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December  2016, 8(4): 413-435. doi: 10.3934/jgm.2016014

The Tulczyjew triple in mechanics on a Lie group

Katarzyna Grabowska 1, and  Marcin Zając 1,

1. 

Department of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland

Received  February 2016 Revised  October 2016 Published  November 2016

Tulczyjew triple for physical systems with configuration manifold equipped with a Lie group structure is constructed and discussed. Systems invariant with respect to group and subgroup actions are considered together with appropriate reductions of the Tulczyjew triple. The theory is applied to free and constrained rigid-body dynamics.
Keywords: Lie groups, classical mechanics, rigid body., Tulczyjew triple, double vector bundles.
Mathematics Subject Classification: Primary: 22E70, 53D05, 53Z05, 70E17, 70H3.
Citation: Katarzyna Grabowska, Marcin Zając. The Tulczyjew triple in mechanics on a Lie group. Journal of Geometric Mechanics, 2016, 8 (4) : 413-435. doi: 10.3934/jgm.2016014
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.  Google Scholar

[2]

J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Universitext, Springer-Verlag, 2000. doi: 10.1007/978-3-642-56936-4.  Google Scholar

[3]

O. Esen and H. Gumral, Tulczyjew's triplet for Lie groups I: Trivializations and reductions, J. Lie Theory, 24 (2014), 1115-1160.  Google Scholar

[4]

O. Esen and H. Gumral, Tulczyjew's triplet for Lie groups II: Dynamics,, arXiv:1503.06566., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. Klein, Espaces variationnels et mécanique, Ann. Inst. Fourier, 12 (1962), 1-124. doi: 10.5802/aif.120.  Google Scholar

[18]

K. Konieczna and P. Urbański, Double vector bundles and duality, Archivum Mathematicum, 35 (1999), 59-95.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

P. Liebermann, Lie algebroids and mechanics, Archivum Mathematicum, 32 (1996), 147-162.  Google Scholar

[22]

E. Martinez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259.  Google Scholar

[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.  Google Scholar

[24]

E. Martinez, Variational calculus on Lie algebroids, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 356-380. doi: 10.1051/cocv:2007056.  Google Scholar

[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.  Google Scholar

[26]

J. Pradines, Geometrie differentielle au-dessus d'un grupoide, C. R. Acad. Sci. Paris, série A, 266 (1968), 1194-1196.  Google Scholar

[27]

G. K. Suslov, Theoretical Mechanics, Gostekhizdat, 1946. Google Scholar

[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.  Google Scholar

[29]

W. M. Tulczyjew, Sur la différentielle de Lagrange, C. R. Acad. Sci. Paris., 280 (1975), 1295-1298.  Google Scholar

[30]

W. M. Tulczyjew, Les sous-varietes lagrangiennes et la dynamique hamiltonienne, C.R. Acad. Sc. Paris, 283 (1976), 15-18.  Google Scholar

[31]

W. M. Tulczyjew, Les sous-varietes lagrangiennes et la dynamique lagrangienne, C.R. Acad. Sc. Paris, 283 (1976), 675-678.  Google Scholar

[32]

W. M. Tulczyjew, The legendre transformation, Ann. Inst. H. Poincare, 27 (1977), 101-114.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

P. Urbański, Double vector bundles in classical mechanics, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 405-421.  Google Scholar

[36]

A. Weinstein, Lagrangian mechanics and grupoids, Fields Inst. Comm., 7 (1996), 207-231.  Google Scholar

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.  Google Scholar

[2]

J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Universitext, Springer-Verlag, 2000. doi: 10.1007/978-3-642-56936-4.  Google Scholar

[3]

O. Esen and H. Gumral, Tulczyjew's triplet for Lie groups I: Trivializations and reductions, J. Lie Theory, 24 (2014), 1115-1160.  Google Scholar

[4]

O. Esen and H. Gumral, Tulczyjew's triplet for Lie groups II: Dynamics,, arXiv:1503.06566., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. Klein, Espaces variationnels et mécanique, Ann. Inst. Fourier, 12 (1962), 1-124. doi: 10.5802/aif.120.  Google Scholar

[18]

K. Konieczna and P. Urbański, Double vector bundles and duality, Archivum Mathematicum, 35 (1999), 59-95.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

P. Liebermann, Lie algebroids and mechanics, Archivum Mathematicum, 32 (1996), 147-162.  Google Scholar

[22]

E. Martinez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259.  Google Scholar

[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.  Google Scholar

[24]

E. Martinez, Variational calculus on Lie algebroids, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 356-380. doi: 10.1051/cocv:2007056.  Google Scholar

[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.  Google Scholar

[26]

J. Pradines, Geometrie differentielle au-dessus d'un grupoide, C. R. Acad. Sci. Paris, série A, 266 (1968), 1194-1196.  Google Scholar

[27]

G. K. Suslov, Theoretical Mechanics, Gostekhizdat, 1946. Google Scholar

[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.  Google Scholar

[29]

W. M. Tulczyjew, Sur la différentielle de Lagrange, C. R. Acad. Sci. Paris., 280 (1975), 1295-1298.  Google Scholar

[30]

W. M. Tulczyjew, Les sous-varietes lagrangiennes et la dynamique hamiltonienne, C.R. Acad. Sc. Paris, 283 (1976), 15-18.  Google Scholar

[31]

W. M. Tulczyjew, Les sous-varietes lagrangiennes et la dynamique lagrangienne, C.R. Acad. Sc. Paris, 283 (1976), 675-678.  Google Scholar

[32]

W. M. Tulczyjew, The legendre transformation, Ann. Inst. H. Poincare, 27 (1977), 101-114.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

P. Urbański, Double vector bundles in classical mechanics, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 405-421.  Google Scholar

[36]

A. Weinstein, Lagrangian mechanics and grupoids, Fields Inst. Comm., 7 (1996), 207-231.  Google Scholar

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