On the Geometric Structure of Hamiltonian Systems with Ports

Authors

Jochen Merker

Keywords

Dirac structure; Courant algebroid; Port-Hamiltonian system; Symplectic geometry; Poisson manifold; Integrability; 53D17; 70H05; 37J05; 70F20; 70G45

Citation

• Journal: Journal of Nonlinear Science
• Year: 2009
• Volume: 19
• Issue: 6
• Pages: 717–738
• Publisher: Springer Science and Business Media LLC
• DOI: 10.1007/s00332-009-9052-3

BibTeX

@article{Merker_2009,
  title={{On the Geometric Structure of Hamiltonian Systems with Ports}},
  volume={19},
  ISSN={1432-1467},
  DOI={10.1007/s00332-009-9052-3},
  number={6},
  journal={Journal of Nonlinear Science},
  publisher={Springer Science and Business Media LLC},
  author={Merker, Jochen},
  year={2009},
  pages={717--738}
}

Download the bib file

References

• G. Blankenstein. Blankenstein, G., van der Schaft, A.J.: Closedness of interconnected Dirac structures. In: Preprints 4th NOLCOS’98, pp. 381–386. Elsevier, Amsterdam (1998) (1998)
• Bloch, A. M. Nonholonomic Mechanics and Control. Interdisciplinary Applied Mathematics (Springer New York, 2003). doi:10.1007/b97376 – 10.1007/b97376
• H. Bursztyn. Bursztyn, H., Crainic, M., Severa, P.: Quasi-Poisson structures as Dirac structures. Trav. Math. XVI, 41–52 (2005) (2005)
• Cervera, J., van der Schaft, A. J. & Baños, A. Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica vol. 43 212–225 (2007) – 10.1016/j.automatica.2006.08.014
• Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• T. Courant. Courant, T., Weinstein, A.: Beyond Poisson structures, Séminaire Sud-Rhodanien de Géométrie VIII. Trav. En Cours 27, 39–49 (1988) (1988)
• Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM Journal on Control and Optimization vol. 37 54–91 (1998) – 10.1137/s0363012996312039
• I. Dorfman. Dorfman, I.: Dirac Structures and Integrability of Nonlinear Evolution Equations. Wiley, New York (1993) (1993)
• Y. Kosmann-Schwarzbach. Kosmann-Schwarzbach, Y.: Quasi, twisted, and all that…in Poisson geometry and Lie algebroid theory. In: Marsden, J.E., Ratiu, T. (eds.) The Breadth of Symplectic and Poisson Geometry. Progr. Math., vol. 232, pp. 363–389. Birkhäuser, Basel (2005) (2005)
• Z. Liu. Liu, Z., Weinstein, A., Xu, P.: Manin triples for Lie bialgebroids. J. Differ. Geom. 45, 647–574 (1997) (1997)
• Yoshimura, H. & Marsden, J. E. Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems. Journal of Geometry and Physics vol. 57 133–156 (2006) – 10.1016/j.geomphys.2006.02.009
• Roytenberg, D., Courant algebroids, derived brackets, and even symplectic supermanifolds. Ph.D. thesis, University of California, Berkeley (1999)
• Ševera, P. & Weinstein, A. Poisson Geometry with a 3-Form Background. Progress of Theoretical Physics Supplement vol. 144 145–154 (2001) – 10.1143/ptps.144.145
• van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
• Van Der Schaft, A. J. & Maschke, B. M. On the Hamiltonian formulation of nonholonomic mechanical systems. Reports on Mathematical Physics vol. 34 225–233 (1994) – 10.1016/0034-4877(94)90038-8
• A.J. Schaft van der. van der Schaft, A.J., Maschke, B.M.: The Hamiltonian formulation of energy conserving physical systems with external ports. Arch. Elektr. Übertrag. 49, 362–371 (1995) (1995)
• van der Schaft, A. & Maschke, B. Interconnected mechanical systems, part I: geometry of interconnection and implicit Hamiltonian systems. Modelling and Control of Mechanical Systems 1–15 (1997) doi:10.1142/9781848160873_0001 – 10.1142/9781848160873_0001