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Tham khảo tài liệu 'classical mechanics - 3rd ed. - goldstein, poole & safk episode 1 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 3.6 Conditions for Closed Orbits Bertrand s Theorem 91 of the force law r about the cừcular orbit radius . Direct substitution into the force law gives 2 - 7 f dr 3.46 r r i As the radius vector of the particle sweeps completely around the plane u goes through ỊÌ cycles of its oscillation cf. Fig. 3.13 . If fỉ is a rational number the ratio of two integers p q then after q revolutions of the radius vector the orbit would begin to retrace itself so that the orbit is closed. At each such that the inequality in Eq. 3.43 is satisfied it is possible to establish a stable circular orbit by giving the particle an initial energy and angular momentum prescribed by Eqs. 3.40 and 3.41 . The question naturally aiiscs as to what form the force law must take in order that the slightly perturbed orbit about any of these circular orbits should be closed. It is clear that under these conditions fj must not only be a rational number it must also be the same rational number at all distances that a circular orbit is possible. Otherwise since can take on only discrete values the number of oscillatory periods would change discontinuously with 0 and indeed the orbits could not be closed at the discontinuity. With everywhere constant the defining equation for fi2. Eq. 3.46 . becomes in effect a differential equation for the force law f in terms of the independent variable . We can indeed consider Eq. 3.46 to be written in terms of r if we keep in mind that the equation is valid only over the ranges in r for which stable circular orbits are possible. A slight rearrangement of Eq. 3.46 leads to the equation ưin d In r 3.47 FIGURE 3.13 Orbit for motion in a central force deviating slightly from a circular orbit for f 5. 92 Chapter 3 The Central Force Problem which can be immediately integrated to give a force law 3.48 All force laws of this form with f a rational number lead to closed stable orbits for initial conditions that differ only slightly from conditions defining a circular orbit. .
