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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.

[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]

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

[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]

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

[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.

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