Advertisement

Acta Applicandae Mathematicae

, Volume 154, Issue 1, pp 189–230 | Cite as

Quadratic Hamilton–Poisson Systems on \(\mathfrak{se}(1,1)^{*}_{-}\): The Inhomogeneous Case

  • D. I. Barrett
  • R. Biggs
  • C. C. Remsing
Article

Abstract

We consider equivalence, stability and integration of quadratic Hamilton–Poisson systems on the semi-Euclidean Lie–Poisson space \(\mathfrak{se}(1,1)^{*}_{-}\). The inhomogeneous positive semidefinite systems are classified (up to affine isomorphism); there are 16 normal forms. For each normal form, we compute the symmetry group and determine the Lyapunov stability nature of the equilibria. Explicit expressions for the integral curves of a subclass of the systems are found. Finally, we identify several basic invariants of quadratic Hamilton–Poisson systems.

Keywords

Hamilton–Poisson system Lie–Poisson space Lyapunov stability 

Mathematics Subject Classification (2010)

53D17 37J25 

References

  1. 1.
    Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Addison-Wesley, Reading (1978) MATHGoogle Scholar
  2. 2.
    Adams, R.M., Biggs, R., Remsing, C.C.: Single-input control systems on the Euclidean group \(\mathsf{SE}(2)\). Eur. J. Pure Appl. Math. 5, 1–15 (2012) MathSciNetMATHGoogle Scholar
  3. 3.
    Adams, R.M., Biggs, R., Remsing, C.C.: On some quadratic Hamilton–Poisson systems. Appl. Sci. 15, 1–12 (2013) MathSciNetMATHGoogle Scholar
  4. 4.
    Adams, R.M., Biggs, R., Remsing, C.C.: Quadratic Hamilton–Poisson systems on \(\mathfrak{so}(3)^{*}_{-}\): classification and integration. In: Proc. 15th Int. Conf. Geom., Integrability and Quantization, pp. 55–66. Bulgarian Academy of Sciences, Varna (2013) Google Scholar
  5. 5.
    Adams, R.M., Biggs, R., Remsing, C.C.: Two-input control systems on the Euclidean group \(\mathsf{SE}(2)\). ESAIM Control Optim. Calc. Var. 19, 947–975 (2013) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Adams, R.M., Biggs, R., Holderbaum, W., Remsing, C.C.: On the stability and integration of Hamilton–Poisson systems on \(\mathfrak{so}(3)^{*} _{-}\). Rend. Mat. Appl. 37, 1–42 (2016) MathSciNetMATHGoogle Scholar
  7. 7.
    Agrachev, A.A., Sachkov, Y.L.: Control Theory from the Geometric Viewpoint. Springer, Berlin (2004) CrossRefMATHGoogle Scholar
  8. 8.
    Armitage, J.V., Eberlein, W.F.: Elliptic Functions. Cambridge University Press, Cambridge (2006) CrossRefMATHGoogle Scholar
  9. 9.
    Aron, A., Dăniasă, C., Puta, M.: Quadratic and homogeneous Hamilton–Poisson systems on \((\mathfrak{so}(3))^{*}\). Int. J. Geom. Methods Mod. Phys. 4, 1173–1186 (2007) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Aron, A., Pop, C., Puta, M.: Some remarks on \((\mathfrak{sl}(2, \mathbb{R}))^{*}\) and Kahan’s integrator. An. Ştiinţ. Univ. ‘Al.I. Cuza’ Iaşi, Mat. 53, 49–60 (2007) MathSciNetMATHGoogle Scholar
  11. 11.
    Aron, A., Mos, I., Csaky, A., Puta, M.: An optimal control problem on the Lie group \(\mathsf{SO}(4)\). Int. J. Geom. Methods Mod. Phys. 5, 319–327 (2008) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Aron, A., Pop, C., Puta, M.: An optimal control problem on the Lie group \(\mathsf{SE}(2,\mathbb{R}) \times \mathsf{SO}(2)\). Bol. Soc. Mat. Mexicana 15, 129–140 (2009) MathSciNetMATHGoogle Scholar
  13. 13.
    Aron, A., Craioveanu, M., Pop, C., Puta, M.: Quadratic and homogeneous Hamilton–Poisson systems on \(\mathsf{A}^{*}_{3,6,-1}\). Balk. J. Geom. Appl. 15, 1–7 (2010) MathSciNetMATHGoogle Scholar
  14. 14.
    Barrett, D.I., Biggs, R., Remsing, C.C.: Quadratic Hamilton–Poisson systems on \(\mathfrak{se}(1,1)^{*}_{-}\): the homogeneous case. Int. J. Geom. Methods Mod. Phys. 12, 1550011 (2015) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Biggs, J., Holderbaum, W.: Integrable quadratic Hamiltonians on the Euclidean group of motions. Int. J. Geom. Methods Mod. Phys. 16, 301–317 (2010) MathSciNetMATHGoogle Scholar
  16. 16.
    Biggs, R., Remsing, C.C.: A classification of quadratic Hamilton–Poisson systems in three dimensions. In: Proc. 15th Int. Conf. Geom., Integrability and Quantization (GIQ-2013), pp. 67–78. Bulgarian Academy of Sciences, Varna (2013) Google Scholar
  17. 17.
    Biggs, R., Remsing, C.C.: Cost-extended control systems on Lie groups. Mediterr. J. Math. 11, 193–215 (2014) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Biggs, R., Remsing, C.C.: Quadratic Hamilton–Poisson systems in three dimensions: equivalence, stability, and integration. Acta Appl. Math. 148, 1–59 (2017) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Dăniasă, C., Gîrban, A., Tudoran, R.M.: New aspects on the geometry and dynamics of quadratic Hamiltonian systems on \(( \mathfrak{so}(3))^{*}\). Int. J. Geom. Methods Mod. Phys. 8, 1695–1721 (2011) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Holm, D.D., Schmah, T., Stoica, C.: Geometric Mechanics and Symmetry. Oxford University Press, Oxford (2009) MATHGoogle Scholar
  21. 21.
    Jurdjevic, V.: Geometric Control Theory. Cambridge University Press, Cambridge (1997) MATHGoogle Scholar
  22. 22.
    Krishnaprasad, P.S.: Optimal control and Poisson reduction. Technical Research Report T.R. 93-87, Inst. Systems Research, Univ. of Maryland (1993) Google Scholar
  23. 23.
    Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P.: Poisson Structures. Springer, Heidelberg (2013) CrossRefMATHGoogle Scholar
  24. 24.
    Lawden, D.F.: Elliptic Functions and Applications. Springer, New York (1989) CrossRefMATHGoogle Scholar
  25. 25.
    Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, 2nd edn. Springer, New York (1999) CrossRefMATHGoogle Scholar
  26. 26.
    Ortega, J.P., Planas-Bielsa, V., Ratiu, T.S.: Asymptotic and Lyapunov stability of constrained and Poisson equilibria. J. Differ. Equ. 214, 92–127 (2005) MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Pop, C., Aron, A.: Drift-free left invariant control system on \(\mathsf{G}_{4}\) with fewer controls than state variables. An. Univ. Ovidius Constanţa 17, 167–180 (2009) MathSciNetMATHGoogle Scholar
  28. 28.
    Pop, C., Petrişor, C.: Some dynamical aspects on the Lie group \(\mathsf{SO}(4)\). In: The International Conference of Differential Geometry and Dynamical Systems (DGDS-2013), pp. 128–138. Geometry Balkan Press, Bucharest (2014) Google Scholar
  29. 29.
    Tudoran, R.M.: The free rigid body dynamics: generalized versus classic. J. Math. Phys. 54, 072704 (2013) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsRhodes UniversityGrahamstownSouth Africa
  2. 2.Department of Mathematics and Applied MathematicsUniversity of PretoriaPretoriaSouth Africa

Personalised recommendations