On the stroboscopic method of second order

Abstract

It is assumed that the dynamical system can be represented by equations of the form

$$\begin{gathered} \dot x = \varepsilon _i f_i (x,y) \hfill \\ \dot y = u(x,y) + \varepsilon _i g_i (x,y) \hfill \\ \end{gathered} $$

as this is the case for the Lagrange equations in celestial mechanics. The perturbation functionsf _i andg _i may also depend on the timet. The fast angular variabley is now taken as independent variable. Using perturbation theory and expanding in Taylor series the differential equations for the zeroth, first, second, ... order approximations are obtained.

In the stroboscopic method in particular the integration is performed analytically over one revolution, say from perigee to perigee. By the rectification step applied tox andt, the initial values for the next revolution are obtained. It is shown how the second order terms can be determined for the various perturbations occurring in satellite theory. The solution constructed in this way remains valid for thousands of revolutions. An important feature of the method is the small amount of computing time needed compared with numerical integration.