Abstract / p1
Acknowledgements / p5
1 Introduction / p3
　1.1 The Reasons of this Work / p3
　1.2 Overview / p6
2 The Lie Transform / p9
　2.1 Introduction / p9
　2.2 Hamiltonian Mechanics / p10
　2.3 The Lie Transform / p17
3 Magnetic Field Lines Hamiltonian / p25
　3.1 Introduction / p25
　3.2 Magnetic Potential / p26
　3.3 Taylor Expansion of the Potential / p28
　3.4 Cylindrical Limit Approximation / p31
　3.5 Helical Toroidal Potential / p40
　3.6 Integrable Model / p45
　3.7 Conclusion / p47
4 Symplectic Integration / p49
　4.1 Introduction / p49
　4.2 Symplectic Methods / p52
　4.3 The Difference Scheme / p54
　4.4 Accuracy and Stability of the Method / p58
　4.5 Runge-Kutta Formulation / p66
　4.6 Linear Symplectic Methods / p70
　4.7 Conclusion / p72
5 Numerical Results / p75
　5.1 Introduction / p75
　5.2 Toroidal Hamiltonian / p76
6 Concluding Remarks / p85
A Coefficients / p87
　A.1 Introduction / p87
　A.2 The case n=1 / p88
　A.3 The cases n=2,3,4,5,6 / p89
References / p97