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4 Symplectic methods The above properties, and the fact that symplecticity is a characterizing property of Hamiltonian systems, reinforce the search of symplectic methods for their numerical integration. A one-step method y1 “ Φh py0 q is per se a transformation of the phase space. , if J BΦh py0 q BΦh py0 q J “ J. By0 By0 13 A closely related problem in complex dynamics concerns the existence of centers. Given ř k a complex analytic function f pzq “ e2πiα z ` 8 k“2 ak z , with α P p0, 1q, a formidable question in complex dynamics was whether it was possible for f to be conjugate to its linear part.
2016 by Taylor & Francis Group, LLC A primer on line integral methods 27 Multiplying the former equation by 2, and subtracting the latter one, gives Y1 “ py0 ` y1 q{2, so the method can be also rewritten as ˆ ˙ y0 ` y1 y1 “ y0 ` hf . 2 In this equivalent form, it is mostly often referred to as implicit mid-point method. 5 s-Stage trapezoidal methods It is for Hamiltonian problems that the corpus of the theory of line integral methods has been first developed, forming the core part of the present manuscript.
As an example, composing the St¨ormer-Verlet method with the symmetric stepsizes h , 2 ´ r2 ´r2 h , 2 ´ r2 h , 2 ´ r2 r2 “ 3 ? 2, yields a fourth-order symplectic method. In turn, the composition of the latter method with the symmetric stepsizes h , 2 ´ r4 © 2016 by Taylor & Francis Group, LLC ´r4 h , 2 ´ r4 h , 2 ´ r4 r4 “ 5 ? 2, A primer on line integral methods 23 yields a sixth-order symplectic method, and so forth. In general, having got an (even) order p symplectic method, its composistion with the symmetric stepsizes ?