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Modified equations for variational integrators applied to Lagrangians linear in velocities
Geometry of Routh reduction
Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland |
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.
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. Google Scholar |
[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. Google Scholar |
[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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