By Rudolf Dvorak
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38) @S r @ψ where we have identified the known part of the second-order terms proportional to ε r by F r (I, ψ). The system has to be solved recursively for S j with j D 1, . . , r to obtain the k j . 40) k2Z n which can be written as the sum F j (I, ψ) D FN j (I ) C FQ j (I, ψ) where FN j is called the mean and FQ j is called the oscillatory part of F j . 4 Perturbation Theory In terms of the coefficients of the Fourier harmonics the solution of the Eq. 43) ω k where k ω D ω 1 k1 C C ω n k n .
N. 1 Action-angle variables in the hyperplane (p i , q i ). (a) action integral for the librational motion. (b) action integral for the rotational motion. quencies (respective periods) to be the same or rationally dependent. Note, that as consequence of the multidimensionality the orbit itself does not need to be closed even if p j (q j ) is periodic and/or (p j , q j ) are closed. In the following discussion we require systems which are separable in all their arguments q i , t with i D 1, . .
The quantity N, which is normalized between 0 (black) and 1 (white), gives a measure of the so-called Poincaré recurrence time. Arnold tongues are indicated in black and originate from specific Ω for K D 0. 6 The Dissipative Standard Map The dissipative standard map is of the form y nC1 D b y n C c C K f (x n ) x nC1 D x n C y nC1 where f is a periodic function, y 2 R and x 2 T with b, c, K are constants. The quantities b, c 2 RC are called the dissipative and drift parameters, respectively. For b D 1 and c D 0 one recovers with f (x) D sin(x) the classical conservative standard map.
Celestial Dynamics by Rudolf Dvorak