By Vladimir Kanovei
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23) can be written in a Hamiltonian form. First, introduce new variables: M ± = Mx ± iMy . 23) can be rewritten as: ∂W ∂W d ± ∓ 2igMz . 25) if we use Mz = (M2 − M + M − )1/2 . 26) W (Mx , My , Mz ) → W(M + , M − ; (M 2 − M + M − )1/2 ). 26). 27) is related to the generalized Hamiltonian system (Dubrovin et al. (1984)). 27). 23) is not unique and a choice of one or another set of canonical variables is a matter of convenience. 4 Field line behaviour The problem to be considered here is fairly old and has numerous applications.
5). 59) and the interval ∆x(0) = (0, 1). e. there are two possible paths. Evidently Tˆ−n ∆x(0) gives 2n possible paths.
There are different ways how one can introduce smoothed distributions in phase space. Consider finite dynamics in the phase space Γ and let Π be a partitioning of Γ by hypercubes of the volume 2N . We can introduce the number M (Π ) which is a minimal number of the hypercubes that cover full Γ. Let us label all hypercubes by k and p(k) , q(k) are 37 38 CHAOTIC DYNAMICS coordinates of the centre of the k-hypercube. 3) and the equality occurs as → 0. Consider N initial trajectories (particles) and make a snapshot at time instant t.