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Aspects of reduction and transformation of Lagrangian systems with symmetry
1. | Department of Mathematics, Ghent University, Krijgslaan 281 S22, B9000 Ghent, Belgium, Belgium |
2. | Belgian Institute for Space Aeronomy, Ringlaan 3, B1180 Brussels, Belgium |
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics, The Benjamin/Cummings Publishing Company, INC, 1978. |
[2] |
M. Crampin and T. Mestdag, Routh's procedure for non-Abelian symmetry groups, J. Math. Phys., 49 (2008), 032901.
doi: 10.1063/1.2885077. |
[3] |
M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys., 19 (1978), 2388-2399.
doi: 10.1063/1.523597. |
[4] |
M. J. Gotay and J. M. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence problem, Ann. Inst. Henri Poincaré, 30 (1979), 129-142. |
[5] |
S. M. Jalnapurkar and J. E. Marsden, Reduction of Hamilton's variational principle, Dynamics and Stability of Systems, 15 (2000), 287-318.
doi: 10.1080/713603744. |
[6] |
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, volume I and II, Interscience Publishers, 1963. |
[7] |
B. Langerock, E. G. Andrés and F. Cantrijn, Routh reduction and the class of magnetic Lagrangian systems, Journal Of Mathematical Physics, 53 (2012), 062902, 19 pp.
doi: 10.1063/1.4723841. |
[8] |
B. Langerock, F. Cantrijn and J. Vankerschaver, Routhian reduction for quasi-invariant Lagrangians, J. Math. Phys., 51 (2010), 022902.
doi: 10.1063/1.3277181. |
[9] |
B. Langerock and M. C. Lopéz, Routhian reduction for singular Lagrangians, J. Geom. Meth. Mod. Phys., 7 (2010), 1451-1489.
doi: 10.1142/S0219887810004907. |
[10] |
B. Langerock, T. Mestdag and J. Vankerschaver, Routh reduction by stages, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 31pp.
doi: 10.3842/SIGMA.2011.109. |
[11] |
J. E. Marsden, Lectures on Mechanics, Cambridge University Press, 1992. |
[12] |
J. E. Marsden, G. Misiołek, J. P. Ortega, M. Perlmutter and T. S. Ratiu, Hamiltonian Reduction by Stages, volume 1913 of Lecture Notes in Mathematics. Springer, Berlin, 2007. |
[13] |
J. E. Marsden, T. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, J. Math. Phys., 41 (2000), 3379-3429.
doi: 10.1063/1.533317. |
[14] |
J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum, Z. Angew. Math. Phys., 44 (1993), 17-43.
doi: 10.1007/BF00914351. |
[15] |
P. Morando and S. Sammarco, Variational problems with symmetries: A Pfaffian system approach, Acta Appl. Math., 120 (2012), 255-274.
doi: 10.1007/s10440-012-9720-4. |
[16] |
R. W. Sharpe, Differential Geometry, volume 166 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. Cartan's generalization of Klein's Erlangen program, With a foreword by S. S. Chern. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics, The Benjamin/Cummings Publishing Company, INC, 1978. |
[2] |
M. Crampin and T. Mestdag, Routh's procedure for non-Abelian symmetry groups, J. Math. Phys., 49 (2008), 032901.
doi: 10.1063/1.2885077. |
[3] |
M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys., 19 (1978), 2388-2399.
doi: 10.1063/1.523597. |
[4] |
M. J. Gotay and J. M. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence problem, Ann. Inst. Henri Poincaré, 30 (1979), 129-142. |
[5] |
S. M. Jalnapurkar and J. E. Marsden, Reduction of Hamilton's variational principle, Dynamics and Stability of Systems, 15 (2000), 287-318.
doi: 10.1080/713603744. |
[6] |
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, volume I and II, Interscience Publishers, 1963. |
[7] |
B. Langerock, E. G. Andrés and F. Cantrijn, Routh reduction and the class of magnetic Lagrangian systems, Journal Of Mathematical Physics, 53 (2012), 062902, 19 pp.
doi: 10.1063/1.4723841. |
[8] |
B. Langerock, F. Cantrijn and J. Vankerschaver, Routhian reduction for quasi-invariant Lagrangians, J. Math. Phys., 51 (2010), 022902.
doi: 10.1063/1.3277181. |
[9] |
B. Langerock and M. C. Lopéz, Routhian reduction for singular Lagrangians, J. Geom. Meth. Mod. Phys., 7 (2010), 1451-1489.
doi: 10.1142/S0219887810004907. |
[10] |
B. Langerock, T. Mestdag and J. Vankerschaver, Routh reduction by stages, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 31pp.
doi: 10.3842/SIGMA.2011.109. |
[11] |
J. E. Marsden, Lectures on Mechanics, Cambridge University Press, 1992. |
[12] |
J. E. Marsden, G. Misiołek, J. P. Ortega, M. Perlmutter and T. S. Ratiu, Hamiltonian Reduction by Stages, volume 1913 of Lecture Notes in Mathematics. Springer, Berlin, 2007. |
[13] |
J. E. Marsden, T. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, J. Math. Phys., 41 (2000), 3379-3429.
doi: 10.1063/1.533317. |
[14] |
J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum, Z. Angew. Math. Phys., 44 (1993), 17-43.
doi: 10.1007/BF00914351. |
[15] |
P. Morando and S. Sammarco, Variational problems with symmetries: A Pfaffian system approach, Acta Appl. Math., 120 (2012), 255-274.
doi: 10.1007/s10440-012-9720-4. |
[16] |
R. W. Sharpe, Differential Geometry, volume 166 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. Cartan's generalization of Klein's Erlangen program, With a foreword by S. S. Chern. |
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