Problem 3.1 Axiom. The axion is a hypothetical particle which might constitute the dark matter of the Universe and may be key in resolving a subtle puzzle in the strong interactions. The axion has a classical mechanics description in terms of an object subject to a cosine potential, U(x) = Uo(l cos(2x)] . (a) Show that for small x the potential can be . . . Fi ure 3.16: Axion otential. approx1mated by a quadratic functlon, g p k U(x) 2 5x2 . Express k in terms of the axion potential parameters U0 and Q. (h) Determine the solution of the equation of motion if the object is initially at rest at some small x0 r:- 0. Problem 3.2 Composite springs Consider systems with two springs of force constants In and k2, respectively. The task is to determine the force constant of the system if (a) the springs are arranged serially (cf. Fig- ure 3.]? (a)), and if m (b) springs are Parallel: and we force the hor' (a) Serial springs. (b) Parallel springs. izontal bar in Figure 3.17 (b) to remain horizontal. Figure 3.17: Composite springs. Problem 3.3 Resonance The amplitude of a particular solution of a damped driven oscillation is given by A Amax A = SO V(wg - w2)2 + 482 W2 Amax V2 (a) Determine the location, Wmax, and the value, Amax, of the maximum of A. (b) Determine the width of the curve, i.e. the distance of the two points at which W - Wmax W+ A equals Amax/ V/2 (such that A? = Amax/2). Find the leading order result Figure 3.18: Amplitude and resonance. of an expansion in B/wo. Problem 3.4 Fourier integrals Consider functions on the interval [-7/2, 7/2] with some real 7 > 0. The scalar product of two such functions is given by T /2 (f 18) := 2 / dif (1)8(1) . -T/2 Consider the functions Sn (t) := sin(nwt) , Cm(t) := cos(mwt) , where m E No, n E N and w := 2x/T. Show that: T/2 (a) ( snIsm ) = at sin(not) sin(mwt) = 0mn , -T/2 7/2 (b) (calcm) = dt cos(nwt) cos(mwt) = (1 + 8mo)omn , -T/2 T /2 (c) (sn|cm) = IN at sin (not) cos(mut) = 0. -T/2 Recall that F if m = n , omn otherwise