Leibniz-Dirac structures and nonconservative systems with constraints
Authors
Abstract
Although conservative Hamiltonian systems with constraints can be formulated in terms of Dirac structures, a more general framework is necessary to cover also dissipative systems such as gradient and metriplectic systems with constraints. We define Leibniz-Dirac structures which lead to a natural generalization of Dirac and Riemannian structures, for instance. From modeling point of view, Leibniz-Dirac structures make it easy to formulate implicit dissipative Hamiltonian systems. We give their exact characterization in terms of vector bundle maps from the tangent bundle to the cotangent bundle and vice verse. Physical systems which can be formulated in terms of Leibniz-Dirac structures are discussed.
Citation
- Journal: Journal of Geometric Mechanics
- Year: 2013
- Volume: 5
- Issue: 2
- Pages: 167–183
- Publisher: American Institute of Mathematical Sciences (AIMS)
- DOI: 10.3934/jgm.2013.5.167
BibTeX
@article{_ift_i_2013,
title={{Leibniz-Dirac structures and nonconservative systems with constraints}},
volume={5},
ISSN={1941-4897},
DOI={10.3934/jgm.2013.5.167},
number={2},
journal={Journal of Geometric Mechanics},
publisher={American Institute of Mathematical Sciences (AIMS)},
author={Çiftçi, Ünver},
year={2013},
pages={167--183}
}
References
- Abraham, R., Marsden, J. E. & Ratiu, T. Manifolds, Tensor Analysis, and Applications. Applied Mathematical Sciences (Springer New York, 1988). doi:10.1007/978-1-4612-1029-0 – 10.1007/978-1-4612-1029-0
- Balseiro, P. et al. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics vol. 1 1–34 (2009) – 10.3934/jgm.2009.1.1
- G. Blankenstein, A joined geometric structure for Hamiltonian and gradient control systems,. in (2003)
- Blankenstein, G. Geometric modeling of nonlinear RLC circuits. IEEE Transactions on Circuits and Systems I: Regular Papers vol. 52 396–404 (2005) – 10.1109/tcsi.2004.840481
- Bloch, A., Krishnaprasad, P. S., Marsden, J. E. & Ratiu, T. S. The Euler-Poincaré equations and double bracket dissipation. Communications in Mathematical Physics vol. 175 1–42 (1996) – 10.1007/bf02101622
- Bullo, F. & Lewis, A. D. Geometric Control of Mechanical Systems. Texts in Applied Mathematics (Springer New York, 2005). doi:10.1007/978-1-4899-7276-7 – 10.1007/978-1-4899-7276-7
- Bursztyn, H. & Radko, O. Gauge equivalence of Dirac structures and symplectic groupoids. Annales de l’institut Fourier vol. 53 309–337 (2003) – 10.5802/aif.1945
- Bursztyn, H., Cavalcanti, G. R. & Gualtieri, M. Reduction of Courant algebroids and generalized complex structures. Advances in Mathematics vol. 211 726–765 (2007) – 10.1016/j.aim.2006.09.008
- Cendra, H. & Grillo, S. Generalized nonholonomic mechanics, servomechanisms and related brackets. Journal of Mathematical Physics vol. 47 (2006) – 10.1063/1.2165797
- Crouch, P. E. Geometric structures in systems theory. IEE Proceedings D Control Theory and Applications vol. 128 242 (1981) – 10.1049/ip-d.1981.0051
- Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
- 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
- Grabowski, J. & Urbański, P. Lie algebroids and Poisson-Nijenhuis structures. Reports on Mathematical Physics vol. 40 195–208 (1997) – 10.1016/s0034-4877(97)85916-2
- Grabowski, J. & Urbański, P. Algebroids — general differential calculi on vector bundles. Journal of Geometry and Physics vol. 31 111–141 (1999) – 10.1016/s0393-0440(99)00007-8
- Gualtieri, M. Generalized complex geometry. Annals of Mathematics vol. 174 75–123 (2011) – 10.4007/annals.2011.174.1.3
- Jotz, M. & Ratiu, T. S. Dirac Structures, Nonholonomic Systems and Reduction. Reports on Mathematical Physics vol. 69 5–56 (2012) – 10.1016/s0034-4877(12)60016-0
- Liu, Z.-J., Weinstein, A. & Xu, P. Manin triples for Lie bialgebroids. Journal of Differential Geometry vol. 45 (1997) – 10.4310/jdg/1214459842
- Morrison, P. J. A paradigm for joined Hamiltonian and dissipative systems. Physica D: Nonlinear Phenomena vol. 18 410–419 (1986) – 10.1016/0167-2789(86)90209-5
- Nguyen, S. Q. H. & Turski, Ł. A. On the Dirac approach to constrained dissipative dynamics. Journal of Physics A: Mathematical and General vol. 34 9281–9302 (2001) – 10.1088/0305-4470/34/43/312
- Ortega, J.-P. & Planas-Bielsa, V. Dynamics on Leibniz manifolds. Journal of Geometry and Physics vol. 52 1–27 (2004) – 10.1016/j.geomphys.2004.01.002
- van der Schaft, A. J. Implicit Hamiltonian systems with symmetry. Reports on Mathematical Physics vol. 41 203–221 (1998) – 10.1016/s0034-4877(98)80176-6
- 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. Port-Hamiltonian Systems on Graphs. SIAM Journal on Control and Optimization vol. 51 906–937 (2013) – 10.1137/110840091