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Linearised higher variational equations
1. | Department of Mathematics, University of Portsmouth, Lion Gate Bldg, Lion Terrace, Portsmouth PO1 3HF, United Kingdom |
References:
[1] |
M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1984, Reprint of the 1972 edition, Selected Government Publications. |
[2] |
A. Aparicio-Monforte, Méthodes Effectives Pour L'intégrabilité des Systèmes Dynamiques, Ph.D. thesis, Université de Limoges, December, 2010. |
[3] |
A. Aparicio Monforte and J.-A. Weil, A reduction method for higher order variational equations of Hamiltonian systems, Symmetries and related topics in differential and difference equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 549 (2011), 1-15.
doi: 10.1090/conm/549/10850. |
[4] |
_______ and _______, A reduced form for linear differential systems and its application to integrability of Hamiltonian systems}, J. Symbolic Comput., 47 (2012), 192-213.
doi: 10.1016/j.jsc.2011.09.011. |
[5] |
A. Aparicio-Monforte, M. Barkatou, S. Simon and J.-A. Weil, Formal first integrals along solutions of differential systems I, ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, 19-26, ACM, New York, 2011.
doi: 10.1145/1993886.1993896. |
[6] |
M. Audin, Les Systèmes Hamiltoniens et Leur Intégrabilité, Cours Spécialisés, vol. 8, Société Mathématique de France, Paris, 2001. |
[7] |
M. Barkatou, On rational solutions of systems of linear differential equations, J. Symbolic Comput., 28 (1999), 547-567.
doi: 10.1006/jsco.1999.0314. |
[8] |
U. Bekbaev, A matrix representation of composition of polynomial maps,, , ().
|
[9] |
________, A radius of absolute convergence for power series in many variables,, , ().
|
[10] |
________, Matrix representations for symmetric and antisymmetric multi-linear maps,, , ().
|
[11] |
________, An inversion formula for multivariate power series,, , ().
|
[12] |
E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
doi: 10.2307/2300300. |
[13] |
A. Blokhuis and J. J. Seidel, An introduction to multilinear algebra and some applications, Philips J. Res., 39 (1984), 111-120. |
[14] | |
[15] |
J. Casasayas, A. Nunes and N. B. Tufillaro, Swinging Atwood's machine: Integrability and dynamics, J. Phys., 51 (1990), 1693-1702.
doi: 10.1051/jphys:0199000510160169300. |
[16] |
W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-0979-9. |
[17] |
I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, 2008. |
[18] |
S. Lang, Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4613-0041-0. |
[19] |
K. Makino and M. Berz, Suppression of the wrapping effect by Taylor model-based verified integrators: Long-term stabilization by preconditioning, Int. J. Differ. Equ. Appl., 10 (2005), 353-384 (2006). |
[20] |
R. Martínez and C. Simó, Non-integrability of the degenerate cases of the swinging Atwood's machine using higher order variational equations, Discrete Contin. Dyn. Syst., 29 (2011), 1-24.
doi: 10.3934/dcds.2011.29.1. |
[21] |
_______ and _______, Non-integrability of Hamiltonian systems through high order variational equations: summary of results and examples, Regul. Chaotic Dyn., 14 (2009), 323-348.
doi: 10.1134/S1560354709030010. |
[22] |
J. J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems, Progress in Mathematics, Birkhäuser Verlag, Basel, 1999. |
[23] |
J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, Methods Appl. Anal., 8 (2001), 33-95. |
[24] |
J. J. Morales-Ruiz, J.-P. Ramis and C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. École Norm. Sup. (4), 40 (2007), 845-884.
doi: 10.1016/j.ansens.2007.09.002. |
[25] |
J. J. Morales-Ruiz, C. Simó and S. Simon, Algebraic proof of the non-integrability of Hill's problem, Ergodic Theory Dynam. Systems, 25 (2005), 1237-1256.
doi: 10.1017/S0143385704001038. |
[26] |
O. Pujol, J.-P. Pérez, J.-P. Ramis, C. Simó, S. Simon and J.-A. Weil, Swinging Atwood machine: Experimental and numerical results, and a theoretical study, Phys. D, 239 (2010), 1067-1081.
doi: 10.1016/j.physd.2010.02.017. |
[27] |
S. Ramanujan, Notebooks, (2 volumes) Tata Institute of Fundamental Research, Bombay, 1957. |
[28] |
S. Simon, Conditions and evidence for non-integrability in the Friedmann-Robertson-Walker Hamiltonian, Journal of Nonlinear Mathematical Physics, 21 (2014), 1-16.
doi: 10.1080/14029251.2014.894710. |
[29] |
M. van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 328, Springer-Verlag, Berlin, 2003. |
[30] |
N. B. Tufillaro, Integrable motion of a swinging Atwood's machine, Amer. J. Phys., 54 (1986), 142-153.
doi: 10.1119/1.14710. |
[31] |
S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I, Funktsional. Anal. i Prilozhen, 16 (1982), 30-41, 96. |
[32] |
H. Zoladek, The Monodromy Group, Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series) 67, Birkhäuser Verlag, Basel, 2006. |
show all references
References:
[1] |
M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1984, Reprint of the 1972 edition, Selected Government Publications. |
[2] |
A. Aparicio-Monforte, Méthodes Effectives Pour L'intégrabilité des Systèmes Dynamiques, Ph.D. thesis, Université de Limoges, December, 2010. |
[3] |
A. Aparicio Monforte and J.-A. Weil, A reduction method for higher order variational equations of Hamiltonian systems, Symmetries and related topics in differential and difference equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 549 (2011), 1-15.
doi: 10.1090/conm/549/10850. |
[4] |
_______ and _______, A reduced form for linear differential systems and its application to integrability of Hamiltonian systems}, J. Symbolic Comput., 47 (2012), 192-213.
doi: 10.1016/j.jsc.2011.09.011. |
[5] |
A. Aparicio-Monforte, M. Barkatou, S. Simon and J.-A. Weil, Formal first integrals along solutions of differential systems I, ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, 19-26, ACM, New York, 2011.
doi: 10.1145/1993886.1993896. |
[6] |
M. Audin, Les Systèmes Hamiltoniens et Leur Intégrabilité, Cours Spécialisés, vol. 8, Société Mathématique de France, Paris, 2001. |
[7] |
M. Barkatou, On rational solutions of systems of linear differential equations, J. Symbolic Comput., 28 (1999), 547-567.
doi: 10.1006/jsco.1999.0314. |
[8] |
U. Bekbaev, A matrix representation of composition of polynomial maps,, , ().
|
[9] |
________, A radius of absolute convergence for power series in many variables,, , ().
|
[10] |
________, Matrix representations for symmetric and antisymmetric multi-linear maps,, , ().
|
[11] |
________, An inversion formula for multivariate power series,, , ().
|
[12] |
E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
doi: 10.2307/2300300. |
[13] |
A. Blokhuis and J. J. Seidel, An introduction to multilinear algebra and some applications, Philips J. Res., 39 (1984), 111-120. |
[14] | |
[15] |
J. Casasayas, A. Nunes and N. B. Tufillaro, Swinging Atwood's machine: Integrability and dynamics, J. Phys., 51 (1990), 1693-1702.
doi: 10.1051/jphys:0199000510160169300. |
[16] |
W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-0979-9. |
[17] |
I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, 2008. |
[18] |
S. Lang, Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4613-0041-0. |
[19] |
K. Makino and M. Berz, Suppression of the wrapping effect by Taylor model-based verified integrators: Long-term stabilization by preconditioning, Int. J. Differ. Equ. Appl., 10 (2005), 353-384 (2006). |
[20] |
R. Martínez and C. Simó, Non-integrability of the degenerate cases of the swinging Atwood's machine using higher order variational equations, Discrete Contin. Dyn. Syst., 29 (2011), 1-24.
doi: 10.3934/dcds.2011.29.1. |
[21] |
_______ and _______, Non-integrability of Hamiltonian systems through high order variational equations: summary of results and examples, Regul. Chaotic Dyn., 14 (2009), 323-348.
doi: 10.1134/S1560354709030010. |
[22] |
J. J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems, Progress in Mathematics, Birkhäuser Verlag, Basel, 1999. |
[23] |
J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, Methods Appl. Anal., 8 (2001), 33-95. |
[24] |
J. J. Morales-Ruiz, J.-P. Ramis and C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. École Norm. Sup. (4), 40 (2007), 845-884.
doi: 10.1016/j.ansens.2007.09.002. |
[25] |
J. J. Morales-Ruiz, C. Simó and S. Simon, Algebraic proof of the non-integrability of Hill's problem, Ergodic Theory Dynam. Systems, 25 (2005), 1237-1256.
doi: 10.1017/S0143385704001038. |
[26] |
O. Pujol, J.-P. Pérez, J.-P. Ramis, C. Simó, S. Simon and J.-A. Weil, Swinging Atwood machine: Experimental and numerical results, and a theoretical study, Phys. D, 239 (2010), 1067-1081.
doi: 10.1016/j.physd.2010.02.017. |
[27] |
S. Ramanujan, Notebooks, (2 volumes) Tata Institute of Fundamental Research, Bombay, 1957. |
[28] |
S. Simon, Conditions and evidence for non-integrability in the Friedmann-Robertson-Walker Hamiltonian, Journal of Nonlinear Mathematical Physics, 21 (2014), 1-16.
doi: 10.1080/14029251.2014.894710. |
[29] |
M. van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 328, Springer-Verlag, Berlin, 2003. |
[30] |
N. B. Tufillaro, Integrable motion of a swinging Atwood's machine, Amer. J. Phys., 54 (1986), 142-153.
doi: 10.1119/1.14710. |
[31] |
S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I, Funktsional. Anal. i Prilozhen, 16 (1982), 30-41, 96. |
[32] |
H. Zoladek, The Monodromy Group, Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series) 67, Birkhäuser Verlag, Basel, 2006. |
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