June  2013, 5(2): 167-183. doi: 10.3934/jgm.2013.5.167

Leibniz-Dirac structures and nonconservative systems with constraints

1. 

Department of Mathematics, Namik Kemal University, 59030 Tekirdaǧ, Turkey

Received  September 2012 Revised  April 2013 Published  July 2013

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: Ünver Çiftçi. Leibniz-Dirac structures and nonconservative systems with constraints. Journal of Geometric Mechanics, 2013, 5 (2) : 167-183. doi: 10.3934/jgm.2013.5.167
References:
[1]

R. Abraham J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications," Second edition, Applied Mathematical Sciences, 75, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.

[2]

P. Balseiro, M. de León, J. C. Marrero and D. Martín de Diego, The ubiquity of the symplectic Hamiltonian equations in mechanics, J. Geom. Mech., 1 (2009), 1-34. doi: 10.3934/jgm.2009.1.1.

[3]

G. Blankenstein, A joined geometric structure for Hamiltonian and gradient control systems, in "Lagrangian and Hamiltonian Methods for Nonlinear Control 2003," IFAC, Laxenburg, (2003), 51-56.

[4]

G. Blankenstein, Geometric modeling of nonlinear RLC circuits, IEEE Trans. Circuits Syst. I Regul. Pap., 52 (2005), 396-404. doi: 10.1109/TCSI.2004.840481.

[5]

A. Bloch, P. S. Krishnaprasad, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and double bracket dissipation, Comm. Math. Phys., 175 (1996), 1-42. doi: 10.1007/BF02101622.

[6]

F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems," Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005.

[7]

H. Bursztyn and O. Radko, Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier (Grenoble), 53 (2003), 309-337. doi: 10.5802/aif.1945.

[8]

H. Bursztyn, G. R. Cavalcanti and M. Gualtieri, Reduction of Courant algebroids and generalized complex structures, Adv. Math., 211 (2007), 726-765. doi: 10.1016/j.aim.2006.09.008.

[9]

H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets, J. Math. Phys., 47 (2006), 022902, 29 pp. doi: 10.1063/1.2165797.

[10]

P. E. Crouch, Geometric structures in systems theory, Proceedings IEE-D, 128 (1981), 242-252. doi: 10.1049/ip-d.1981.0051.

[11]

T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1.

[12]

M. Dalsmo and A. van der Schaft, On representations and integrability of mathematical structures in energy-conserving physical systems, SIAM J. Control Optim., 37 (1999), 54-91. doi: 10.1137/S0363012996312039.

[13]

J. Grabowski and P. Urbański, Lie algebroids and Poisson-Nijenhuis structures, Rep. Math. Phys., 40 (1997), 195-208. doi: 10.1016/S0034-4877(97)85916-2.

[14]

J. Grabowski and P. Urbański, Algebroids-general differential calculi on vector bundles, J. Geom. Phys., 31 (1999), 111-141. doi: 10.1016/S0393-0440(99)00007-8.

[15]

M. Gualtieri, Generalized complex geometry, Ann. of Math. (2), 174 (2011), 75-123. doi: 10.4007/annals.2011.174.1.3.

[16]

M. Jotz and T. S. Ratiu, Dirac structures, nonholonomic systems and reduction, Rep. Math. Phys., 69 (2012), 5-56. doi: 10.1016/S0034-4877(12)60016-0.

[17]

Z. Liu, A. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Differ. Geom., 45 (1997), 547-574.

[18]

P. J. Morrison, A paradigm for joined Hamiltonian and dissipative systems, Phys. D, 18 (1986), 410-419. doi: 10.1016/0167-2789(86)90209-5.

[19]

S. Q. H. Nguyen and L. A. Turski, On the Dirac approach to constrained dissipative dynamics, J. Phys. A, 34 (2001), 9281-9302. doi: 10.1088/0305-4470/34/43/312.

[20]

J.-P. Ortega and V. Planas-Bielsa, Dynamics on Leibniz manifolds, J. Geom. Phys., 52 (2004), 1-27. doi: 10.1016/j.geomphys.2004.01.002.

[21]

A. J. van der Schaft, Implicit Hamiltonian systems with symmetry, Rep. Math. Phys., 41 (1998), 203-221. doi: 10.1016/S0034-4877(98)80176-6.

[22]

A. J. van der Schaft, "$L_2$-gain and Passivity Techniques in Nonlinear Control," Second edition, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 2000. doi: 10.1007/978-1-4471-0507-7.

[23]

A. J. van der Schaft and B. M. Maschke, Port-Hamiltonian systems on graphs, SIAM J. Control Optim., 51 (2013), 906-937. doi: 10.1137/110840091.

show all references

References:
[1]

R. Abraham J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications," Second edition, Applied Mathematical Sciences, 75, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.

[2]

P. Balseiro, M. de León, J. C. Marrero and D. Martín de Diego, The ubiquity of the symplectic Hamiltonian equations in mechanics, J. Geom. Mech., 1 (2009), 1-34. doi: 10.3934/jgm.2009.1.1.

[3]

G. Blankenstein, A joined geometric structure for Hamiltonian and gradient control systems, in "Lagrangian and Hamiltonian Methods for Nonlinear Control 2003," IFAC, Laxenburg, (2003), 51-56.

[4]

G. Blankenstein, Geometric modeling of nonlinear RLC circuits, IEEE Trans. Circuits Syst. I Regul. Pap., 52 (2005), 396-404. doi: 10.1109/TCSI.2004.840481.

[5]

A. Bloch, P. S. Krishnaprasad, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and double bracket dissipation, Comm. Math. Phys., 175 (1996), 1-42. doi: 10.1007/BF02101622.

[6]

F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems," Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005.

[7]

H. Bursztyn and O. Radko, Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier (Grenoble), 53 (2003), 309-337. doi: 10.5802/aif.1945.

[8]

H. Bursztyn, G. R. Cavalcanti and M. Gualtieri, Reduction of Courant algebroids and generalized complex structures, Adv. Math., 211 (2007), 726-765. doi: 10.1016/j.aim.2006.09.008.

[9]

H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets, J. Math. Phys., 47 (2006), 022902, 29 pp. doi: 10.1063/1.2165797.

[10]

P. E. Crouch, Geometric structures in systems theory, Proceedings IEE-D, 128 (1981), 242-252. doi: 10.1049/ip-d.1981.0051.

[11]

T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1.

[12]

M. Dalsmo and A. van der Schaft, On representations and integrability of mathematical structures in energy-conserving physical systems, SIAM J. Control Optim., 37 (1999), 54-91. doi: 10.1137/S0363012996312039.

[13]

J. Grabowski and P. Urbański, Lie algebroids and Poisson-Nijenhuis structures, Rep. Math. Phys., 40 (1997), 195-208. doi: 10.1016/S0034-4877(97)85916-2.

[14]

J. Grabowski and P. Urbański, Algebroids-general differential calculi on vector bundles, J. Geom. Phys., 31 (1999), 111-141. doi: 10.1016/S0393-0440(99)00007-8.

[15]

M. Gualtieri, Generalized complex geometry, Ann. of Math. (2), 174 (2011), 75-123. doi: 10.4007/annals.2011.174.1.3.

[16]

M. Jotz and T. S. Ratiu, Dirac structures, nonholonomic systems and reduction, Rep. Math. Phys., 69 (2012), 5-56. doi: 10.1016/S0034-4877(12)60016-0.

[17]

Z. Liu, A. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Differ. Geom., 45 (1997), 547-574.

[18]

P. J. Morrison, A paradigm for joined Hamiltonian and dissipative systems, Phys. D, 18 (1986), 410-419. doi: 10.1016/0167-2789(86)90209-5.

[19]

S. Q. H. Nguyen and L. A. Turski, On the Dirac approach to constrained dissipative dynamics, J. Phys. A, 34 (2001), 9281-9302. doi: 10.1088/0305-4470/34/43/312.

[20]

J.-P. Ortega and V. Planas-Bielsa, Dynamics on Leibniz manifolds, J. Geom. Phys., 52 (2004), 1-27. doi: 10.1016/j.geomphys.2004.01.002.

[21]

A. J. van der Schaft, Implicit Hamiltonian systems with symmetry, Rep. Math. Phys., 41 (1998), 203-221. doi: 10.1016/S0034-4877(98)80176-6.

[22]

A. J. van der Schaft, "$L_2$-gain and Passivity Techniques in Nonlinear Control," Second edition, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 2000. doi: 10.1007/978-1-4471-0507-7.

[23]

A. J. van der Schaft and B. M. Maschke, Port-Hamiltonian systems on graphs, SIAM J. Control Optim., 51 (2013), 906-937. doi: 10.1137/110840091.

[1]

Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020

[2]

Jacques Demongeot, Dan Istrate, Hajer Khlaifi, Lucile Mégret, Carla Taramasco, René Thomas. From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2121-2134. doi: 10.3934/dcdss.2020181

[3]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

[4]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure and Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017

[5]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097

[6]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185

[7]

María Barbero Liñán, Hernán Cendra, Eduardo García Toraño, David Martín de Diego. Morse families and Dirac systems. Journal of Geometric Mechanics, 2019, 11 (4) : 487-510. doi: 10.3934/jgm.2019024

[8]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

[9]

Hernán Cendra, María Etchechoury, Sebastián J. Ferraro. An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems. Journal of Geometric Mechanics, 2014, 6 (2) : 167-236. doi: 10.3934/jgm.2014.6.167

[10]

Delio Mugnolo, René Pröpper. Gradient systems on networks. Conference Publications, 2011, 2011 (Special) : 1078-1090. doi: 10.3934/proc.2011.2011.1078

[11]

Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201

[12]

Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855

[13]

Santiago Capriotti. Dirac constraints in field theory and exterior differential systems. Journal of Geometric Mechanics, 2010, 2 (1) : 1-50. doi: 10.3934/jgm.2010.2.1

[14]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785

[15]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i

[16]

Siniša Slijepčević. Extended gradient systems: Dimension one. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 503-518. doi: 10.3934/dcds.2000.6.503

[17]

Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2019-2034. doi: 10.3934/dcdss.2019130

[18]

Kais Ammari, Eduard Feireisl, Serge Nicaise. Polynomial stabilization of some dissipative hyperbolic systems. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4371-4388. doi: 10.3934/dcds.2014.34.4371

[19]

Giuseppe Da Prato. Transition semigroups corresponding to Lipschitz dissipative systems. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 177-192. doi: 10.3934/dcds.2004.10.177

[20]

Russell Johnson, Carmen Núñez. Remarks on linear-quadratic dissipative control systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 889-914. doi: 10.3934/dcdsb.2015.20.889

2020 Impact Factor: 0.857

Metrics

  • PDF downloads (93)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]