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Decomposition of almost Poisson structure of non-self-adjoint dynamical systems

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Abstract

Non-self-adjoint dynamical systems, e.g., nonholonomic systems, can admit an almost Poisson structure, which is formulated by a kind of Poisson bracket satisfying the usual properties except for the Jacobi identity. A general theory of the almost Poisson structure is investigated based on a decomposition of the bracket into a sum of a Poisson one and an almost Poisson one. The corresponding relation between Poisson structure and symplectic structure is proved, making use of Jacobiizer and symplecticizer. Based on analysis of pseudo-symplectic structure of constraint submanifold of Chaplygin’s nonholonomic systems, an almost Poisson bracket for the systems is constructed and decomposed into a sum of a canonical Poisson one and an almost Poisson one. Similarly, an almost Poisson structure, which can be decomposed into a sum of canonical one and an almost “Lie-Poisson” one, is also constructed on an affine space with torsion whose autoparallels are utilized to describe the free motion of some non-self-adjoint systems. The decomposition of the almost Poisson bracket directly leads to a decomposition of a dynamical vector field into a sum of usual Hamiltionian vector field and an almost Hamiltonian one, which is useful to simplifying the integration of vector fields.

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References

  1. Santilli R M. Foundations of Theoretical Mechanics I. New York: Springer-Verlag, 1978

    MATH  Google Scholar 

  2. Sarlet W, Prince G E, Crampin M. Adjoint symmetries for time- dependent second-order equations. J Phys A: Math Gen, 1990, 23(8): 1335–1347

    Article  MATH  MathSciNet  Google Scholar 

  3. Guo Y X, Shang M, Mei F X. Poincare-Cartan integral invariant of non-conservative dynamical systems. Int J Theor Phys, 1999, 38(3): 1017–1027

    Article  MATH  MathSciNet  Google Scholar 

  4. Marsden J E, Ratiu T S. Introduction to Mechanics and Symmetry. 2nd ed. NewYork: Springer-Verlag, 1999

    MATH  Google Scholar 

  5. Santilli R M. Foundations of Theoretical Mechanics. New York: Springer-Verlag, 1983

    MATH  Google Scholar 

  6. van der Schaft A J, Maschke B M. The Hamiltonian formulation of energy conserving physical systems with external ports. Int J Electron Commun, 1995, 49(3): 362–371

    Google Scholar 

  7. Guo Y X, Liu S X, Liu C, et al. Influence of nonholonomic constraints on variations, symplectic structure and dynamics of mechanical systems. J Math Phys, 2007, 48(7): 082901

    Google Scholar 

  8. Bloch A M, Baillieul J, Crouch P, et al. Nonholonomic Mechanics and Control. London: Springer, 2003

    MATH  Google Scholar 

  9. Guo Y X, Luo S K, Mei F X. Progress of geometric dynamics of nonholonomic constrained mechanical systems: Lagrange theory and other aspects (in Chinese). Adv Mech, 2004, 34(3): 477–492

    Google Scholar 

  10. Mei F X, Liu D, Luo Y. Advanced Analytical Mechanics (in Chinese). Beijing: Press of Beijing Institute of Technology, 1991

    Google Scholar 

  11. Neimark Ju I, Fufaev N A. Dynamics of Nonholonomic Systems. New York: American Mathematical Society, 1972

    MATH  Google Scholar 

  12. Shapiro I L. Physical aspects of the space-time torsion. Phys Rep, 2002, 357(1): 113

    Article  MATH  MathSciNet  Google Scholar 

  13. Hehl F W, McCrea J D, Mielke E W. Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance. Phys Rep, 1995, 258(1): 1–171

    Article  MathSciNet  Google Scholar 

  14. Hammond R T. Torsion gravity. Rep Prog Phys, 2002, 65(5): 599–649

    Article  MathSciNet  Google Scholar 

  15. Guo Y X, Wang Y, Chee G Y, et al. Nonholonomic versus vakonomic dynamics on a Riemann-Cartan manifold. J Math Phys, 2005, 46(5): 062902

    Google Scholar 

  16. Guo Y X, Song Y B, Zhang X B, et al. Almost-Poisson structure for autoparallels on Riemann-Cartan spacetime. Chin Phys Lett, 2003, 20(8): 1192–1195

    Article  Google Scholar 

  17. Maulbetsch C, Shabonov S V. The inverse variational problem for autoparallels. J Phys A: Math Gen, 1999, 32: 5355–5366

    Article  MATH  Google Scholar 

  18. Zecca A. Dirac equation in space-time with torsion. Int J Theor Phys, 2002, 41(3): 421–428

    Article  MATH  MathSciNet  Google Scholar 

  19. Kleinert H. Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion. 2002, arXiv: gr-qc/0203029v1

  20. Fiziev P, Kleinert H. Motion of a Rigid Body in Body-Fixed Coordinate System—for Autoparrallel Trajectories in Spaces with Torsion. 1995, arXiv:hep-th/9503075v1

  21. van der Schaft A J, Maschke B M. On the Hamiltonian formulation of nonholonomic mechanical systems. Rep Math Phys, 1994, 34(2): 225–233

    Article  MATH  MathSciNet  Google Scholar 

  22. Koon W S, Marsden J E. Poisson reduction for nonholonomic mechanical systems with symmetry. Rep Math Phys, 1998, 42(1): 101–133

    Article  MATH  MathSciNet  Google Scholar 

  23. Austin M A, Krishnaprasad P S, Wang L S. Almost Poisson integration of rigid body systems. J Comput Phys, 1993, 107(1): 105–117

    Article  MATH  MathSciNet  Google Scholar 

  24. Le Blanc A. Quasi-Poisson structures and integrable systems related to the moduli space of flat connections on a punctured Riemann sphere. J Geom Phys, 2007, 57(8): 1631–1652

    Article  MATH  MathSciNet  Google Scholar 

  25. Cantrijn F, de León M, de Diego D M. On almost-Poisson structures in nonholonomic mechanics. Nonlinearity, 1999, 12(3): 721–737

    Article  MATH  MathSciNet  Google Scholar 

  26. Cantrijn F, de León M, Marrero J C, et al. On almost-Poisson structures in nonholonomic mechanics: II. The time-dependent framework. Nonlinearity, 2000, 13(4): 1379–1409

    Article  MATH  MathSciNet  Google Scholar 

  27. Mei F X. The algebraic structure and poisson’s theory for the equations of motion of non-holonomic systems. J Appl Math Mech, 1998, 62(1): 155–158

    Article  MathSciNet  Google Scholar 

  28. García-Naranjo L. Reduction of almost Poisson brackets for nonholonomic systems on Lie groups. Regul Chaot Dynam, 2007, 12(4): 365–388

    Article  Google Scholar 

  29. Tang X. Deformation quantization of pseudo-symplectic (Poisson) groupoids. Geom Funct Anal, 2006, 16(3): 731–766

    MATH  MathSciNet  Google Scholar 

  30. Li H F. Strict Quantizations of Almost Poisson Manifolds. Commun Math Phys, 2005, 257(2): 257–272

    Article  MATH  Google Scholar 

  31. Chen K C. Noncanonical Poisson brackets for elastic and micromorphic solids. Int J Solid Struct, 2007, 44(24): 7715–7730

    Article  Google Scholar 

  32. Cendra H, Grillo S. Generalized nonholonomic mechanics, servomechanisms and related brackets. J Math Phys, 2006, 47(2): 022902

    Google Scholar 

  33. José F, Cariñena J F, da Costa J M N, et al. Internal deformation of Lie algebroids and symplectic realizations. J Phys A: Math Gen, 2006, 39(22): 6897–6918

    Article  MATH  Google Scholar 

  34. Patrick G W. Variational development of the semi-symplectic geometry of nonholonomic mechanics. Rep Math Phys, 2007, 59(2): 145–184

    Article  MATH  MathSciNet  Google Scholar 

  35. Guo Y X, Luo S K, Shang M, et al. Birkhoffian formulation of nonholonomic constrained systems. Rep Math Phys, 2001, 47(3): 313–322

    Article  MATH  MathSciNet  Google Scholar 

  36. Guo Y X, Mei F X. Integrability for Pfaffian constrained systems: a geometrical theory. Acta Mech Sinica, 1998, 14(1): 85–91

    Google Scholar 

  37. Kleinert H, Pelster A. Autoparallels from a new action principle. Gen Relat Grav, 1999, 31(9): 1439–1447

    Article  MATH  MathSciNet  Google Scholar 

  38. Shabanov S V. Constrained systems and analytical mechanics in spaces with torsion. J Phys A: Math Gen, 1998, 31(22): 5177–5190

    Article  MATH  MathSciNet  Google Scholar 

  39. Kleinert H, Shabonov S V. Theory of Brownian motion of a massive particle in spaces with curvature and torsion. J Phys A: Math Gen, 1998, 31(34): 7005–7009

    Article  MATH  Google Scholar 

  40. Maulbetsch C, Shabanov S V. The inverse variational problem for autoparallels. J Phys A: Math Gen, 1999, 32(28): 5355–5366

    Article  MATH  MathSciNet  Google Scholar 

  41. Kleinert H, Shabanov S V. Space with torsion from embedding, and the special role of autoparallel trajectories. Phys Lett B, 1998, 428(2): 315–321

    MathSciNet  Google Scholar 

  42. Fiziev P P, Kleinert H. New action principle for classical particle trajectories in spaces with torsion. Europhys Lett, 1996, 35(2): 241–246

    Article  Google Scholar 

  43. Shashikanth B N, Sheshmani A, Kelly S D, et al. Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape. Theor Comput Fluid Dynam, 2008, 22(1): 37–64

    Article  Google Scholar 

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Authors and Affiliations

  1. College of Physics, Liaoning University, Shenyang, 110036, China

    YongXin Guo, ShiXing Liu & Peng Chang

  2. Department of Applied Mechanics, Beijing Institute of Technology, Beijing, 100081, China

    Chang Liu & ShiXing Liu

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  1. YongXin Guo
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  2. Chang Liu
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  3. ShiXing Liu
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  4. Peng Chang
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Correspondence to YongXin Guo.

Additional information

Supported by the National Natural Science Foundation of China (Grant Nos. 10872084, 10472040), the Outstanding Young Talents Training Fund of Liaoning Province of China (Grant No. 3040005) and the Research Program of Higher Education of Liaoning Province of China (Grant No. 2008S098)

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Guo, Y., Liu, C., Liu, S. et al. Decomposition of almost Poisson structure of non-self-adjoint dynamical systems. Sci. China Ser. E-Technol. Sci. 52, 761–770 (2009). https://doi.org/10.1007/s11431-009-0038-z

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  • DOI: https://doi.org/10.1007/s11431-009-0038-z

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