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Clebsch optimal control formulation in mechanics
A theoretical framework for backward error analysis on manifolds
1.  Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WA, United Kingdom 
References:
[1] 
R. Abraham, J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications," volume 75 of "Applied Mathematical Sciences," SpringerVerlag, New York, second edition, 1988. 
[2] 
V. I. Arnold, "Mathematical Methods of Classical Mechanics," volume 60 of "Graduate Texts in Mathematics," SpringerVerlag, New York, second edition, 1989, Translated from the Russian by K. Vogtmann and A. Weinstein. 
[3] 
G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of neartotheidentity symplectic mappings with application to symplectic integration algorithms, J. Statist. Phys., 74 (1994), 11171143. doi: 10.1007/BF02188219. 
[4] 
M. P. Calvo, A. Murua and J. M. SanzSerna, Modified equations for ODEs, In "Chaotic Numerics (Geelong, 1993)," volume 172 of "Contemp. Math.," pages 6374, Amer. Math. Soc., Providence, RI, 1994. 
[5] 
E. Cartan, Les groupes de transformations continus, infinis, simples, Ann. Sci. École Norm. Sup. (3), 26 (1909), 93161. 
[6] 
D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math. (2), 92 (1970), 102163. doi: 10.2307/1970699. 
[7] 
O. Gonzalez, D. J. Higham and A. M. Stuart, Qualitative properties of modified equations, IMA J. Numer. Anal., 19 (1999), 169190. doi: 10.1093/imanum/19.2.169. 
[8] 
E. Hairer, Global modified Hamiltonian for constrained symplectic integrators, Numer. Math., 95 (2003), 325336. doi: 10.1007/s0021100204287. 
[9] 
E. Hairer and C. Lubich, The lifespan of backward error analysis for numerical integrators, Numer. Math., 76 (1997), 441462. doi: 10.1007/s002110050271. 
[10] 
E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," volume 31 of "Springer Series in Computational Mathematics," SpringerVerlag, Berlin, 2002. Structurepreserving algorithms for ordinary differential equations. 
[11] 
E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I," volume 8 of "Springer Series in Computational Mathematics," SpringerVerlag, Berlin, second edition, 1993. Nonstiff problems. 
[12] 
A. Iserles, H. Z. MuntheKaas, S. P. Nørsett and A. Zanna, Liegroup methods, In "Acta Numerica, 2000," volume 9 of "Acta Numer.," pages 215365, Cambridge Univ. Press, Cambridge, 2000. 
[13] 
J. M. Lee, "Introduction to Smooth Manifolds," volume 218 of "Graduate Texts in Mathematics," SpringerVerlag, New York, 2003. 
[14] 
R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), 341434. doi: 10.1017/S0962492902000053. 
[15] 
H. Omori, "Infinitedimensional Lie Groups," volume 158 of "Translations of Mathematical Monographs," American Mathematical Society, Providence, RI, 1997, Translated from the 1979 Japanese original and revised by the author. 
[16] 
R. S. Palais, "Foundations of Global Nonlinear Analysis," W. A. Benjamin, Inc., New YorkAmsterdam, 1968. 
[17] 
S. Reich, "Numerical Integration of the Generatized Euler Equations," Technical report, Vancouver, BC, Canada, Canada, 1993. 
[18] 
S. Reich, On higherorder semiexplicit symplectic partitioned RungeKutta methods for constrained Hamiltonian systems, Numer. Math., 76 (1997), 231247. doi: 10.1007/s002110050261. 
[19] 
S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), 15491570 (electronic). doi: 10.1137/S0036142997329797. 
[20] 
R. Schmid, Infinitedimensional Lie groups with applications to mathematical physics, J. Geom. Symmetry Phys., 1 (2004), 54120. 
show all references
References:
[1] 
R. Abraham, J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications," volume 75 of "Applied Mathematical Sciences," SpringerVerlag, New York, second edition, 1988. 
[2] 
V. I. Arnold, "Mathematical Methods of Classical Mechanics," volume 60 of "Graduate Texts in Mathematics," SpringerVerlag, New York, second edition, 1989, Translated from the Russian by K. Vogtmann and A. Weinstein. 
[3] 
G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of neartotheidentity symplectic mappings with application to symplectic integration algorithms, J. Statist. Phys., 74 (1994), 11171143. doi: 10.1007/BF02188219. 
[4] 
M. P. Calvo, A. Murua and J. M. SanzSerna, Modified equations for ODEs, In "Chaotic Numerics (Geelong, 1993)," volume 172 of "Contemp. Math.," pages 6374, Amer. Math. Soc., Providence, RI, 1994. 
[5] 
E. Cartan, Les groupes de transformations continus, infinis, simples, Ann. Sci. École Norm. Sup. (3), 26 (1909), 93161. 
[6] 
D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math. (2), 92 (1970), 102163. doi: 10.2307/1970699. 
[7] 
O. Gonzalez, D. J. Higham and A. M. Stuart, Qualitative properties of modified equations, IMA J. Numer. Anal., 19 (1999), 169190. doi: 10.1093/imanum/19.2.169. 
[8] 
E. Hairer, Global modified Hamiltonian for constrained symplectic integrators, Numer. Math., 95 (2003), 325336. doi: 10.1007/s0021100204287. 
[9] 
E. Hairer and C. Lubich, The lifespan of backward error analysis for numerical integrators, Numer. Math., 76 (1997), 441462. doi: 10.1007/s002110050271. 
[10] 
E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," volume 31 of "Springer Series in Computational Mathematics," SpringerVerlag, Berlin, 2002. Structurepreserving algorithms for ordinary differential equations. 
[11] 
E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I," volume 8 of "Springer Series in Computational Mathematics," SpringerVerlag, Berlin, second edition, 1993. Nonstiff problems. 
[12] 
A. Iserles, H. Z. MuntheKaas, S. P. Nørsett and A. Zanna, Liegroup methods, In "Acta Numerica, 2000," volume 9 of "Acta Numer.," pages 215365, Cambridge Univ. Press, Cambridge, 2000. 
[13] 
J. M. Lee, "Introduction to Smooth Manifolds," volume 218 of "Graduate Texts in Mathematics," SpringerVerlag, New York, 2003. 
[14] 
R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), 341434. doi: 10.1017/S0962492902000053. 
[15] 
H. Omori, "Infinitedimensional Lie Groups," volume 158 of "Translations of Mathematical Monographs," American Mathematical Society, Providence, RI, 1997, Translated from the 1979 Japanese original and revised by the author. 
[16] 
R. S. Palais, "Foundations of Global Nonlinear Analysis," W. A. Benjamin, Inc., New YorkAmsterdam, 1968. 
[17] 
S. Reich, "Numerical Integration of the Generatized Euler Equations," Technical report, Vancouver, BC, Canada, Canada, 1993. 
[18] 
S. Reich, On higherorder semiexplicit symplectic partitioned RungeKutta methods for constrained Hamiltonian systems, Numer. Math., 76 (1997), 231247. doi: 10.1007/s002110050261. 
[19] 
S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), 15491570 (electronic). doi: 10.1137/S0036142997329797. 
[20] 
R. Schmid, Infinitedimensional Lie groups with applications to mathematical physics, J. Geom. Symmetry Phys., 1 (2004), 54120. 
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