Generalized Duality, Hamiltonian Formalism and New Brackets
DOI:
https://doi.org/10.15407/mag10.02.189Анотація
Показано, що будь-яка сингулярна лагранжева теорія: 1) може бути сформульована без залучення зв’язків за допомогою Клеро-версії гамільтонового формалізму; 2) приводить до спеціального вигляду неабелевої калібрувальної теорії, яка подібна до пуассонової калібрувальної теорії; 3) може бути сформульована як багаточасова класична динаміка. Узагальнення перетворення Лежандра на випадок нульового гессіана проведено з використанням змішаного (обгортаючого/загального) розв’язку багатовимірного рівняння Клеро. Рівняння руху записуються в гамільтоновій формі за допомогою введення нових антисиметричних дужок. Відзначено, що будь-яка класична система з виродженим лагранжіаном еквівалентна багатовимірній класичній динаміці. На закінчення наведено взаємовідношення представленого формалізму й теорії зв’язків Дірака.
Mathematics Subject Classification: 37J05, 44A15, 49K20, 70H45.
Ключові слова:
зв’язки Дірака, неабелева калібрувальна теорія, вироджений лагранжіан, гессіан, перетворення Лежандра, багатовимірне рівняння Клеро, калібрувальна свобода, дужка Пуассона, багатовимірна динамікаПосилання
R. Abraham and J.E. Marsden, Foundations of Mechanics. Benjamin–Cummings, Reading, 1978.
P. Alart, O. Maisonneuve, and R. T. Rockafellar, Nonsmooth Mechanics and Analysis: Theoretical and Numerical Advances. Springer–Verlag, Berlin, 2006. https://doi.org/10.1007/0-387-29195-4
D.V. Alekseevskij, A.M. Vinogradov, and V.V. Lychagin, Geometry I: Basic Ideas and Concepts of Differential Geometry. Springer, New York, 1991.
V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations. Springer–Verlag, New York, 1988. https://doi.org/10.1007/978-3-662-11832-0
V.I. Arnold, Mathematical Methods of Classical Mechanics. Springer, Berlin, 1989. https://doi.org/10.1007/978-1-4757-2063-1
D. Baleanu, Fractional Hamiltonian Analysis of Irregular Systems. — Signal Proc. 86 (2006), 2632–2636. https://doi.org/10.1016/j.sigpro.2006.02.008
L. Bates and J. Śniatycki, Nonholonomic reduction. — Rep. Math. Phys. 32 (1993), 99– 115. https://doi.org/10.1016/0034-4877(93)90073-N
C. Battle, J. Gomis, J.M. Pons, and N. Roman-Roy, On the Legendre Transformation for Singular Lagrangians and Related Topics. — J. Phys. A: Math. Gen. 20 (1987), 5113–5123. https://doi.org/10.1088/0305-4470/20/15/027
P. Bueken, Multi-Hamiltonian Formulation for a Class of Degenerate Completely Integrable Systems. — J. Math. Phys. 37 (1996), 2851–2862. https://doi.org/10.1063/1.531543
G. Caratù, G. Marmo, A. Simoni, B. Vitale, and F. Zaccaria, Lagrangian and Hamiltonian Eormalisms: an Analysis of Classical Mechanics on Tangent and Cotangent Bundles. — Nuovo Cim. B31 (1976), 152–172. https://doi.org/10.1007/BF02730325
J.F. Carinena, Theory of Singular Lagrangians. — Fortsch. Physik 38 (1990), 641–679. https://doi.org/10.1002/prop.2190380902
R. Cawley, Determination of the Hamiltonian in the Presence of Constraints. — Phys. Rev. Lett. 42 (1979), 413–416. https://doi.org/10.1103/PhysRevLett.42.413
H. Cendra, D.D. Holm, M.J.V. Hoyle, and J.E. Marsden, The Maxwell–Vlasov Equations in Euler–Poincare form. — J. Math. Phys. 39 (1998), 3138–3157. https://doi.org/10.1063/1.532244
N.H. Christ and T.D. Lee, Operator Ordering and Feynman Rules in Gauge Theories. — Phys. Rev. D22 (1980), 939–958. https://doi.org/10.1103/PhysRevD.22.939
F.H. Clarke, Extremal Arcs and Extended Hamiltonian Systems. — Trans. Amer. Math. Soc. 231 (1977), 349–367. https://doi.org/10.1090/S0002-9947-1977-0442784-4
P. Dazord, Mécanique Hamiltonienne en Presence de Constraintes. — Illinois J. Math. 38 (1994), 148–1175.
Quantum Fields and Strings: A Cource for Mathematicians, P. Deligne, P. Etingof, D.S. Freed, L.C. Jeffrey, D. Kazhdan, J.W. Morgan, D.R. Morrison, and E. Witten (Eds.), Vol. 1, 2. AMS, Providence, 1999.
A.A. Deriglazov, Analysis of Constrained Theories Without Use of Primary Constraints. — Phys. Lett. B626 (2005), 243–248. https://doi.org/10.1016/j.physletb.2005.08.114
P.A.M. Dirac, Lectures on Quantum Mechanics. Yeshiva University, New York, 1964.
R.G. Di Stefano, A Modification of Dirac’s Method of Hamiltonian Analysis. — Phys. Rev. D27 (1983), 1752–1765. https://doi.org/10.1103/PhysRevD.27.1752
D. Dominici, G. Longhi, J. Gomis, and J.M. Pons, Hamilton–Jacobi Theory for Constrained Systems. — J. Math. Phys. 25 (1984), 2439–2452. https://doi.org/10.1063/1.526452
P. Droz-Vincent, Is Interaction Possible without Heredity? — Lett. Nuovo Cim. 1 (1969), 839–841. https://doi.org/10.1007/BF02753981
S. Duplij, Some Abstract Properties of Semigroups Appearing in Superconformal Theories. — Semigroup Forum 54 (1997), 253–260 (hep-th/9505179). https://doi.org/10.1007/BF02676606
S. Duplij, Analysis of Constraint Systems Using the Clairaut Equation. Proceedings of 5th Mathematical Physics Meeting: Summer School in Modern Mathematical Physics, 6–17 July 2008, B. Dragovich and Z. Rakic (Eds.), Institute of Physics, Belgrade, 2009, 217– 225.
S. Duplij, A New Hamiltonian Formalism for Singular Lagrangian Theories. — J. Kharkov National Univ., ser. Nuclei, Particles and Fields 969 (2011), 34–39.
D.B. Fairlie, P. Fletcher, and C.K. Zachos, Infinite-Dimensional Algebras and a Trigonometric Basis for the Classical Lie Algebras. — J. Math. Phys. 31 (1990), 1088–1094. https://doi.org/10.1063/1.528788
W. Fenchel, On Conjugate Convex Functions. — Canad. J. Math. 1 (1949), 73–77. https://doi.org/10.4153/CJM-1949-007-x
E. Floratos, J. Iliopoulos, and G. Tiktopoulos, A Note on SU (∞) Classical Yang–Mills Theories. — Phys. Lett. 217 (1989), 285–288. https://doi.org/10.1016/0370-2693(89)90867-8
G. Giachetta, L. Mangiarotti, and G. Sardanashvily, New Lagrangian and Hamiltonian Methods in Field Theory. World Scientific, Singapore, 1997. https://doi.org/10.1142/2199
D.M. Gitman and I.V. Tyutin, Canonical Quantization of Constrained Fields. Nauka, Moskow, 1986.
D.M. Gitman and I.V. Tyutin, Quantization of Fields with Constraints. Springer–Verlag, Berlin, 1990. https://doi.org/10.1007/978-3-642-83938-2
D.M. Gitman and I.V. Tyutin, Hamiltonization of Theories with Degenerate Coordinates. — Nucl. Phys. B630 (2002), 509–527. https://doi.org/10.1016/S0550-3213(02)00170-0
R. Goebel and R.T. Rockafellar, Generalized Conjugacy in Hamilton–Jacobi Theory for Fully Convex Lagrangians. — J. Convex Analysis 9 (2002), 463–474.
S.A. Gogilidze, A.M. Khvedelidze, D.M. Mladenov, and H.-P. Pavel, Hamiltonian Reduction of SU (2) Dirac–Yang–Mills Mechanics. — Phys. Rev. D57 (1998), 7488–7500. https://doi.org/10.1103/PhysRevD.57.7488
S.A. Gogilidze, A.M. Khvedelidze, and V.N. Pervushin, On Abelianization of First Class Constraints. — J. Math. Phys. 37 (1996), 1760–1771. https://doi.org/10.1063/1.531478
H. Goldstein, Classical Mechanics. Addison–Wesley, Reading, 1990.
W. Greiner and J. Reinhardt, Field Quantization. Springer, Berlin, 1996. https://doi.org/10.1007/978-3-642-61485-9
M. Henneaux and C. Teitelboim, Quantization of Gauge Systems. Princeton University Press, Princeton, 1994.
N. Hitchin, Generalized Calabi–Yau manifolds. — Quart. J. Math. Oxford Ser. 54 (2003), 281–308. https://doi.org/10.1093/qmath/hag025
A. Ioffe, Euler–Lagrange and Hamiltonian Formalisms in Dynamic Optimization. — Trans. Amer. Math. Soc. 349 (1997), 2871–2900. https://doi.org/10.1090/S0002-9947-97-01726-1
S. Izumiya, Systems of Clairaut Type. — Colloq. Math. 66 (1994), 219–226. https://doi.org/10.2996/kmj/1138040057
E. Kamke, Üeber Clairautsche Differentialgleichung. — Math. Zeit. 27 (1928), 623–639. https://doi.org/10.1007/BF01171118
A.M. Khvedelidze, On the Hamiltonian Formulation of Gauge Theories in Terms of Physical Variables. — J. Math. Sci. 119 (2004), 513–555. https://doi.org/10.1023/B:JOTH.0000009374.82259.de
A. Komar, Constraint Formalism of Classical Mechanics. — Phys. Rev. D18 (1978), 1881– 1886. https://doi.org/10.1103/PhysRevD.18.1881
A. Konechny and A. Schwarz, Introduction to Matrix Theory and Noncommutative Geometry. — Phys. Rep. 360 (2002), 353–465. https://doi.org/10.1016/S0370-1573(01)00096-5
O. Krupkova, Hamiltonian Field Theory. — J. Geom. Phys. 43 (2002), 93–132. https://doi.org/10.1016/S0393-0440(01)00087-0
O. Krupkova and D. Smetanova, Legendre Transformation for Regularizable Lagrangians in Field Theory. — Lett. Math. Phys. 58 (2001), 189–204. https://doi.org/10.1023/A:1014548309187
G. Longhi, L. Lusanna, and J.M. Pons, On the Many-time Formulation of Classical Particle Dynamics. — J. Math. Phys. 30 (1989), 1893–1912. https://doi.org/10.1063/1.528224
F. Loran, Non-Abelianizable First Class Constraints. — Commun. Math. Phys. 254 (2005), 167–178. https://doi.org/10.1007/s00220-004-1248-7
L. Lusanna, The Second Noether Theorem as the Basis of the Theory of Singular Lagrangians and Hamiltonian Constraints. — Riv. Nuovo Cim. 14 (1991), 1–75. https://doi.org/10.1007/BF02810161
V.K. Maltsev, On Canonical Formulation of Field Theories with Singular Lagrangians. — JETP Lett. 27 (1978), 473–475.
C.M. Marle, Géométrie des Systèmes Méchaniques à Liaisons Actives. Symplectic Geometry and Mathematical Physics, P. Donato, C. Duval, J. Elhadad, and G. M. Tuynman (Eds.), Birkhäuser, Boston, 1991, 260–287. https://doi.org/10.1007/978-1-4757-2140-9_13
G. Marmo, G. Mendella, and W. Tulczyjew, Constrained Hamiltonian Systems as Implicit Differential Equations. — J. Phys. A30 (1997), 277–293. https://doi.org/10.1088/0305-4470/30/1/020
G. Marmo, E.J. Saletan, A. Simoni, and B. Vitale, Dynamical Systems: a Differential Geometric Approach to Symmetry and Reduction. J. Wiley, Chichester, 1985.
M.R. Menzo and W.M. Tulczyjew, Infinitesimal Symplectic Relations and Deneralized Hamiltonian Dynamics. — Ann. Inst. Henri Poincaré A4 (1978), 349–367.
O. Mišković and J. Zanelli, Dynamical Structure of Irregular Constrained Systems. — J. Math. Phys. 44 (2003), 3876–3887. https://doi.org/10.1063/1.1601299
J.M. Pons, On Dirac’s Incomplete Analysis of Gauge Transformations. — Stud. Hist. Philos. Mod. Phys. 36 (2005), 491–518. https://doi.org/10.1016/j.shpsb.2005.04.004
A. Randono, Canonical Lagrangian Dynamics and General Relativity. — Class. Quant. Grav. 25 (2008), 205017. https://doi.org/10.1088/0264-9381/25/20/205017
T. Regge and C. Teitelboim, Constrained Hamiltonian Systems. Academia Nazionale dei Lincei, Rome, 1976.
R.T. Rockafellar, Conjugates and Legendre Transforms of Convex Functions. — Canad. J. Math. 19 (1967), 200–205. https://doi.org/10.4153/CJM-1967-012-4
R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, 1970. https://doi.org/10.1515/9781400873173
R.T. Rockafellar, Equivalent Subgradient Versions of Hamiltonian and Euler–Lagrange Equations in Variational Analysis. — SIAM J. Control Opt. 34 (1996), 1300–1315. https://doi.org/10.1137/S0363012994273465
M.V. Saveliev and A.M. Vershik, New Examples of Continuum Graded Lie Algebras. — Phys. Lett. A143 (1990), 121–128. https://doi.org/10.1016/0375-9601(90)90662-8
S. Sternberg, Legendre Transformation of Curves. — Proc. Amer. Math. Soc. 5 (1954), 942–945. https://doi.org/10.1090/S0002-9939-1954-0066704-X
E.C. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective. Wiley, New York, 1974.
K. Sundermeyer, Constrained Dynamics. Springer–Verlag, Berlin, 1982.
W.M. Tulczyjew, The Legendre Transformation. — Ann. Inst. Henri Poincaré A27 (1977), 101–114.
W.M. Tulczyjew, Geometric Formulation of Physical Theories. Bibliopolis, Naples, 1989.
W.M. Tulczyjew and P. Urbański, A Slow and Careful Legendre Transformation for Singular Lagrangians. — Acta Phys. Pol. B30 (1999), 2909–2977.
C. Vallee, M. Hjiaj, D. Fortune, and G. de Saxce, Generalized Legendre–Fenchel Transformation. — Adv. Mech. Math. 6 (2004), 289–312. https://doi.org/10.1007/978-90-481-9577-0_17
A.J. van der Schaft, Implicit Hamiltonian Systems with Symmetry. — Rep. Math. Phys. 41 (1998), 203–221. https://doi.org/10.1016/S0034-4877(98)80176-6
A.M. Wang and T.N. Ran, Corrected Forms of Physical Supplementary Conditions and Gauge Conditions in Singular Lagrangian Systems. — Phys. Rev. Lett. 73 (1994), 2011– 2014. https://doi.org/10.1103/PhysRevLett.73.2011
S. Weinberg, The Quantum Theory of Fields. Vols. 1, 2, 3. Cambridge University Press, Cambridge, 1995–2000.
A. Wipf, Hamilton’s Formalism for Systems with Constraints. In: Canonical Gravity: From Classical to Quantum, J. Ehlers and H. Friedrich (Eds.), Springer, Heidelberg, 1994, 22–58. https://doi.org/10.1007/3-540-58339-4_14
H. Yoshimura and J.E. Marsden, Dirac Structures and the Legendre Transformation for Implicit Lagrangian and Hamiltonian Systems. In: Lagrangian and Hamiltonian Methods for Nonlinear Control, F. Bullo (Ed.), Springer, Berlin, 2007, 233–247.
C. Zachos, Hamiltonian Flows, SU (∞), SO(∞), U Sp(∞), and Strings, In: Differen-tial Geometric Methods in Theoretical Physics: Physics and Geometry, L.-L. Chau andW. Nahm (Eds.), NATO ASI, Plenum Press, New York, 1990, 423–430. https://doi.org/10.1007/978-1-4684-9148-7_43
