By V. I. Arnold (auth.)
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1• • Figure 38 Forces of interaction Such forces are called forces of interaction (example: the force of universal gravitation). If all forces acting on a point of the system are forces of interaction, then the system is said to be closed. By definition, the force acting on the i-th point of a closed system is n Fi = 44 L Fij' j= 1 j*i 10: Motions of a system of n points The vector Fij is the force with which the j-th point acts on the i-th. ji is the magnitude of the force and eij is the unit vector in the direction from the i-th point to the j-th point.
Hint. Cf. Section 2D below. PROBLEM. We now look at the case r max = 00. If lim r .... oo U(r) = lim r .... lfthe initial energy E is larger than V, then the point goes to infinity with finite velocity r 00 = J2(E - V). We notice that if U(r) approaches its limit slower than r- 2 , then the effective potential V will be attracting at infinity (here we assume that the potential V is attracting at infinity). If, as r -+ 0, IU(r) I does not grow faster than M 2 /2r 2 , then rmin -+ and the orbit never approaches the center.
We look at a eylinder with base 2A l and a band ofwidth 2A 2 • We draw on the band a sine wave with period 2nAdw and amplitude A 2 and wind the band onto the eylinder (Figure 19). The orthogonal projeetion oft he sinusoid XJ Figure 19 Construction of a Lissajous figure 25 2: Investigation of the equations of motion wound around thc. eylinder onto the Xi> X2 plane gives the desired orbit, ealled a Lissajous figure. Lissajous figures ean eonveniently be seen on an oseilloseope whieh displays independent harmonie oseillations on the horizontal and vertieal axes.