Journal of Generalized Lie Theory and Applications

Time transformation and reversibility of Nambu--Poisson systems

Klas MODIN

Full-text: Open access

Abstract

A time transformation technique for Nambu--Poisson systems is developed, and its structural properties are examined. The approach is based on extension of the phase space $\mathscr{P}$ into $\overline{\mathscr{P}}=\mathscr{P}\times\mathbb{R}$, where the additional variable controls the time-stretching rate. It is shown that time transformation of a system on $\mathscr{P}$ can be realised as an extended system on $\overline{\mathscr{P}}$, with an extended Nambu-Poisson structure. In addition, reversible systems are studied in conjunction with the Nambu-Poisson structure. The application in mind is adaptive numerical integration by splitting of Nambu-Poisson Hamiltonians. As an example, a novel integration method for the rigid body problem is presented and analysed.

Article information

Source
J. Gen. Lie Theory Appl., Volume 3, Number 1 (2009), Article ID S080103, 39-52.

Dates
First available in Project Euclid: 19 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.jglta/1319028474

Digital Object Identifier
doi:10.4303/jglta/S080103

Mathematical Reviews number (MathSciNet)
MR2486609

Zentralblatt MATH identifier
1167.37042

Subjects
Primary: 37M99: None of the above, but in this section 22E70: Applications of Lie groups to physics; explicit representations [See also 81R05, 81R10] 53Z05: Applications to physics 34A26: Geometric methods in differential equations

Citation

MODIN, Klas. Time transformation and reversibility of Nambu--Poisson systems. J. Gen. Lie Theory Appl. 3 (2009), no. 1, Article ID S080103, 39--52. doi:10.4303/jglta/S080103. https://projecteuclid.org/euclid.jglta/1319028474


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References

  • D. Alekseevsky and P. Guha. On decomposability of Nambu-Poisson tensor. Acta Math. Univ. Comenian. (N.S.), 65 (1996), 1–9.
  • S. Blanes and C. J. Budd. Adaptive geometric integrators for Hamiltonian problems with approximate scale invariance. SIAM J. Sci. Comput., 26 (2005), 1089–1113 (electronic).
  • S. D. Bond and B. J. Leimkuhler. Time-transformations for reversible variable stepsize integration. Numer. Algorithms, 19 (1998), 55–71,
  • R. Chatterjee and L. Takhtajan. Aspects of classical and quantum Nambu mechanics. Lett. Math. Phys. 37 (1996), 475–482.
  • S. Codriansky, R. Navarro, and M. Pedroza. The Liouville condition and Nambu mechanics. J. Phys. A, 29 (1996), 1037–1044.
  • P. Gautheron. Some remarks concerning Nambu mechanics. Lett. Math. Phys. 37 (1996),103–116.
  • E. Hairer. Variable time step integration with symplectic methods. Appl. Numer. Math., 25 (1007), 219–227.
  • E. Hairer, C. Lubich, and G. Wanner. Geometric numerical integration. Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, Second Ed., 2006.
  • E. Hairer and G. Söderlind. Explicit, time reversible, adaptive step size control. SIAM J. Sci. Comput. 26 (2005), 1838–1851 (electronic).
  • W. Huang and B. Leimkuhler. The adaptive Verlet method. SIAM J. Sci. Comput. 18 (1997), 239–256.
  • R. Ibáñez, M. de León, J. C. Marrero, and D. Martin de Diego. Dynamics of generalized Poisson and Nambu-Poisson brackets. J. Math. Phys. 38 (1997), 2332–2344.
  • J. S. W. Lamb and J. A. G. Roberts. Time-reversal symmetry in dynamical systems: a survey. Phys. D, 112 (1998), 1–39.
  • G. Marmo, G. Vilasi, and A. M. Vinogradov. The local structure of $n$-Poisson and $n$-Jacobi manifolds. J. Geom. Phys. 25 (1998), 141–182.
  • J. E. Marsden and T. S. Ratiu. Introduction to mechanics and symmetry, Texts in Applied Mathematics, 17, Springer-Verlag, New York, Second Ed., 1999.
  • R. I. McLachlan. On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16 (1995),151–168.
  • R. I. McLachlan and G. R. W. Quispel. What kinds of dynamics are there? Lie pseudogroups, dynamical systems and geometric integration. Nonlinearity, 14 (2001), 1689–1705.
  • R. I. McLachlan and G. R. W. Quispel. Splitting methods. Acta Numer. 11 (2002), 341–434, 2002.
  • S. Mikkola and P. Wiegert. Regularizing time transformations in symplectic and composite integration. Celestial Mech. Dynam. Astronom. 82 (2002), 375–390.
  • K. Modin. On explicit adaptive symplectic integration of separable Hamiltonian systems. Journal of Multibody Dynamics, 2008 (to be published).
  • K. Modin and C. Führer. Time-step adaptivity in variational integrators with application to contact problems. ZAMM Z. Angew. Math. Mech. 86 (2006), 785–794.
  • P. Morando. Liouville condition, Nambu mechanics, and differential forms. J. Phys. A, 29 (1996), L329–L331.
  • N. Nakanishi. On Nambu-Poisson manifolds. Rev. Math. Phys. 10 (1998), 499–510.
  • Y. Nambu. Generalized Hamiltonian dynamics. Phys. Rev. D, 7 (1973), 2405–2412.
  • M. Preto and S. Tremaine. A class of symplectic integrators with adaptive timestep for separable hamiltonian systems. Astron. J. 118 (1999), 2532–2541.
  • R. Schmid. Infinite dimensional Lie groups with applications to mathematical physics. J. Geom. Symmetry Phys. 1 (2004), 54–120.
  • D. Stoffer. Variable steps for reversible integration methods. Computing, 55 (1995), 1–22.
  • L. Takhtajan. On foundation of the generalized Nambu mechanics. Comm. Math. Phys. 160 (1994), 295–315.
  • I. Vaisman. A survey on Nambu-Poisson brackets. Acta Math. Univ. Comenian. (N.S.), 68 (1999), 213–241.
  • I. Vaisman. Nambu-Lie groups. J. Lie Theory, 10 (2000), 181–194.