Download PDF by Badger R.M.: Absolute Intensities in the Hydrogen-Chloride Rotation

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Extra resources for Absolute Intensities in the Hydrogen-Chloride Rotation Spectrum

Example text

5) It now becomes evident that the change of variables to positions and momenta has uncovered a symmetry in their function (or, rather, an antisymmetry, since we also have a change of sign). We will see shortly that this antisymmetry, together with the smoothness of the solutions with respect to perturbation of the initial data, has important ramifications for the flow map of the system. When discussing Hamiltonian systems, it is often convenient to use the notation z := (q, p)T , with q, p ∈ Rd , z ∈ R2d , and to introduce the 2d × 2d canonical structure matrix J, J := 0 −Id +Id .

RUNGE–KUTTA d METHODS FOR dt z = f (z) s z n+1 n = z + ∆t bi f (Zi ), i=1 where, for i = 1, 2, . . , s, s Zi = z n + ∆t aij f (Zj ). j=1 The number of stages s and the constant coefficients {bi }, {aij } completely characterize a Runge–Kutta method. In general, such a method is implicit and leads to a nonlinear system in the s internal stage variables Zi . In some cases the formulas are layered in such a way that the first stage variable Z1 is given explicitly in terms of z n , then Z2 is determined from z n and Z1 , etc.

N is determined by the powers of the The growth of error through the powers [R] eigenvalues of that matrix. A numerical method is asymptotically stable if the growth of the solution for a linear model problem is asymptotically bounded. A sufficient condition for asymptotic stability is that the eigenvalues of the method are (i) in the unit disk in the complex plane, and (ii) simple (not repeated) if on the unit circle. Since the eigenvalues of Euler’s method are both of modulus greater than one, their powers grow exponentially fast and the method is unstable.

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Absolute Intensities in the Hydrogen-Chloride Rotation Spectrum by Badger R.M.


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