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Leibniz-Dirac structures and nonconservative systems with constraints
1. | Department of Mathematics, Namik Kemal University, 59030 Tekirdaǧ, Turkey |
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. |
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