-
Previous Article
On strain measures and the geodesic distance to $SO_n$ in the general linear group
- JGM Home
- This Issue
-
Next Article
The Frank tensor as a boundary condition in intrinsic linearized elasticity
The Tulczyjew triple in mechanics on a Lie group
1. | Department of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland |
References:
[1] |
R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems, Birkhäuser Verlag, Basel, 1997.
doi: 10.1007/978-3-0348-8891-2. |
[2] |
J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Universitext, Springer-Verlag, 2000.
doi: 10.1007/978-3-642-56936-4. |
[3] |
O. Esen and H. Gumral, Tulczyjew's triplet for Lie groups I: Trivializations and reductions, J. Lie Theory, 24 (2014), 1115-1160. |
[4] |
O. Esen and H. Gumral, Tulczyjew's triplet for Lie groups II: Dynamics,, arXiv:1503.06566., ().
|
[5] |
L. C. García-Naranjo, A. J. Maciejewski, J. C. Marrero and M. Przybylska, The inhomogeneous Suslov problem, Phys. Let. A, 378 (2014), 2389-2394.
doi: 10.1016/j.physleta.2014.06.026. |
[6] |
E. Garcia-Torano Andrés, E. Guzmán, J. C. Marrero and T. Mestdag, Reduced dynamics and Lagrangian submanifolds of symplectic manifolds, J. Phys. A: Math. Theor., 47 (2014), 225203, 24pp.
doi: 10.1088/1751-8113/47/22/225203. |
[7] |
K. Grabowska, A Tulczyjew triple for classical fields, J. Phys. A: Math. Theor., 45 (2012), 145207, 35pp.
doi: 10.1088/1751-8113/45/14/145207. |
[8] |
K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204 (25pp).
doi: 10.1088/1751-8113/41/17/175204. |
[9] |
K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics, J. Geom. Phys., 61 (2011), 2233-2253.
doi: 10.1016/j.geomphys.2011.06.018. |
[10] |
K. Grabowska, J. Grabowski and P. Urbański, Geometrical Mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575.
doi: 10.1142/S0219887806001259. |
[11] |
K. Grabowska and L. Vitagliano, Tulczyjew triples in Higher derivative field theory, J Geom. Mech., 7 (2015), 1-33.
doi: 10.3934/jgm.2015.7.1. |
[12] |
J. Grabowski and G. Marmo, Deformed Tulczyjew triples and Legendre transform, Geometrical structures for physical theories, I (Vietri, 1996), Rend. Sem. Mat. Univ. Politec. Torino, 54 (1996), 279-294. |
[13] |
J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305.
doi: 10.1016/j.geomphys.2009.06.009. |
[14] |
J. Grabowski and P. Urbański, Tangent lifts of Poisson and related structures, J. Phys. A, 28 (1995), 6743-6777.
doi: 10.1088/0305-4470/28/23/024. |
[15] |
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. |
[16] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré quations and semidirect products with applications to continuum theories, Adv. in Math., 137 (1998), 1-81.
doi: 10.1006/aima.1998.1721. |
[17] |
J. Klein, Espaces variationnels et mécanique, Ann. Inst. Fourier, 12 (1962), 1-124.
doi: 10.5802/aif.120. |
[18] |
K. Konieczna and P. Urbański, Double vector bundles and duality, Archivum Mathematicum, 35 (1999), 59-95. |
[19] |
M. de León, J. C. Marrero and E. Martinez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A, 38 (2005), R241-R308.
doi: 10.1088/0305-4470/38/24/R01. |
[20] |
P. Liebermann and Ch. M. Marle, Symplectic Geometry and Analytical Mechanics, Reidel Publishing Company, Dordrecht, 1987.
doi: 10.1007/978-94-009-3807-6. |
[21] |
P. Liebermann, Lie algebroids and mechanics, Archivum Mathematicum, 32 (1996), 147-162. |
[22] |
E. Martinez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320.
doi: 10.1023/A:1011965919259. |
[23] |
E. Martinez, Geometric formulation of Mechanics on Lie algebroids, in Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, (1999), Publicaciones de la RSME, 2 (2001), 209-222. |
[24] |
E. Martinez, Variational calculus on Lie algebroids, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 356-380.
doi: 10.1051/cocv:2007056. |
[25] |
E. Martinez, T. Mestdag and W. Sarlet, Lie algebroid structures and Lagrangian systems on affine bundles, J. Geom. Phys., 44 (2002), 70-95.
doi: 10.1016/S0393-0440(02)00114-6. |
[26] |
J. Pradines, Geometrie differentielle au-dessus d'un grupoide, C. R. Acad. Sci. Paris, série A, 266 (1968), 1194-1196. |
[27] | |
[28] |
W. M. Tulczyjew, Hamiltonian systems, Lagrangian systems, and the Legendre transformation, Symposia Mathematica, Vol. XII (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973), Academic Press, London, (1974), 247-258. |
[29] |
W. M. Tulczyjew, Sur la différentielle de Lagrange, C. R. Acad. Sci. Paris., 280 (1975), 1295-1298. |
[30] |
W. M. Tulczyjew, Les sous-varietes lagrangiennes et la dynamique hamiltonienne, C.R. Acad. Sc. Paris, 283 (1976), 15-18. |
[31] |
W. M. Tulczyjew, Les sous-varietes lagrangiennes et la dynamique lagrangienne, C.R. Acad. Sc. Paris, 283 (1976), 675-678. |
[32] |
W. M. Tulczyjew, The legendre transformation, Ann. Inst. H. Poincare, 27 (1977), 101-114. |
[33] |
W. M. Tulczyjew, Geometric Formulation of Physical Theories. Statics and Dynamics of Mechanical Systems, Monographs and Textbooks in Physical Science. Lecture Notes, 11, Bibliopolis, Naples, 1989. |
[34] |
W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians, The Infeld Centennial Meeting, Warsaw, (1998), Acta Phys. Polon. B, 30 (1999), 2909-2978. |
[35] |
P. Urbański, Double vector bundles in classical mechanics, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 405-421. |
[36] |
A. Weinstein, Lagrangian mechanics and grupoids, Fields Inst. Comm., 7 (1996), 207-231. |
show all references
References:
[1] |
R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems, Birkhäuser Verlag, Basel, 1997.
doi: 10.1007/978-3-0348-8891-2. |
[2] |
J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Universitext, Springer-Verlag, 2000.
doi: 10.1007/978-3-642-56936-4. |
[3] |
O. Esen and H. Gumral, Tulczyjew's triplet for Lie groups I: Trivializations and reductions, J. Lie Theory, 24 (2014), 1115-1160. |
[4] |
O. Esen and H. Gumral, Tulczyjew's triplet for Lie groups II: Dynamics,, arXiv:1503.06566., ().
|
[5] |
L. C. García-Naranjo, A. J. Maciejewski, J. C. Marrero and M. Przybylska, The inhomogeneous Suslov problem, Phys. Let. A, 378 (2014), 2389-2394.
doi: 10.1016/j.physleta.2014.06.026. |
[6] |
E. Garcia-Torano Andrés, E. Guzmán, J. C. Marrero and T. Mestdag, Reduced dynamics and Lagrangian submanifolds of symplectic manifolds, J. Phys. A: Math. Theor., 47 (2014), 225203, 24pp.
doi: 10.1088/1751-8113/47/22/225203. |
[7] |
K. Grabowska, A Tulczyjew triple for classical fields, J. Phys. A: Math. Theor., 45 (2012), 145207, 35pp.
doi: 10.1088/1751-8113/45/14/145207. |
[8] |
K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204 (25pp).
doi: 10.1088/1751-8113/41/17/175204. |
[9] |
K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics, J. Geom. Phys., 61 (2011), 2233-2253.
doi: 10.1016/j.geomphys.2011.06.018. |
[10] |
K. Grabowska, J. Grabowski and P. Urbański, Geometrical Mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575.
doi: 10.1142/S0219887806001259. |
[11] |
K. Grabowska and L. Vitagliano, Tulczyjew triples in Higher derivative field theory, J Geom. Mech., 7 (2015), 1-33.
doi: 10.3934/jgm.2015.7.1. |
[12] |
J. Grabowski and G. Marmo, Deformed Tulczyjew triples and Legendre transform, Geometrical structures for physical theories, I (Vietri, 1996), Rend. Sem. Mat. Univ. Politec. Torino, 54 (1996), 279-294. |
[13] |
J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305.
doi: 10.1016/j.geomphys.2009.06.009. |
[14] |
J. Grabowski and P. Urbański, Tangent lifts of Poisson and related structures, J. Phys. A, 28 (1995), 6743-6777.
doi: 10.1088/0305-4470/28/23/024. |
[15] |
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. |
[16] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré quations and semidirect products with applications to continuum theories, Adv. in Math., 137 (1998), 1-81.
doi: 10.1006/aima.1998.1721. |
[17] |
J. Klein, Espaces variationnels et mécanique, Ann. Inst. Fourier, 12 (1962), 1-124.
doi: 10.5802/aif.120. |
[18] |
K. Konieczna and P. Urbański, Double vector bundles and duality, Archivum Mathematicum, 35 (1999), 59-95. |
[19] |
M. de León, J. C. Marrero and E. Martinez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A, 38 (2005), R241-R308.
doi: 10.1088/0305-4470/38/24/R01. |
[20] |
P. Liebermann and Ch. M. Marle, Symplectic Geometry and Analytical Mechanics, Reidel Publishing Company, Dordrecht, 1987.
doi: 10.1007/978-94-009-3807-6. |
[21] |
P. Liebermann, Lie algebroids and mechanics, Archivum Mathematicum, 32 (1996), 147-162. |
[22] |
E. Martinez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320.
doi: 10.1023/A:1011965919259. |
[23] |
E. Martinez, Geometric formulation of Mechanics on Lie algebroids, in Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, (1999), Publicaciones de la RSME, 2 (2001), 209-222. |
[24] |
E. Martinez, Variational calculus on Lie algebroids, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 356-380.
doi: 10.1051/cocv:2007056. |
[25] |
E. Martinez, T. Mestdag and W. Sarlet, Lie algebroid structures and Lagrangian systems on affine bundles, J. Geom. Phys., 44 (2002), 70-95.
doi: 10.1016/S0393-0440(02)00114-6. |
[26] |
J. Pradines, Geometrie differentielle au-dessus d'un grupoide, C. R. Acad. Sci. Paris, série A, 266 (1968), 1194-1196. |
[27] | |
[28] |
W. M. Tulczyjew, Hamiltonian systems, Lagrangian systems, and the Legendre transformation, Symposia Mathematica, Vol. XII (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973), Academic Press, London, (1974), 247-258. |
[29] |
W. M. Tulczyjew, Sur la différentielle de Lagrange, C. R. Acad. Sci. Paris., 280 (1975), 1295-1298. |
[30] |
W. M. Tulczyjew, Les sous-varietes lagrangiennes et la dynamique hamiltonienne, C.R. Acad. Sc. Paris, 283 (1976), 15-18. |
[31] |
W. M. Tulczyjew, Les sous-varietes lagrangiennes et la dynamique lagrangienne, C.R. Acad. Sc. Paris, 283 (1976), 675-678. |
[32] |
W. M. Tulczyjew, The legendre transformation, Ann. Inst. H. Poincare, 27 (1977), 101-114. |
[33] |
W. M. Tulczyjew, Geometric Formulation of Physical Theories. Statics and Dynamics of Mechanical Systems, Monographs and Textbooks in Physical Science. Lecture Notes, 11, Bibliopolis, Naples, 1989. |
[34] |
W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians, The Infeld Centennial Meeting, Warsaw, (1998), Acta Phys. Polon. B, 30 (1999), 2909-2978. |
[35] |
P. Urbański, Double vector bundles in classical mechanics, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 405-421. |
[36] |
A. Weinstein, Lagrangian mechanics and grupoids, Fields Inst. Comm., 7 (1996), 207-231. |
[1] |
Brennan McCann, Morad Nazari. Control and maintenance of fully-constrained and underconstrained rigid body motion on Lie groups and their tangent bundles. Journal of Geometric Mechanics, 2022, 14 (1) : 29-55. doi: 10.3934/jgm.2022002 |
[2] |
Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017 |
[3] |
Eckhard Meinrenken. Quotients of double vector bundles and multigraded bundles. Journal of Geometric Mechanics, 2021 doi: 10.3934/jgm.2021027 |
[4] |
Jordi-Lluís Figueras, Àlex Haro. Triple collisions of invariant bundles. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2069-2082. doi: 10.3934/dcdsb.2013.18.2069 |
[5] |
Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247 |
[6] |
Yaobang Ye, Zongyu Zuo, Michael Basin. Robust adaptive sliding mode tracking control for a rigid body based on Lie subgroups of SO(3). Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1823-1837. doi: 10.3934/dcdss.2022010 |
[7] |
Jean-Claude Zambrini. Stochastic deformation of classical mechanics. Conference Publications, 2013, 2013 (special) : 807-813. doi: 10.3934/proc.2013.2013.807 |
[8] |
Jamie Cruz, Miguel Gutiérrez. Spiral motion in classical mechanics. Conference Publications, 2009, 2009 (Special) : 191-197. doi: 10.3934/proc.2009.2009.191 |
[9] |
Cristina Stoica. An approximation theorem in classical mechanics. Journal of Geometric Mechanics, 2016, 8 (3) : 359-374. doi: 10.3934/jgm.2016011 |
[10] |
Sujit Nair, Naomi Ehrich Leonard. Stable synchronization of rigid body networks. Networks and Heterogeneous Media, 2007, 2 (4) : 597-626. doi: 10.3934/nhm.2007.2.597 |
[11] |
Juan Carlos Marrero, D. Martín de Diego, Diana Sosa. Variational constrained mechanics on Lie affgebroids. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 105-128. doi: 10.3934/dcdss.2010.3.105 |
[12] |
Javier Pérez Álvarez. Invariant structures on Lie groups. Journal of Geometric Mechanics, 2020, 12 (2) : 141-148. doi: 10.3934/jgm.2020007 |
[13] |
André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351 |
[14] |
Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517 |
[15] |
Sebastián Ferrer, Francisco J. Molero. Andoyer's variables and phases in the free rigid body. Journal of Geometric Mechanics, 2014, 6 (1) : 25-37. doi: 10.3934/jgm.2014.6.25 |
[16] |
Eliot Fried. New insights into the classical mechanics of particle systems. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1469-1504. doi: 10.3934/dcds.2010.28.1469 |
[17] |
Cesare Tronci. Momentum maps for mixed states in quantum and classical mechanics. Journal of Geometric Mechanics, 2019, 11 (4) : 639-656. doi: 10.3934/jgm.2019032 |
[18] |
Håkon Hoel, Anders Szepessy. Classical Langevin dynamics derived from quantum mechanics. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 4001-4038. doi: 10.3934/dcdsb.2020135 |
[19] |
Rémi Lassalle, Jean Claude Zambrini. A weak approach to the stochastic deformation of classical mechanics. Journal of Geometric Mechanics, 2016, 8 (2) : 221-233. doi: 10.3934/jgm.2016005 |
[20] |
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 |
2020 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]