, symplectic geometry, Lagrangian, Hamiltonian, primary constraints, secondary constraints, non-holonomic systems, vaconomic systems, non-isotropic optic media, Fermat's principle.
Mots-clés : distributions, Poincaré–Helmholtz action

Bibliographie

[1] P. A. M. Dirac, “Generalized Hamiltonian dynamics”, Canad. J. Math., 2:2 (1950), 129–148 | DOI | MR | Zbl

[2] P. A. M. Dirac, “Generalized Hamiltonian dynamics”, Proc. Roy. Soc. London Ser. A, 246 (1958), 326–332 | DOI | MR | Zbl

[3] J. L. Anderson and P. G. Bergmann, “Constraints in covariant field theories”, Phys. Rev. (2), 83:5 (1951), 1018–1025 | DOI | MR | Zbl

[4] B. V. Medvedev, “P. A. M. Dirac and the logical foundations of quantum theory. II”: P. A. M. Dirac, Collected works, v. III, Fizmatlit, Moscow, 2004, 649–650 (Russian)

[5] V. V. Kozlov, General theory of vortices, 2nd revised and augmented ed., Institute for Computer Studies, Moscow–Izhevsk, 2013, 324 СЃ. ; English transl. of 1st ed. Dynamical systems X. General theory of vortices, Encyclopaedia Math. Sci., 67, Springer-Verlag, Berlin, 2003, viii+184 pp. | MR | Zbl | DOI | MR | Zbl

[6] L. C. Young, Lectures on the calculus of variations and optimal control theory, W. B. Saunders Co., Philadelphia–London–Toronto, ON, 1969, xi+331 pp. | MR | Zbl

[7] A. J. Hanson, T. Regge, and C. Teitelboim, Constrained Hamiltonian systems, Accad. Naz. Lincei, Rome, 1976, 135 pp.

[8] V. V. Nesterenko and A. M. Chervyakov, “Some properties of constraints in theories with degenerate Lagrangians”, Theoret. and Math. Phys., 64:1 (1985), 701–707 | DOI | MR | Zbl

[9] L. Faddeev and R. Jackiw, “Hamiltonian reduction of unconstrained and constrained systems”, Phys. Rev. Lett., 60:17 (1988), 1692–1694 | DOI | MR | Zbl

[10] B. M. Barbashov, “Hamiltonian formalism for Lagrangian systems with prescribed constraints”, Fiz. Elementarnykh Chastits i Atom. Yadra, 34:1 (2003), 5–42 (Russian)

[11] G. D. Birkhoff, Dynamical systems, Amer. Math. Soc. Colloq. Publ., 9, Amer. Math. Soc., New York, 1927, viii+295 pp. | MR | Zbl

[12] E. Newman and P. G. Bergmann, “Lagrangians linear in the “velocities””, Phys. Rev. (2), 99:2 (1955), 587–592 | DOI | MR | Zbl

[13] F. A. Berezin, “Hemiltonian formalizm in the general Lagrange problem”, Uspekhi Mat. Nauk, 29:3(177) (1974), 183–184 (Russian) | MR | Zbl

[14] V. I. Arnold, V. V. Kozlov, and A. I. Neĭshtadt, Mathematical aspects of classical and celestial mechanics, Encyclopaedia Math. Sci., 3, Dynamical systems. III, 3rd ed., Springer-Verlag, Berlin, 2006, xiv+518 pp. | DOI | MR | Zbl

[15] M. V. Deryabin, “The Dirac Hamiltonian formalism and the realization of constraints by small masses”, J. Appl. Math. Mech., 64:1 (2000), 35–39 | DOI | MR | Zbl

[16] D. R. Merkin, Giroscopic systems, 2nd ed., Nauka, Moscow, 1974, 344 pp. (Russian) | Zbl

[17] V. V. Strygin and V. A. Sobolev, Separation of motions by the method of integral manifolds, Nauka, Moscow, 1988, 256 pp. (Russian) | MR | Zbl

[18] A. Yu. Ishlinskii, Mechanics of gyroscolic systems, 2nd ed., USSR Academy of Sciences publishing house, Moscow, 1963, 482 pp. (Russian) | Zbl

[19] V. V. Kozlov, “The dynamics of systems with large gyroscopic forces and the realization of constraints”, J. Appl. Math. Mech., 78:3 (2014), 213–219 | DOI | MR | Zbl

[20] A. V. Vlakhova, “Nonholonomic motions of gyroscopic and wheeled systems”, Moscow Univ. Mech. Bull., 68:5 (2013), 126–132 | DOI | Zbl

[21] A. V. Vlakhova, “The dynamics of systems with rolling and gyroscopic systems with small generalized velocities and the realization of constraints”, J. Appl. Math. Mech., 78:6 (2014), 568–579 | DOI | MR | Zbl

[22] E. Cartan, Leçons sur les invariants intégraux, Hermann, Paris, 1922, x+210 pp. | MR | Zbl

[23] A. F. Filippov, “Differential equations with discontinnuous right-hand side”, Amer. Math. Soc. Transl. Ser. 2, 42, Amer. Math. Soc., Providence, RI, 1964, 199–231 | DOI | MR | Zbl

[24] V. I. Utkin, Sliding modes and their application in variable structure systems, Mir, Moscow, 1978, 257 pp. | Zbl

[25] P. Appell, Traité de mécanique rationnelle, v. I, 5-e éd., Gauthier-Villars, Paris, 1926, 619 pp. ; v. II, 6-e éd., 1953, 575 pp. | Zbl

[26] V. V. Kozlov, “On the integration theory of equations of nonholonomic mechanics”, Regul. Chaotic. Dyn., 7:2 (2002), 161–176 | DOI | MR | Zbl

[27] V. V. Kozlov, “Quadratic conservation laws for equations of mathematical physics”, Russian Math. Surveys, 75:3 (2020), 445–494 | DOI | MR | Zbl

[28] G. A. Bliss, Lectures on the calculus of variations, Univ. of Chicago Press, Chicago, IL, 1946, ix+296 pp. | MR | Zbl

[29] V. V. Kozlov, “Dynamics of systems with nonintegrable constraints. I”, Moscow Univ. Mech. Bull., 37:3-4 (1982), 27–34 ; II, 37:3-4 (1982), 74–80 ; III, 38:3 (1983), 40–51 ; IV. Integral principles, 42:5 (1987), 40–49 ; V. Freedom principle and ideal constraints condition, 43:6 (1988), 23–29 | MR | Zbl | MR | Zbl | MR | Zbl | MR | Zbl | MR | Zbl

[30] R. S. Strichartz, “Sub-Riemannian geometry”, J. Differential Geom., 24:2 (1986), 221–263 | DOI | MR | Zbl

[31] P. A. Griffiths, Exterior differential systems and the calculus of variations, Progr. Math., 25, Birkhäuser, Boston, MA, 1983, ix+335 pp. | DOI | MR | Zbl

[32] A. M. Vershik and V. Ya. Gershkovich, “Nonholonomic dynamical systems, geometry of distributions and variational problems”, Dynamical systems VII, Encyclopaedia Math. Sci., 16, Springer, Berlin, 1994, 1–81 | DOI | MR | MR | Zbl

[33] A. A. Agrachev, “Topics in sub-Riemannian geometry”, Russian Math. Surveys, 71:6 (2016), 989–1019 | DOI | MR | Zbl

[34] H. Hertz, Die Principien der Mechanik in neuem Zusammenhange dargestellt, Gesammelte Werke, III, J. A. Barth, Leipzig, 1894, xxix+312 pp. | Zbl