Geometry of Routh reduction

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March 2019, 11(1): 23-44. doi: 10.3934/jgm.2019002

Geometry of Routh reduction

Katarzyna Grabowska ^, and Paweƚ Urbański

Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland

* Corresponding author

Received February 2018 Revised December 2018 Published January 2019

Fund Project: Research founded by the Polish National Science Centre grant under the contract number DEC-2012/06/A/ST1/00256.

The Routh reduction for Lagrangian systems with cyclic variable is presented as an example of a Lagrangian reduction. It appears that the Routhian, which is a generating object of reduced dynamics, is not a function any more but a section of a bundle of affine values.

Keywords: Routh reduction, symmetries, Lagrangian mechanics, Hamiltonian mechanics, Tulczyjew triple.

Mathematics Subject Classification: Primary: 51P05, 70H33, 37J15; Secondary: 70H03.

Citation: Katarzyna Grabowska, Paweƚ Urbański. Geometry of Routh reduction. Journal of Geometric Mechanics, 2019, 11 (1) : 23-44. doi: 10.3934/jgm.2019002

References:

[1]	L. Adamec, A route to routh the classical setting, Journal of Nonlinear Mathematical Physics, 18 (2011), 87-107. doi: 10.1142/S1402925111001180.
[2]	V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 1997.
[3]	S. L. Bażański, The Jacobi variational principle revisited, in Classical and Quantum Integrability (Warsaw, 2001), Banach Center Publ., 59 (2003), 99-111. doi: 10.4064/bc59-0-4.
[4]	S. Benenti, Hamiltonian Structures and Generating Families, Universitext, Springer, 2011. doi: 10.1007/978-1-4614-1499-5.
[5]	M. Crampin and T. Mestdag, Routh procedure for non-Abelian symmetry groups, J. Math. Phys., 49 (2008), 032901, 28 pp. doi: 10.1063/1.2885077.
[6]	J.-P. Dufour, Introduction aux tissus, (preprint), Séminaire GETODIM, (1991), 55-76.
[7]	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.
[8]	K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Poisson and Jacobi structures, J. Geom. Phys., 52 (2004), 398-446. doi: 10.1016/j.geomphys.2004.04.004.
[9]	K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Euler-Lagrange equations, J. Geom. Phys., 57 (2007), 1984-1998. doi: 10.1016/j.geomphys.2007.04.003.
[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 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.
[13]	J. Grabowski, M. Rotkiewicz and P. Urbanski, Double Affine Bundles, J. Geom. Phys., 60 (2010), 581-598. doi: 10.1016/j.geomphys.2009.12.008.
[14]	J. Grabowski and P. Urbanski, Tangent lifts of Poisson and related structures, J. Phys. A: Math. Gen., 28 (1995), 6743-6777. doi: 10.1088/0305-4470/28/23/024.
[15]	J. Grabowski and P. Urbanski, Algebroids general differential calculi on vector bundles, J. Geom. Phys, 31 (1999), 111-141. doi: 10.1016/S0393-0440(99)00007-8.
[16]	K. Konieczna and P. Urbański, Double vector bundles and duality, Arch. Math. (Brno), 35 (1999), 59-95.
[17]	B. Langerock, E. García-Toraño Andrés and F. Cantrijn, Routh reduction and the class of magnetic Lagrangian systems, J. Math. Phys., 53 (2012), 062902, 19pp. doi: 10.1063/1.4723841.
[18]	B. Langerock, F. Cantrijn and J. Vankerschaver, Routhian reduction for quasi-invariant Lagrangians, J. Math. Phys., 51 (2010), 022902, 20pp. doi: 10.1063/1.3277181.
[19]	P. Liebermann and Ch. M. Marle, Symplectic Geometry and Analytical Mechanics, Reidel Publishing Company, Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.
[20]	J. E. Marsden, T. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, Journal of Mathematical Physics, 41 (2000), 3379-3429. doi: 10.1063/1.533317.
[21]	T. Mestdag, Finsler geodesics of Lagrangian systems through Routh reduction, Mediterranean Journal of Mathematics, 13 (2016), 825-839. doi: 10.1007/s00009-014-0505-z.
[22]	J. Pradines, Fibrés Vectoriels Doubles et Calcul Des Jets Non-Holonomes, Notes polycopiés Amiens, 1977 (in French).
[23]	E. J. Routh, Stability of a Given State of Motion, Halsted Press, New York, 1877.
[24]	W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sc. Paris, 283 (1976), 15-18.
[25]	W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sc. Paris, 283 (1976), 675-678.
[26]	W. M. Tulczyjew, The Legendre transformation, Ann. Inst. Henri Poincaré, 27 (1977), 101-114.
[27]	W. M. Tulczyjew, Geometric Formulation of Physical Theories, Bibliopolis, Naples, 1989.
[28]	W. M. Tulczyjew and P. Urbański, An affine framework for the dynamics of charged particles, Atti Accad. Sci. Torino, Suppl., 126 (1992), 257-265.
[29]	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.
[30]	W. M. Tulczyjew, P. Urbański and S. Zakrzewski, A pseudocategory of principal bundles, Atti Accad. Sci. Torino, 122 (1988), 66-72.
[31]	P. Urbański, An affine framework for analytical mechanics, in Classical and quantum integrability (Warsaw, 2001), Banach Center Publ., 59 (2003), 257-279. doi: 10.4064/bc59-0-14.
[32]	P. Urbański, Double vector bundles in classical mechanics, Rend. Sem. Mat. Univ. Poi. Torino, 54 (1996), 405-421.

show all references

References:

[1]	L. Adamec, A route to routh the classical setting, Journal of Nonlinear Mathematical Physics, 18 (2011), 87-107. doi: 10.1142/S1402925111001180.
[2]	V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 1997.
[3]	S. L. Bażański, The Jacobi variational principle revisited, in Classical and Quantum Integrability (Warsaw, 2001), Banach Center Publ., 59 (2003), 99-111. doi: 10.4064/bc59-0-4.
[4]	S. Benenti, Hamiltonian Structures and Generating Families, Universitext, Springer, 2011. doi: 10.1007/978-1-4614-1499-5.
[5]	M. Crampin and T. Mestdag, Routh procedure for non-Abelian symmetry groups, J. Math. Phys., 49 (2008), 032901, 28 pp. doi: 10.1063/1.2885077.
[6]	J.-P. Dufour, Introduction aux tissus, (preprint), Séminaire GETODIM, (1991), 55-76.
[7]	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.
[8]	K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Poisson and Jacobi structures, J. Geom. Phys., 52 (2004), 398-446. doi: 10.1016/j.geomphys.2004.04.004.
[9]	K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Euler-Lagrange equations, J. Geom. Phys., 57 (2007), 1984-1998. doi: 10.1016/j.geomphys.2007.04.003.
[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 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.
[13]	J. Grabowski, M. Rotkiewicz and P. Urbanski, Double Affine Bundles, J. Geom. Phys., 60 (2010), 581-598. doi: 10.1016/j.geomphys.2009.12.008.
[14]	J. Grabowski and P. Urbanski, Tangent lifts of Poisson and related structures, J. Phys. A: Math. Gen., 28 (1995), 6743-6777. doi: 10.1088/0305-4470/28/23/024.
[15]	J. Grabowski and P. Urbanski, Algebroids general differential calculi on vector bundles, J. Geom. Phys, 31 (1999), 111-141. doi: 10.1016/S0393-0440(99)00007-8.
[16]	K. Konieczna and P. Urbański, Double vector bundles and duality, Arch. Math. (Brno), 35 (1999), 59-95.
[17]	B. Langerock, E. García-Toraño Andrés and F. Cantrijn, Routh reduction and the class of magnetic Lagrangian systems, J. Math. Phys., 53 (2012), 062902, 19pp. doi: 10.1063/1.4723841.
[18]	B. Langerock, F. Cantrijn and J. Vankerschaver, Routhian reduction for quasi-invariant Lagrangians, J. Math. Phys., 51 (2010), 022902, 20pp. doi: 10.1063/1.3277181.
[19]	P. Liebermann and Ch. M. Marle, Symplectic Geometry and Analytical Mechanics, Reidel Publishing Company, Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.
[20]	J. E. Marsden, T. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, Journal of Mathematical Physics, 41 (2000), 3379-3429. doi: 10.1063/1.533317.
[21]	T. Mestdag, Finsler geodesics of Lagrangian systems through Routh reduction, Mediterranean Journal of Mathematics, 13 (2016), 825-839. doi: 10.1007/s00009-014-0505-z.
[22]	J. Pradines, Fibrés Vectoriels Doubles et Calcul Des Jets Non-Holonomes, Notes polycopiés Amiens, 1977 (in French).
[23]	E. J. Routh, Stability of a Given State of Motion, Halsted Press, New York, 1877.
[24]	W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sc. Paris, 283 (1976), 15-18.
[25]	W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sc. Paris, 283 (1976), 675-678.
[26]	W. M. Tulczyjew, The Legendre transformation, Ann. Inst. Henri Poincaré, 27 (1977), 101-114.
[27]	W. M. Tulczyjew, Geometric Formulation of Physical Theories, Bibliopolis, Naples, 1989.
[28]	W. M. Tulczyjew and P. Urbański, An affine framework for the dynamics of charged particles, Atti Accad. Sci. Torino, Suppl., 126 (1992), 257-265.
[29]	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.
[30]	W. M. Tulczyjew, P. Urbański and S. Zakrzewski, A pseudocategory of principal bundles, Atti Accad. Sci. Torino, 122 (1988), 66-72.
[31]	P. Urbański, An affine framework for analytical mechanics, in Classical and quantum integrability (Warsaw, 2001), Banach Center Publ., 59 (2003), 257-279. doi: 10.4064/bc59-0-14.
[32]	P. Urbański, Double vector bundles in classical mechanics, Rend. Sem. Mat. Univ. Poi. Torino, 54 (1996), 405-421.

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