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Lie systems: theory, generalisations, and applications / J.F.Cariñena, J. de Lucas; Institute of Mathematics, Polish Academy of Sciences. - Warszawa: Instytut matematyczny PAN, 2011. - ii, 161 p. - Ref.: p.152-162. - (Dissertationes mathematicae; 479) - ISSN 0012-3862
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1 The theory of Lie systems .................................... 6
1.1 Motivation and general scheme of the work ............... 6
1.2 Historical introduction ................................. 8
1.3 Fundamentals about Lie systems and superposition
rules .................................................. 11
1.4 Geometric approach to superposition rules .............. 22
1.5 Geometric Lie Theorem .................................. 27
1.6 Determination of superposition rules ................... 28
1.7 Mixed superposition rules and constants of motion ...... 30
1.8 Differential geometry on Hilbert spaces ................ 32
1.9 Quantum Lie systems .................................... 34
1.10 Superposition rules for second and higher-order
differential equations ................................. 36
1.11 Superposition rules for PDEs ........................... 39
2 SODE Lie systems ............................................ 41
2.1 The harmonic oscillator with t-dependent frequency ..... 41
2.2 Generalised Ermakov system ............................. 46
2.3 Milne-Pinney equation .................................. 48
2.4 A new superposition rule for the Milne-Pinney
equation ............................................... 50
2.5 Painlevé-Ince equations and other SODE Lie systems ..... 54
2.6 Mixed superposition rules and Ermakov systems .......... 57
2.7 Relations between the new and the known superposition
rule ................................................... 60
2.8 A new mixed superposition rule for the Pinney
equation ............................................... 61
3 Applications of quantum Lie systems ......................... 63
3.1 The reduction method in quantum mechanics .............. 64
3.2 Interaction picture and Lie systems .................... 66
3.3 The method of unitary transformations .................. 67
3.4 t-dependent operators for quantum Lie systems .......... 68
3.5 Initial examples ....................................... 68
3.6 Quadratic Hamiltonians ................................. 69
3.7 Particular cases ....................................... 71
3.8 Nonsolvable Hamiltonians and particular instances ...... 73
3.9 Reduction in quantum mechanics ......................... 74
4 Integrability conditions for Lie systems .................... 76
4.1 Integrability of Riccati equations ..................... 77
4.2 Transformation laws of Riccati equations ............... 78
4.3 Lie structure of an equation describing
transformations of Lie systems ......................... 79
4.4 Description of some known integrability conditions ..... 81
4.5 Integrability and reduction ............................ 85
4.6 Linearisation of Riccati equations ..................... 88
5 Lie integrability in classical physics ...................... 90
5.1 TDHO as a SODE Lie system .............................. 90
5.2 Transformation laws of Lie equations on SL(2,  ) ....... 92
5.3 Description of some known integrability conditions ..... 93
5.4 Some applications of integrability conditions to
TDHOs .................................................. 95
5.5 Integrable TDHOs and t-dependent constants of motion ... 98
5.6 Applications to two-dimensional TDHOs .................. 99
6 Integrability in quantum mechanics ......................... 101
6.1 Spin Hamiltonians ..................................... 102
6.2 Lie structure of an equation describing
transformations of Lie systems ........................ 103
6.3 Integrability conditions for SU(2) Schrödinger
equations ............................................. 106
6.4 Application of integrability conditions in a SU(2)
Schrödinger equation .................................. 107
6.5 Applications to physics ............................... 109
7 The theory of quasi-Lie schemes and Lie families ........... 111
7.1 Introduction .......................................... 111
7.2 Generalised flows and t-dependent vector fields ....... 113
7.3 Quasi-Lie systems and schemes ......................... 115
7.4 t-dependent superposition rules ....................... 119
8 Applications of quasi-Lie schemes and Lie families ......... 123
8.1 Dissipative Milne-Pinney equations .................... 123
8.2 Nonlinear oscillators ................................. 128
8.3 Dissipative Mathews-Lakshmanan oscillators ............ 131
8.4 The Emden equation .................................... 132
8.5 t-dependent constants of motion and particular
solutions for Emden equations ......................... 135
8.6 Applications of particular solutions to study Emden
equations ............................................. 136
8.7 The Kummer-Liouville transformation for a general
Emden-Fowler equation ................................. 139
8.8 Constants of motion for sets of Emden-Fowler
equations ............................................. 140
8.9 A t-dependent superposition rule for Abel equations ... 143
8.10 Lie families and second-order differential
equations ............................................. 146
9 Conclusions and outlook .................................... 150
References .................................................... 152
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