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

Leibniz-Dirac structures and nonconservative systems with constraints

Ünver Çiftçi 1,

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.
Keywords: gradient systems., dissipative Hamiltonian systems, Dirac manifolds.
Mathematics Subject Classification: 53D05, 70F25, 70G4.
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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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