By Andre Mercier
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For a more detailed analysis we have to use quantitative, for instance numerical, methods which are not discussed here. 5) with f(x) Lipschitz-continuous in lR. We assume that a. F(x) = J~ f(s)ds is an odd function. b. F(x) ----+ +00 as x ----+ 00 and there exists a constant (3 > 0 such that for x > (3, F(x) > 0 and monotonically increasing. c. There exists a constant a > 0 such that for 0 < x < a, F(x) < O. 11 In the figure the typical behaviour of F(x) has been sketched where the number of oscillations of F(x) between -(3 and (3 has been chosen at random.
1. 0 It is now possible to prove an important result for minimal sets in IR2. 4. 4:i; = f(x) in ]R2, then M is a critical point or a periodic orbit. Proof. The set M is not empty, bounded and invariant and so contains at least one orbit f. The limitsets a(r) and w(r) are also contained in M. If M contains a critical point, the minimal character of M means that M has to be identified with this critical point. If M contains no critical points, , and w(r) consist of ordinary points. Because of the minimal character of M we have that, C w(r)' Choose a point p E , and a transversallo of, in p.
A 24 2 Autonomous equations The small conductor will be put into motion in the x-direction but it remains parallel to the long conductor; without deformation of the spring, its position is x = o. The fixed position of the long conductor is given by x = a. The equation of motion of the small conductor is .. k x - 2Iil mx+ -= 0 a-x with k positive and x < a. a. Show that, putting A = 2Iil/k, the equation can be written as Ii _ k x 2 m ax + A = 0 a-x - b. Put the equation in the frame-work of Hamiltonian systems.