Authors

Izu Vaisman

Abstract

A big-isotropic structure E is an isotropic subbundle of TM⊕TM, endowed with the metric defined by pairing. The structure E is said to be integrable if the Courant bracket [X,Y]∊ΓE, ∀X,Y∊ΓE. Then, necessarily, one also has [X,Z]∊ΓE⊥, ∀Z∊ΓE⊥ [Vaisman, I., “Isotropic subbundles of TM⊕TM,” Int. J. Geom. Methods Mod. Phys. 4, 487–516 (2007)]. A weak-Hamiltonian dynamical system is a vector field XH such that (XH,dH)∊ΓE⊥(H∊C∞(M)). We obtain the explicit expression of XH and of the integrability conditions of E under the regularity condition dim(prT*ME)=const. We show that the port-controlled, Hamiltonian systems (in particular, constrained mechanics) [Dalsmo, M. and van der Schaft, A. J., “On representations and integrability of mathematical structures in energy conserving physical systems,” SIAM J. Control Optim. 37, 54–91 (1998)] may be interpreted as weak-Hamiltonian systems. Finally, we give reduction theorems for weak-Hamiltonian systems and a corresponding corollary for constrained mechanical systems.

Citation

  • Journal: Journal of Mathematical Physics
  • Year: 2007
  • Volume: 48
  • Issue: 8
  • Pages:
  • Publisher: AIP Publishing
  • DOI: 10.1063/1.2769145

BibTeX

@article{Vaisman_2007,
  title={{Weak-Hamiltonian dynamical systems}},
  volume={48},
  ISSN={1089-7658},
  DOI={10.1063/1.2769145},
  number={8},
  journal={Journal of Mathematical Physics},
  publisher={AIP Publishing},
  author={Vaisman, Izu},
  year={2007}
}

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References