By Rosu H.C.
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2 2 The corresponding equation of motion is 58 (23) ·· m x +kx = F (t) , or ·· x +w2x = F (t)/m , (24) where w is the frequency of the proper oscillations. The general solution of this equation is the sum of the solution of the homogeneous equation and a particular solution of the nonhomogeneous equation x = xh + xp . We shall study the case in which the external force is periodic in time of frequency γ forma F (t) = f cos(γt + β) . The particular solution of (24) is sought in the form x1 = b cos(γt + β) and by substituting it one finds that the relationship b = f /m(w2 − γ 2) should be fulfilled.
I3 I1 sin2 θ (88) Now, since the system is conservative, another integral of motion is the energy 1 1 E = T + V = I1(φ˙ 2 sin2 θ + θ˙2 ) + I3 (φ˙ cos θ + ϕ) ˙ 2 + M gl cos θ. 2 2 The quantity I3 ω3 = pϕ is an integral of motion. Multiplying this constant by pϕ ω3 we get I3 pϕ ω32 = p2ϕ ω3 I32ω33 = p2ϕ ω3 1 I3 ω32 = 2 50 1 p2ϕ . 2 I3 The quantity 12 I3 ω32 is a constant. Therefore, we can define the quantity 1 E´ = E − I3 ω32 = const. 2 1 ˙2 1 ˙ 2 2 I1θ + I1 φ sin θ + M gl cos θ , = 2 2 wherefrom we can identify 1 V (θ) = φ˙ 2 sin2 θ + M gl cos θ 2 pφ − pϕ cos θ 1 V (θ) = I1 2 I1 sin2 θ 2 sin2 θ + M gl cos θ.
Goldstein, More on the prehistory of the Laplace-Runge-Lenz vector, Am. J. Phys. 44, 1123 (1976) 31 3. THE RIGID BODY Forward: Due to its particular features, the study of the motion of the rigid body has generated several interesting mathematical techniques and methods. In this chapter, we briefly present the basic rigid body concepts. 1 Definition A rigid body (RB) is defined as a system of particles whose relative distances are forced to stay constant during the motion. 2 Degrees of freedom In order to describe the general motion of a RB in the three-dimensional space one needs six variables, for example the three coordinates of the center of mass measured with respect to an inertial frame and three angles for labeling the orientation of the body in space (or of a fixed system within the body with the origin in the center of mass).