-
Previous Article
When is a control system mechanical?
- JGM Home
- This Issue
-
Next Article
Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion
Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes
1. | Unidad asociada ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain |
References:
[1] |
R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, Benjamin/Cummings, Reading, Massachusetts, 1978. |
[2] |
F. Cantrijn, J. Cortés, M. de León and D. Martín de Diego, On the geometry of generalized Chaplygin systems, Math. Proc. Cambridge Philos. Soc., 132 (2002), 323-351.
doi: 10.1017/S0305004101005679. |
[3] |
T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661.
doi: 10.2307/2001258. |
[4] |
S. Evens, J.-H. Lu and A. Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford, 50 (1999), 417-436.
doi: 10.1093/qjmath/50.200.417. |
[5] |
Y. Fedorov, L. García-Naranjo and J. C. Marrero, Hamiltonian dynamics on skew-symmetric algebroids, unimodularity and preservation of volumes in nonholonomic mechanics,, in preparation., ().
|
[6] |
P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra, 129 (1990), 194-230.
doi: 10.1016/0021-8693(90)90246-K. |
[7] |
B. Jovanovic, Nonholonomic geodesic flows on Lie groups and the integrable Suslov problem on $SO(4)$, J. Phys. A: Math. Gen., 31 (1998), 1415-22.
doi: 10.1088/0305-4470/31/5/011. |
[8] |
V. V. Kozlov, Invariant measures of the Euler-Poincaré equations on Lie algebras, Funkt. Anal. Prilozh., 22 69-70 (Russian); English trans.: Funct. Anal. Appl., 22 (1988), 58-59.
doi: 10.1007/BF01077727. |
[9] |
V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics, Regular and Chaotic Dynamics, 7 (2002), 161-176.
doi: 10.1070/RD2002v007n02ABEH000203. |
[10] |
M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38 (2005), R241-R308.
doi: 10.1088/0305-4470/38/24/R01. |
[11] |
A. Lewis, Reduction of simple mechanical systems, Mechanics and symmetry seminars, University of Warwick, 1997, http://www.mast.queensu.ca/~andrew/notes/abstracts/1997a.html. |
[12] |
A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées, J. Differential Geometry, 12 (1977), 253-300. |
[13] |
K. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids," London Mathematical Society Lecture Note Series 213, Cambridge University Press, 2005. |
[14] |
E. Martínez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320.
doi: 10.1023/A:1011965919259. |
[15] |
J. E. Marsden and T. Ratiu, "Introduction to Mechanics with Symmetry," Texts in Applied Mathematics, 17, Springer-Verlag, 1994. |
[16] |
J. P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics, 222, Birkhäuser Boston, Inc., Boston, MA, 2004. |
[17] |
J. P. Ostrowski, "The Mechanics and Control of Undulatory Robotic Locomotion," PhD Thesis, California Institute of Technology, 1995. |
[18] |
A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231. |
[19] |
A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys., 23 (1997), 379-394.
doi: 10.1016/S0393-0440(97)80011-3. |
[20] |
D. V. Zenkov and A. M. Bloch, Invariant measures of nonholonomic flows with internal degrees of freedom, Nonlinearity, 16 (2003), 1793-1807.
doi: 10.1088/0951-7715/16/5/313. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, Benjamin/Cummings, Reading, Massachusetts, 1978. |
[2] |
F. Cantrijn, J. Cortés, M. de León and D. Martín de Diego, On the geometry of generalized Chaplygin systems, Math. Proc. Cambridge Philos. Soc., 132 (2002), 323-351.
doi: 10.1017/S0305004101005679. |
[3] |
T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661.
doi: 10.2307/2001258. |
[4] |
S. Evens, J.-H. Lu and A. Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford, 50 (1999), 417-436.
doi: 10.1093/qjmath/50.200.417. |
[5] |
Y. Fedorov, L. García-Naranjo and J. C. Marrero, Hamiltonian dynamics on skew-symmetric algebroids, unimodularity and preservation of volumes in nonholonomic mechanics,, in preparation., ().
|
[6] |
P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra, 129 (1990), 194-230.
doi: 10.1016/0021-8693(90)90246-K. |
[7] |
B. Jovanovic, Nonholonomic geodesic flows on Lie groups and the integrable Suslov problem on $SO(4)$, J. Phys. A: Math. Gen., 31 (1998), 1415-22.
doi: 10.1088/0305-4470/31/5/011. |
[8] |
V. V. Kozlov, Invariant measures of the Euler-Poincaré equations on Lie algebras, Funkt. Anal. Prilozh., 22 69-70 (Russian); English trans.: Funct. Anal. Appl., 22 (1988), 58-59.
doi: 10.1007/BF01077727. |
[9] |
V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics, Regular and Chaotic Dynamics, 7 (2002), 161-176.
doi: 10.1070/RD2002v007n02ABEH000203. |
[10] |
M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38 (2005), R241-R308.
doi: 10.1088/0305-4470/38/24/R01. |
[11] |
A. Lewis, Reduction of simple mechanical systems, Mechanics and symmetry seminars, University of Warwick, 1997, http://www.mast.queensu.ca/~andrew/notes/abstracts/1997a.html. |
[12] |
A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées, J. Differential Geometry, 12 (1977), 253-300. |
[13] |
K. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids," London Mathematical Society Lecture Note Series 213, Cambridge University Press, 2005. |
[14] |
E. Martínez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320.
doi: 10.1023/A:1011965919259. |
[15] |
J. E. Marsden and T. Ratiu, "Introduction to Mechanics with Symmetry," Texts in Applied Mathematics, 17, Springer-Verlag, 1994. |
[16] |
J. P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics, 222, Birkhäuser Boston, Inc., Boston, MA, 2004. |
[17] |
J. P. Ostrowski, "The Mechanics and Control of Undulatory Robotic Locomotion," PhD Thesis, California Institute of Technology, 1995. |
[18] |
A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231. |
[19] |
A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys., 23 (1997), 379-394.
doi: 10.1016/S0393-0440(97)80011-3. |
[20] |
D. V. Zenkov and A. M. Bloch, Invariant measures of nonholonomic flows with internal degrees of freedom, Nonlinearity, 16 (2003), 1793-1807.
doi: 10.1088/0951-7715/16/5/313. |
[1] |
Raquel Caseiro, Camille Laurent-Gengoux. Modular class of Lie $ \infty $-algebroids and adjoint representations. Journal of Geometric Mechanics, 2022 doi: 10.3934/jgm.2022008 |
[2] |
Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213 |
[3] |
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 |
[4] |
Franco Cardin, Alberto Lovison. Finite mechanical proxies for a class of reducible continuum systems. Networks and Heterogeneous Media, 2014, 9 (3) : 417-432. doi: 10.3934/nhm.2014.9.417 |
[5] |
Dennise García-Beltrán, José A. Vallejo, Yurii Vorobiev. Lie algebroids generated by cohomology operators. Journal of Geometric Mechanics, 2015, 7 (3) : 295-315. doi: 10.3934/jgm.2015.7.295 |
[6] |
Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure and Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703 |
[7] |
Robert L. Griess Jr., Ching Hung Lam. Groups of Lie type, vertex algebras, and modular moonshine. Electronic Research Announcements, 2014, 21: 167-176. doi: 10.3934/era.2014.21.167 |
[8] |
Adriano Da Silva, Alexandre J. Santana, Simão N. Stelmastchuk. Topological conjugacy of linear systems on Lie groups. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3411-3421. doi: 10.3934/dcds.2017144 |
[9] |
Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004 |
[10] |
Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81 |
[11] |
Madeleine Jotz Lean, Kirill C. H. Mackenzie. Transitive double Lie algebroids via core diagrams. Journal of Geometric Mechanics, 2021, 13 (3) : 403-457. doi: 10.3934/jgm.2021023 |
[12] |
K. C. H. Mackenzie. Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids. Electronic Research Announcements, 1998, 4: 74-87. |
[13] |
Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3905-3919. doi: 10.3934/dcdsb.2020268 |
[14] |
Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2855-2878. doi: 10.3934/cpaa.2019128 |
[15] |
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 |
[16] |
Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1857-1870. doi: 10.3934/dcdss.2020461 |
[17] |
Y. Kabeya, Eiji Yanagida, Shoji Yotsutani. Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems. Communications on Pure and Applied Analysis, 2002, 1 (1) : 85-102. doi: 10.3934/cpaa.2002.1.85 |
[18] |
Yusi Fan, Chenrui Yao, Liangyun Chen. Structure of sympathetic Lie superalgebras. Electronic Research Archive, 2021, 29 (5) : 2945-2957. doi: 10.3934/era.2021020 |
[19] |
Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623 |
[20] |
Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684 |
2020 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]