Study_of_the_Apsidal_Precession_of_the_Physical_Sy

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/259312490 Study of the Apsidal Precession of the Physical Symmetrical Pendulum Article in Journal of Applied Mechanics · December 2013 DOI: 10.1115/1.4029470 · Source: arXiv CITATIONS 0 READS 43 3 authors , including: Some of the authors of this publication are also working on these related projects: Theoretical constraints and gauge theories View project Rodolfo A. Diaz National University of Colombia 65 PUBLICATIONS 398 CITATIONS SEE PROFILE All content following this page was uploaded by Rodolfo A. Diaz on 20 October 2014. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. arXiv:1312.4019v1 [physics.class-ph] 14 Dec 2013Study of the apsidal precession of the Physical Symmetrical Pendulum. H´ ector R. Maya1,2∗, Rodolfo A. Diaz2†, William J. Herrera2‡ 1Universidad de C´ ordoba, Departamento de F´ ısica. Monter´ ıa, Colombia. 2Universidad Nacional de Colombia, Departamento de F´ ısica. Bogot´ a, Colombia. December 17, 2013 Abstract We study the apsidal precession of a Physical Symmetrical Pe ndulum (Allais’ precession) as a generalization of the precession corresponding tothe Ideal Spherical Pendulum( Airy’s Precession). Based on the Hamilton-Jacobi formalis m and using the technics of variation of parameters along with the averaging method, we obtain approximate solutions, in terms of which the motion of both systems admits a simple ge ometrical description. The method developed in this paper is considerably simpler than the standard one in terms of elliptical functions and the numerical agreement with the exact solutions is excellent. In addition, the present p rocedure permits to show clearly the origin of the Airy’s and Allais’ precession, as well as the effect of the spin of the Physical Pendulum on the Allais’ precession. Further, the method can be extended to the study of the asymmetrical pe ndulum in which an exact solution is not possible anymore. PACS: 45.40.-f, 83.10.Ff, 45.20.Jj, 47.10.Df Keywords : Ideal spherical pendulum, Physical symmetrical pendulum , Hamilton-Jacobi formalism, Averaging method, Apsidal precession. 1 Introduction A physical symmetrical pendulum is a particular case of the symmetr ical top in which the center of mass (CM) is located below the fixed point, and the precession and spin can be considered as small perturbations with respect to the nutation∗. Lagrange [1]; Poisson [2]; Golubev [3] and Leimanis [4], found exact solut ions (in terms of elliptical functions) for the dynamics and kinematics of the symmetrical top under the action of gravity. On the other hand, Johansen and Kane [5] obtained an approximate solution for the Ideal Spherical Pendu lum (ISP) using the method of averaging by using canonical variables. In addition, Miles [6] analyzed the response of t he ISP under a harmonic excitation, while Hemp and Sethana [7] studied the dynamics of the ISP when the support unde rgoes vertical motion. More closely related with this paper are the articles of Airy [8], Olsson [9 ] and Synge [10]. By using different methods ofapproximation,these authorsfound the angularfrecuencyof precessionthat undergoesthe apsidalaxisofthe projection of the ISP trajectory (the so-called Airy’s precession). In these studies it is assumed that the motion starts with small initial amplitudes. More recently, Gusev, Rudenko and Vinogradov [ 11], considered this precession when the pendulum is submitted to small perturbations coming from the anisotropy of t he support and they found an analytical formula for the angular frequency of the plane of oscillation as a function of the initial conditions and the anisotropy of the support. As for the Physical Symmetrical Pendulum (PSP), the most usual a nalysis of its dynamics is greatly simplified due to the assumption of spin dominance, in which the nutation and prece ssion are considered small perturbations [12]. By contrast, we study the symmetrical top (i.e. the PSP) considering it as a pendulum with a fixed point, in a regime of nutation dominance. We shall assume that the pendulum is released n ear to the surface of the earth with small initial ∗[email protected] †[email protected] ‡[email protected] ∗In the most usual scenario of the symmetrical top, the spin is the dominant part of its motion. 1 2 H´ ector R. Maya, Rodolfo A. Diaz, W. J. Herrera amplitudes and a small transverse initial velocity, but without initial s pin (with respect to the earth). On the other hand, in an inertial reference frame the PSP has a small correction to the initial precession and spin (owing to the rotation of the earth), but they are very small (of the order of 10−4rad/seg) and are kept small at all times, from which the nutation becomes the dominant motion. In this regime of initial conditions the P SP describes trajectories that are approximately elliptical and that precess very slowly. In the case of the ISP this eff ect is called “Airy’s precession” while in the case of a physical pendulum (which is our case) we shall call it “Allais’ prece ssion” [13]. The results of this study represent a first approximation to the dynamics of the paraconical pendulum, originally designed by Allais, and currently used widely by many researchers in the characterization of gravitationa l anomalies during eclipses [14]. As we shall see, the spin introduces a significant correction to the precession of the PS P (with respect to the precession of the ISP). Our paper is distributed as follows: In section 2, we study the Ideal Spherical Pendulum, using the Hamilton-Jacobi approach combined with the technics of variation of parameters an d the averaging method. These approximate results are numerically compared with the exact solution showing an excellent agreement between them even for long times. In this approach, the Airy’s precession appears naturally. The met hods developed in this section are applied to the symmetrical physical pendulum in section 3, and once again an excelle nt agreement with the exact solution even for large times is apparent. The corresponding Allais’ precession appears cle arly from the formalism, as well as the correction to this precession coming from the spin. Section 4 shows our conclusion s and appendix 6 shows some few technical details. 2 Apsidal precession in the ideal spherical pendulum We now study the origin of the Apsidal precession of the ideal spher ical pendulum, keeping in mind that our aim is just to establish the general framework to apply a similar procedure for the physical symmetrical pendulum. We assume that the (small) initial amplitudes are of the order of 0 .1 rad (experimental conditions). Let us consider a system of axes XYZfixed on an inertial reference frame and the spherical pendulum of massmand lengthl,as displayed in Fig. 1 Zmg0 YX ϕTθ lx y Figure 1: Inertial system XYZ and the spherical pendulum. The angular coordinates θ andϕ are shown, as well as the weight and tension . The natural coordinates that exhibit the symmetries of the IPS ar e the spherical coordinates θandϕ. The Lagrangian in such coordinates becomes L=1 2ml2/parenleftig ˙θ2+ ˙ϕ2sin2θ/parenrightig +mglcosθ. (1) The associated canonically conjugate momenta are given by pθ=∂L ∂˙θ=ml2˙θ , pϕ=ml2˙ϕsin2θ, (2) sinceϕis cyclic, its conjugate momentum pϕis constant and can be identified as the z−component of the angular Apsidal precession of the Physical Pendulum 3 momentum. The Hamiltonian of the system reads h=˙θpθ+ ˙ϕ pϕ−L, (3a) h=1 2ml2/parenleftigg p2 θ+p2 ϕ sin2θ/parenrightigg −mglcosθ, (3b) or in dimensionless units H=1 2/parenleftigg P2 θ+P2 ϕ sin2θ/parenrightigg −cosθ, (4) where we have introduced the following definitions H≡h mgl, ω0≡/radicalbiggg l, τ≡ω0t , Pθ≡pθ ml2ω0, Pϕ≡pϕ ml2ω0. (5) Now we introduce a canonical transformation: ( θ,ϕ,Pθ,Pϕ)→/parenleftbig¯θ,ϕ,¯Pθ,Pϕ/parenrightbig which keeps unaltered the variables associated with the precession (so that the constant of motion is s till apparent), and permits to eliminate the circular functions from this Hamiltonian. An appropriate generating functio n of type II [12] for this Canonical Transformation (CT) reads F2(θ,ϕ,¯Pθ,Pϕ) =¯Pθsinθ+¯Pϕϕ. (6) The formulas of transformation for the variables that describe th e nutation are given by ¯θ=∂F2 ∂¯Pθ= sinθ , Pθ=∂F2 ∂θ=¯Pθcosθ, (7) while the new variables that describe the precession are identical to the old ones. Therefore, we continue using the same symbolsϕandPϕfor them. Using this result in Eq. (4) we obtain the new Hamiltonian ¯H=1 2/bracketleftigg ¯P2 θ(1−¯θ2)+P2 ϕ ¯θ2/bracketrightigg −/radicalbig 1−¯θ2. (8) Now, since we are interested in a regime of small initial amplitudes (i.e. s mall values of θ0) and since ¯θ≡sinθ, we also have small values of the coordinate ¯θ. Consequently, we can expand the radical in power series and keep terms up to fourth order in ¯θ. We do this because this is the lowest order in which the apsidal prece ssion appears. By doing such an expansion and omitting constant terms in the Hamiltonian we obtain ¯H=¯P2 θ 2+¯θ2 2+P2 ϕ 2¯θ2−¯P2 θ¯θ2 2+¯θ4 8, (9) We identify the terms up to second order in the momenta and/or var iables with the non-perturbed Hamiltonian (let us recall that all momenta and variables are dimensionless), while higher order terms are identified with the Hamiltonian of perturbation H0=1 2¯P2 θ+¯θ2 2+P2 ϕ 2¯θ2, (10a) H1=−¯P2 θ¯θ2 2+¯θ4 8. (10b) We shall use these Hamiltonians to determine the approximate solutio ns for the ISP. To do it, we shall apply the Hamilton-Jacobi formalism along with the technics of variation of par ameters as well as the method of averaging. 2.1 Restricted Hamilton-Jacobi Method for the non-perturb ed Hamiltonian We start by finding the exact solution for the Hamiltonian H0by means of the implementation of the restricted Hamilton- Jacobi (RHJ) method, that is for the Hamilton’s Characteristic Fun ction [12]. Equation (10a) shows that ϕis cyclic in H0, hence we can write the Hamilton’s Characterisitc Function Win the form W=W¯θ+α2ϕ, (11) 4 H´ ector R. Maya, Rodolfo A. Diaz, W. J. Herrera wherePϕ≡α2is the canonical conjugate momentum associated with ϕ,and the RHJ equation for H0becomes 1 2/parenleftbigg∂W ∂¯θ/parenrightbigg2 +¯θ2 2+α2 2 2¯θ2=α1, (12) whereα1correspondsto the numerical value of H0i.e. the mechanical energy of the system described by this Hamiltonia n α1=1 2¯P2 θ0+¯θ2 0 2+α2 2 2¯θ2 0(13) Equation (12) is an ordinary differential equation for W¯θthat can be solved directly dW¯θ d¯θ=±/radicalbig −α2 2+2α1¯θ2−¯θ4 ¯θ, (14) and the function Wgiven by (11) becomes W=α2ϕ±/integraldisplay/radicalbig −α2 2+2α1¯θ2−¯θ4 ¯θd¯θ. (15) The new angular variables are obtained from τ+β1=∂W ∂α1=±/integraldisplay¯θ/radicalbig −α2 2+2α1¯θ2−¯θ4d¯θ=∓1 2arcsin/parenleftbiggα1−¯θ2 k/parenrightbigg , (16) k2≡α2 1−α2 2 (17) the initial conditions that we shall consider [Equations (32a) with θ0<<1 Rad], combined with Eq. (13) lead to α2 1>α2 2, and 0<θ <π/ 2. The first condition says that k2is a positive constant while the second condition leads us to preserve only the upper sign in Eqs. (16). Further, the sign of the integral a ssociated with the equation for ∂W/∂α 2is chosen accordingly β2=∂W ∂α2=ϕ−/integraldisplayα2 ¯θ/radicalbig −α2+2α1¯θ2−¯θ4d¯θ, (18) in order to evaluate this integral we use Eq. (16) to define the varia ble u=τ+β1=−1 2arcsin/parenleftbiggα1−¯θ2 k/parenrightbigg (19) substituting (19) in (18) we can write ϕas ϕ=β2+/integraldisplayα2 α1+ksin(2u)du=β2+arctan/bracketleftbigg1 α2(k+α1tanu)/bracketrightbigg (20a) ϕ=β2+arctan/braceleftbigg1 α2[k+α1tan(τ+β1)]/bracerightbigg (20b) From Eqs. (16), (14) and (20) we have ¯θ=/radicalbig α1+ksin[2(β1+τ)], (21a) ¯Pθ=dW¯θ d¯θ=kcos[2(β1+τ)] ¯θ, (21b) ϕ=β2+arctan/braceleftbigg1 α2[k+α1tan(β1+τ)]/bracerightbigg , (21c) these equations along with Pϕ=α2,form the solution for the non-perturbed Hamiltonian H0. Apsidal precession of the Physical Pendulum 5 2.2 Method of averaging using canonical variables for the co mplete Hamiltonian The next step is to obtain an approximate solution for the complete H amiltonian ¯Hof Eq. (9), based on the exact solution (21) for the non-perturbed Hamiltonian H0. To do this, we shall use the method of averaging using canonical variables. Our approach is a variation of the approximation propose d by K. F. Johansen and T. R. Kane [5]. The goal of such an approach is to obtain a set of approximate equations of eas y solution for the canonical variables, by applying the averaging method in the version proposed by Krylov-Bogoliubov-Mit ropolsky [17]. As we shall see, this technics is based on the variation of parameters and “the fast integration”. In the framework of the Hamilton-Jacobi equation for Hamilton’s Pr incipal Function, it is clear that the transforma- tions (21) are associated with a generating function of type II give n by S0=S0(¯θ,ϕ,α 1,α2,τ) =W(¯θ,ϕ,α 1,α2)−α1τ (22) such that Pj=∂S0 ∂qj, βj=∂S0 ∂αj. (23) Note that at this step we are using the Hamilton-Jacobi formalism fo r the Hamilton’s principal function S. This is a natural choice since at this moment what we pretend is to see the wa y in which H1modifies the exact solution (21) of H0. We can do this by demanding that the associated CT reduces the to tal Hamiltonian H0+H1to the Hamiltonian H1. Of course, it means that the new Hamiltonian must be numerically differ ent from the old one, and it is possible only if the generating function depends on time explicitly as is the case of S0.We can then propose a generating function Sfor a canonical transformation to a new set of canonical variables Q1,Q2,P1andP2if we replace in S0the constants α1and α2byP1andP2respectively, that is S=S0/parenleftbig¯θ,ϕ,P 1,P2,τ/parenrightbig , (24) where nowPiandQihave become variables (this is the method of variation of parameter s). The transformations induced by (24) are obviously (21) but with the replacements αi→Pi, βi→Qi ¯θ=/radicalbig P1+ksin[2(Q1+τ)], (25a) ¯Pθ=kcos[2(Q1+τ)]/radicalbig P1+ksin[2(Q1+τ)], Pϕ=P2 (25b) ϕ=Q2+arctan/bracketleftbigg1 P2(k+P1tan(Q1+τ))/bracketrightbigg , (25c) where k=/radicalig P2 1−P2 2. (26) The new Hamiltonian Kas a function of Q1, Q2, P1andP2is given by K=H0+H1+∂S ∂t=H1(Q,P), (27) where we have used H0=P1and∂S/∂t=−P1.Expressing H1[Eq. (10b)] in terms of these new variables, yields K=−¯P2 θ¯θ2 2+¯θ4 8(28a) K=1 8{P1+ksin[2(Q1+τ)]}2−1 2k2cos2[2(Q1+τ)], (28b) Consequently, we can obtain an exact solution for the Hamiltonian (9 ), by solving the Hamilton equations for Q1,Q2,P1 andP2as a function of τand substituting them in Eq. (25). Nevertheless, we recall that it is not our goal. Rather, we shall obtain a set of approximate equations of motion with easy solut ion in terms of elementary functions. For this, we first consider in Eq. (25), that QiandPiare functions that vary very slowly within a period πof theθcoordinate†(as it is customary in the quasi-harmonic approximation). Hence, an app roximate solution for the canonical variables of the †This is a reasonable ansatz since those variables are consta nt when we use the non-perturbed Hamiltonian H0. The time evolution of these variables arises from the introduction of the (much smaller ) Hamiltonian H1. 6 H´ ector R. Maya, Rodolfo A. Diaz, W. J. Herrera complete Hamiltonian (9) is still given by (25) but with QiandPirepresenting solutions of the Hamilton equations with Kreplaced by its average over a period T, this approximation yields /an}bracketle{tK/an}bracketri}ht=1 T/integraldisplayT 0K(Q,P,t)dt=1 16/parenleftbig −3k2+2P2 1/parenrightbig , (29a) /an}bracketle{tK/an}bracketri}ht=1 16/parenleftbig 3P2 2−P2 1/parenrightbig (29b) where we have neglected the variations of all QiandPiwithin a period, and kis given by (26). Note that the new coor- dinates are all cyclic in this averaged Hamiltonian. Consequently, und er this approximation all new canonical momenta are kept constant. The equations of motion for these new coordin ates become ˙P1=−∂/an}bracketle{tK/an}bracketri}ht ∂Q1= 0,˙P2=−∂/an}bracketle{tK/an}bracketri}ht ∂Q2= 0 (30a) ˙Q1=∂/an}bracketle{tK/an}bracketri}ht ∂P1=−P1 8,˙Q2=∂/an}bracketle{tK/an}bracketri}ht ∂P2=3P2 8. (30b) The integration of these equations is straightforward P1=α1=H0, P2=α2=Pϕ, (31a) Q1=−P1 8τ+Q10, Q2=3P2 8τ+Q20, (31b) the constants Q10andQ20are obtained by using the initial conditions and evaluating (25) and (3 1b) atτ= 0. 2.3 Approximate solution for the ideal spherical pendulum Let us find an approximate solution for the ideal spherical pendulum associated with the elliptical mode characterized by initial conditions in which the pendulum is released with a small initial am plitudeθ0and a small initial precession but without initial nutation (initial conditions with respect to the ear th). As we have discussed, owing to the rotation of the earth the initial precession with respect to an inertial referen ce frame is slightly different. Therefore, in an inertial reference frame the initial conditions become θ(0) =θ0, ϕ(0) = 0,˙θ(0) = 0,˙ϕ(0) = ˙ϕ0, (32a) we also take l= 1m. According with Eq. (7), the initial conditions (32a) are transform ed into ¯θ(0) = sinθ0,¯Pθ(0) =Pθ(0)secθ0 (33) By applying the initial conditions (32a) in Eqs. (2, 5) we see that Pθ(0) = 0. Moreover, by combining Eqs. (13, 17, 31a, 31b), we obtain the following expressions for the constants α2=˙ϕ0 ω0sin2θ0=˙ϕ0 ω0¯θ2 0, α1=¯θ2 0 2+α2 2 2¯θ2 0(34a) k=¯θ2 0 2−α2 2 2¯θ2 0(34b) Q10=π 4, Q20=−arctan/parenleftbigg¯θ2 0 α2/parenrightbigg , (34c) The new coordinates (31b) are Q1=−α1 8τ+π 4, Q2=3P2 8τ−arctan/parenleftbigg¯θ2 0 α2/parenrightbigg , (35) and the approximate solutions (25a) and (25c) yield ¯θ=/radicalbigg α1+kcos/bracketleftig 2/parenleftig 1−α1 8/parenrightig τ/bracketrightig , (36a) ϕ=Q2+arctan/bracketleftbigg1 α2(k+α1tan(Q1+τ))/bracketrightbigg , (36b) Apsidal precession of the Physical Pendulum 7 In Figure 2 we show the solutions obtained with the approximation (36 ) and the exact solutions for the initial conditions θ0= 0.1rad, ˙ϕ0= 1.00167 rad/s. In part Aof this figure we superpose the graphics of the polar angle ¯θand sinθ. Note that they cannot be distinguished. In part Bit is shown the exact (increasing) azimuthal angle and the approxima te one (monotonic piecewise). It is observed that both graphics coincide f or 0≤τ≤τπ/2whereτπ/2corresponds to the value of τfor which the phase of the function tan xin (36b) is equal to π/2. It is clear that the discontinuities in the derivative appear for values (2 n+1)π/2 of the argument where n= 1,2,... 302304306308310Τ0.040.050.060.070.080.090.10ΘA , 123456Τ /Minus2246/CurlΨPhiB Figure 2: (A) Graphical form of the approximate solution for ¯θwith 300 ≤τ≤310. The graphics of the exact solution cannot be distinguished from the approximate one. (B) Graph ical form of the (monotonic piecewise) approximate solutio n for the angle ϕ, and of the smooth approximate solution after the introduct ion of the γangle, for 0≤τ≤6. The approximate (smooth) and exact solutions are superposed. However, we can correct such a problem by introducing the angle γsuch that arctan/bracketleftbiggk+α1tan(Q1+τ) α2/bracketrightbigg = (Q1+τ)+arctanγ, (37) solving for γwe obtain γ=2kcos2(Q1+τ)+(α1−α2)sin(2(Q1+τ)) (α1+α2)−(α1−α2)cos(2(Q1+τ))+ksin(2(Q1+τ)). (38) Note that by means of the identity (37), the strictly increasing pha se (Q1+τ) has been extracted from the argument of the function tan x,and we have introduced a phase γthat oscillates and remains always finite as argument of the function arctanx. Substituting in (36b) we finally get ϕ=Q2+(Q1+τ)+arctanγ, (39) and the parameter Q20in Eq. (34c) can be rewritten in terms of the initial conditions as Q20=−π 4+arctan/parenleftbiggα2−¯θ2 0 ¯θ2 0+α2/parenrightbigg (40) Figure 3 shows the superposition of the graphics for the azimuthal angleϕ(curve) obtained from the exact solution and the approximation (39), for 500 ≤τ≤510. Like in the case of the polar angle we observe that the exact an d approximate solutions cannot be distinguished from each other. The interval to plot was chosen in order to exhibit the asymptotic behavior of the approximate solution. The period of motion is by definition twice the period of the coordinate ¯θ(36a), such that in this approximation the period (in dimensionless units) is given by T=4π 2/parenleftbig 1−α1 8/parenrightbig=16π 8−α1, (41) and for our current example it yields T= 2.00847s, whose deviation with respect to the exact period is one par t of 106 [16]. From Eq. (36b) it is followed that the approximate solution for ˙ ϕhas the same period Tofθ. The straight line shown in Fig. 3 corresponds to the linear approximation for the func tionϕ(τ) given by ˜ϕa(τ)≡ϕ0+ϕ(T)−ϕ(0) Tτ=1 8(8−α1+3α2)τ. (42) 8 H´ ector R. Maya, Rodolfo A. Diaz, W. J. Herrera 502 504 506 508 510Τ502504506508510/CurlΨPhi Figure 3: Graphical form of the approximate values of the angle ϕfor500≤τ≤510. The straight line corresponds to the linear approximation given by (42). The approximate and exact solutions are superposed. Finally, we obtain the average angular velocity of the apsidal preces sion. That is, the quotient between the angular excess over 2πand the period of the motion ωa=ϕ(T)−ϕ(0)−2π T=3 8α2, (43) and using Eq. (34a) we see that it coincides with the angular velocity in the Airy’s apsidal precession [8] ωa=3 8α2=3 8˙ϕ0 ω0sin2θ0. (44) 2.3.1 Geometrical interpretation of the solution We conclude the study of the dynamics of the ISP by expressing the solutions obtained so far in terms of the cartesian coordinates. As we shall see, it permits us to give a simple geometrica l interpretation to the projection of the motion on theXYplane, when we use the approximate solutions (36). In terms of the spherical coordinates, the dimensionless cartesian coordinates yield q1≡x l= sinθcosϕ , q2≡y l= sinθsinϕ, (45) and replacing the approximate solutions for θandϕ, we obtain an approximate solution for the dimensionless cartesian coordinates (see appendix)/bracketleftbigg q1 q2/bracketrightbigg =/bracketleftbigg cos(ωaτ)−sin(ωaτ) sin(ωaτ) cos(ωaτ)/bracketrightbigg/bracketleftbigg¯θ0cos(ωτ)/parenleftbig α2/¯θ0/parenrightbig sin(ωτ)/bracketrightbigg (46) Figure4showstheprojectionofthe trajectoryonthe XYplanefortheinitial conditions θ0= 0.1rad, ˙ϕ0= 1.00167Rad/s and 30nT≤t≤(30n+ 1)Twithn= 0,1,2; using the approximation (46) and the exact solution. Once again, the comparison shows that the superposition of both solutions makes t hem indistinguishable. It is easy to see that the parametric equations (46) describe an ellip se of semiaxes aandbwith angular frequency ω (in standard units), given by a=l¯θ0=lsinθ0, b=lα2 ¯θ0, (47) ω=ω0/parenleftig 1−α1 8/parenrightig , α1=¯θ2 0 2+α2 2 2¯θ2 0, (48) the matrix of rotation in Eq. (46) shows that such an ellipse precess es counterclockwise with angular frequency ωa=/parenleftbigg3 8α2/parenrightbigg ω0=3 8ab l2ω0, (49) Apsidal precession of the Physical Pendulum 9 /Minus0.10/Minus0.05 0.05 0.10X/LParen1m/RParen1 /Minus0.04/Minus0.020.020.04Y/LParen1m/RParen1 T30T60T Figure 4: Projection of the trajectory in the XY plane for30nT≤t≤(30n+1)Twithn= 0,1,2,using the approximate solution (46). The approximate and exact solutions cannot b e distinguished. the last expression is the so-called apsidal Airy’s precession [8]. As a matter of consistency, when ˙ ϕ0= 0,andωa= 0, the difference of phase α1/8 given by Eq. (48) coincides with the relative variation of first order of the frequency of a plane pendulum with finite amplitude θ0[12]. When ˙ ϕ0/ne}ationslash= 0, equation (34a) shows that α2/ne}ationslash= 0, so that this variation is corrected by α2 2/(16¯θ2 0) as an effect of the precession. Finally, we should emphasize that th e expression obtained by us for the apsidal Airy’s precession does not depend on the a priori as sumption that Airy makes with respect to the elliptical trajectories of the projection of the pendulum motion [8]. Such a fe ature is deduced in our framework in a natural manner as a consequence of the slow variation of the new coordinates. We p oint out that the method also provides the correction to the period up to second order in the coordinates. 3 Physical Symmetrical Pendulum We define a PhysicalSymmetrical Pendulum (PSP) as a rigid solid with an axis ofsymmetry that can rotate freely around a fixed point (support) located at one edge of the symmetry axis an d in which the center of mass (CM) is located below the fixed point. An example is a disc supported by a cylindrical rod hun g on an edge, as shown in Fig. 5. A comparison with the symmetrical top shows us that indeed we are dealing with the same system, and the only difference is the range in which nutation motion occurs: 0 ≤θ≤π/2 for the PSP, and π/2≤θ≤πfor the top‡. Y ψ Z_cm mθ gX Zϕ ϕ ΦX_ 0Y_ Figure 5: Physical Symetrical Pendulum hung on an edge. θ,Φandψare the Euler angles used in the description of the motion. The spherical coordinate ϕis shown as well as its relation with the Euler angle Φ. ‡We are using the convention of positive direction of the Z−axis in the direction of the gravitational field (i.e. downwa rds). 10 H´ ector R. Maya, Rodolfo A. Diaz, W. J. Herrera We use the Euler angles θ,Φ, ψas generalized coordinates. As shown in Fig. 5, these angles provide the orientation of the system of axes ¯X¯Y¯Zfixed to the pendulum, with respect to the inertial system of axes XYZ. We can note that there is a difference of phase of π/2 between the Euler angle Φ and the azimuthal angle ϕof the spherical coordinates, that is Φ =ϕ+π/2. (50) The components of the angular velocity ωin the basis of axes ¯X¯Y¯Zfixed to the pendulum, yield [12] ω¯x=˙Φsinθsinψ+˙θcosψ, ω¯y=˙Φsinθcosψ−˙θsinψ, ω¯z=˙Φcosθ+˙ψ (51) it is clear that in the system of axis ¯X¯Y¯Z, the position rcmof the CM, gives rcm=l¯k (52) wherelis the distance from the fixed point to the CM, while in the inertial syst em of axisXYZ, it becomes rcm=lsinθsinΦi−lsinθcosΦj+lcosθk. (53) Finally, the Lagrangian of the system is given by L=1 2/bracketleftbigg I¯x/parenleftig ˙θ2+˙Φ2sin2θ/parenrightig +I¯z/parenleftig ˙Φcosθ+˙ψ/parenrightig2/bracketrightbigg +mglcosθ, (54) whereI¯xandI¯zare the moments of inertia of the pendulum with respect to the axes ¯Xand¯Zfixed to the body. The constant canonical momenta associated with the cyclic coordinate s Φ andψare given by pΦ=∂L ∂˙Φ=I¯x˙Φsin2θ+I¯zcosθ/parenleftig ˙ψ+˙Φcosθ/parenrightig (55) pψ=∂L ∂˙ψ=I¯z/parenleftig ˙Φcosθ+˙ψ/parenrightig (56) 3.1 Hamiltonian of the PSP The Lagrangian (54) is a homogenous function of second degree, t he transformation from cartesian to generalized coordi- nates does not depend on time, and the potential does not depend on the generalized velocities. Thus, the Hamiltonian hbecomes the total energy of the system and can be expressed in c anonical variables as follows h=1 2/parenleftigg p2 θ I¯x+(pψcosθ−pΦ)2 I¯xsin2θ+p2 ψ I¯z/parenrightigg −mglcosθ, (57) or in dimensionless units H=1 2/parenleftigg P2 θ+(PΦ−Pψcosθ)2 sin2θ+αP2 ψ/parenrightigg −cosθ, (58) where we have used the definitions Ω0≡/radicalbigg mgl I¯x, τ≡Ω0t , H≡h mgl, Pψ,Φ≡pψ,Φ√mglI¯x, α≡I¯x I¯z(59) the parameter αaccounts on the shape of the pendulum. For a PSP like the one repre sented in Figure 5, α>1 and its ellipsoid of inertia is prolate. In this study we are interested in this cas e. Whenα<1 we would have an oblate ellipsoid of inertia while α= 1 corresponds to an spherical ellipsoid. Now we proceed in a way similar to the case of the ISP. That is, we intro duce a generating function similar to (6), that leaves the variables of precession and spin unaltered F2(θ,Φ,ψ) =¯Pθsinθ+¯PΦΦ+¯Pψψ, (60) Apsidal precession of the Physical Pendulum 11 which leads to the following transformation between canonical coor dinates ¯θ= sinθ , Pθ=¯Pθcosθ, (61) and the Hamiltonian becomes ¯H=1 2 ¯P2 θ/parenleftbig 1−¯θ2/parenrightbig +/parenleftig PΦ−Pψ/radicalbig 1−¯θ2/parenrightig2 ¯θ2+αP2 ψ −/radicalbig 1−¯θ2, (62) where we have preserved the symbols of the variables that describ e precession and spin. Expanding up to fourth order in ¯θand neglecting constant terms, the Hamiltonian becomes ¯H=1 2 ¯P2 θ/parenleftbig 1−¯θ2/parenrightbig +/bracketleftig PΦ−Pψ/parenleftig 1−¯θ2 2/parenrightig/bracketrightig2 2¯θ2+αP2 ψ +¯θ2 2+¯θ4 8, (63) for future purposes, it is convenient the following separation ¯H=¯H0+¯H1 (64) ¯H0≡¯P2 θ 2+¯θ2 2+(PΦ−Pψ)2 2¯θ2−¯P2 θ¯θ2 2+¯θ4 8, (65) ¯H1≡Pψ 2[PΦ+Pψ(α−1)]+P2 ψ 8¯θ2, (66) It is easy to realize that this separation is not consequent with the p erturbation theory, since in both ¯H0and¯H1there are terms of order two and four. Indeed this separation has the a im of matching ¯H0with the Hamiltonian of the ISP (9), instead of implementing a standard perturbation theory. To do that, we introduce a new generating function G2/parenleftbig Φ,ψ,¯PΦ,¯Pψ/parenrightbig = Φ(¯PΦ+¯Pψ)+ψ¯Pψ, (67) which leaves invariant the variables that describe the nutation and t ransforms the precession and spin variables as follows ¯Φ =∂G2 ∂¯PΦ= Φ, PΦ=∂G2 ∂Φ=¯PΦ+¯Pψ (68a) ¯ψ=∂G2 ∂¯Pψ=ψ+Φ, Pψ=∂G2 ∂ψ=¯Pψ. (68b) and the new Hamiltonian becomes ¯H0=¯P2 θ 2+¯θ2 2+¯P2 Φ 2¯θ2−¯P2 θ¯θ2 2+¯θ4 8, (69a) ¯H1=¯Pψ 2(¯PΦ+α¯Pψ)+¯P2 ψ 8¯θ2. (69b) Since (69a) coincides with (9), we can considerequations (25) asth e solution ofthe Hamiltonian ¯H0and then we construct an additional CT that permits to reduce the new Hamiltonian to ¯H1alone. This is the task of the next section. 3.2 Variation of Parameters According with the results of the previous section, we consider the approximate solution (25) as the solution of (69a), that is ¯θ=/radicalbig P1+ksin[2(Q1+τ)], (70a) ¯Pθ=kcos[2(Q1+τ)] ¯θ,¯PΦ=P2, (70b) ¯Φ =Q2+arctan/bracketleftbigg1 P2(k+P1tan(Q1+τ))/bracketrightbigg , (70c) 12 H´ ector R. Maya, Rodolfo A. Diaz, W. J. Herrera with Q1=−P1 8τ+Q10, Q2=3P2 8τ+Q20, (71a) P1=¯P2 θ0 2+¯θ2 0 2+¯P2 Φ0 2¯θ2 0, (71b) P2=¯PΦ=PΦ−Pψ, (71c) k2=P2 1−P2 2. (71d) This solution suggests to construct a last CT to the new variables Γ i,Πi,withi= 1,2,replacing the constants Qi0by Γi,andPiby Πi(variation of parameters), while leaving unaltered the variables tha t describe the spin degrees of freedom Q1=−Π1 8τ+Γ1, Q2=3Π2 8τ+Γ2, (72a) Q3= Γ3=¯ψ, P1= Π1, P2= Π2,Π3=¯Pψ (72b) Indeed, the CT given by (72) is a particular case of a more general c lass of canonical transformations ( Q, P)→(Γ,Π) that we call Linear Transformations (see appendix). There is a gen erating function of type II associated with these transformation S(Q1,Q2,Q3,Π1,Π2,Π3,τ), and the new Hamiltonian Kas a function of Γ iand Πiis obtained from K=¯H0+¯H1+∂S ∂τ=¯H1(Γ,Π), (73) where we have used our a priori assumption that (70) is the exact s olution for ¯H0,that is ¯H0+∂S ∂τ= 0. (74) such that ¯H1described by (69b) in terms of these new variables, is the new Hamilto nian K=¯H1(Γ,Π,) =1 2Π3(Π2+αΠ3)+Π2 3 8{Π1+ksin[2(ω1τ+Γ1)]}, (75) whereω1= 1−Π1/8.It is clear that Γ 1is still a slowly varying coordinate, such that we can neglect its variat ion within a period of time T= 2π/ω1. Carrying out a “fast integration” with respect to τwe see that the oscillating term vanishes and the averaged Hamiltonian becomes /an}bracketle{tK/an}bracketri}ht=1 2Π3(Π2+αΠ3)+1 8Π2 3Π1, (76) it is observed that the coordinates Γ are cyclic within this averaged H amiltonian so that the new momenta are constant (within our approximation) and equal to their initial values Π1=P1,Π2=P2,Π3=Pψ (77) while the equations of motion for the new coordinates are ˙Γ1=∂/an}bracketle{tK/an}bracketri}ht ∂Π1=1 8Π2 3,˙Γ2=∂/an}bracketle{tK/an}bracketri}ht ∂Π2=Π3 2, (78a) ˙Γ3=∂/an}bracketle{tK/an}bracketri}ht ∂Π3=Π2 2+αΠ3+1 4Π3Π1, (78b) integrating out we have Γ1=1 8P2 ψτ+Q10, (79a) Γ2=Pψ 2τ+Q20, (79b) Γ3=/parenleftbiggP2 2+αPψ+1 4PψP1/parenrightbigg τ+¯ψ0. (79c) Apsidal precession of the Physical Pendulum 13 By virtue of these results, the relations (70) are still approximate solutions for the variables ¯θ,¯Pθ,¯Φ of the system described by (63), where P1,P2,andkare the constants defined by (71b)-(71d), while the time evolution of the functions Qiin (72) yields Q1=1 8/parenleftbig P2 ψ−P1/parenrightbig τ+Q10, (80a) Q2=/parenleftbigg3P2 8+Pψ 2/parenrightbigg τ+Q20. (80b) As for the spin angle ψ, it can be obtained from (68b) ψ=¯ψ−Φ = Γ 3−Φ. (81) It is important to point out that despite the multiple canonical trans formations carried out to reduce the exact Hamiltonian (58) to the approximate Hamiltonian (76), the new coord inates (Γ 1,Γ2,Γ3) has an apparent physical description with respect to the original coordinates ( θ,Φ, ψ), that is: the coordinate Γ 1still describes the nutation, the coordinateΓ 2still describes the precessionand Γ 3describes the spin. As we shall seelater, this fact simplifies consider ably the geometricalanalysis ofthe solutions. Now, we shall obtain the a pproximatesolutions associatedwith the elliptic mode with this method, and then we compare them with the exact solutions . We are interested in the geometrical interpretation of these solutions and then verify the globality of them (that is we int end to check how close are the approximatesolutions with respect to the exact solutions for long intervals of time). 3.3 Approximate solution for the PSP The initial conditions that characterize the elliptical mode in an inertia l reference frame are given by θ(0) =θ0,Φ(0) =π/2, ψ(0) = 0, ˙θ(0) = 0,˙Φ(0) =˙Φ0,˙ψ(0) = 0, (82a) for these conditions, we have again that Pθ(0) = 0,and in terms of the new coordinates they transform according with (61) into ¯θ0= sinθ0,¯Pθ0=Pθ(0)secθ0. (83) The dimensionless constants of motion PΦandPψare determined from these conditions and equations (55, 56, 59) PΦ=pΦ√mglI¯x=˙Φ0√mglI¯x/bracketleftbig I¯x¯θ2 0+I¯z/parenleftbig 1−¯θ2 0/parenrightbig/bracketrightbig , (84a) Pψ=pψ√mglI¯x=I¯z√mglI¯x/parenleftbigg ˙Φ0/radicalig 1−¯θ2 0/parenrightbigg , (84b) and the constants P1, P2, Q10, Q20are obtained from (71) P1=P2 2 2¯θ2 0+¯θ2 0 2, P2=PΦ−Pψ, (85a) k=¯θ2 0 2−P2 2 2¯θ2 0. (85b) Q10=π 4, Q20=π 2−arctan/parenleftbiggP1+k P2/parenrightbigg , (86) The coordinate ¯θ, is given by (70a) ¯θ=/radicaligg P1+kcos/bracketleftbigg1 4/parenleftig 8−P1+P2 ψ/parenrightig τ/bracketrightbigg , (87) 14 H´ ector R. Maya, Rodolfo A. Diaz, W. J. Herrera while the azimuthal angle Φ given by (70c) presents the same discont inuities observed in (36b), it is worked out in a similar way of the previous case by introducing the auxiliary function γ=2kcos2(Q1+τ)+(P1−P2)sin[2(Q1+τ)] (P1+P2)−(P1−P2)cos[2(Q1+τ)]+ksin[2(Q1+τ)], (88) and the parameter Q20in (86) can be rewritten in terms of the initial conditions as Q20=π 4−arctan/parenleftbigg¯θ2 0−P2 ¯θ2 0+P2/parenrightbigg , (89) such that the approximate solution for the azimuthal angle become s Φ =Q2+(Q1+τ)+arctanγ. (90) And the spin angle ψcan be obtained from (81). The period of the system, that by definition is twice the period of the angle¯θ, is given by T=16π 8−P1+P2 ψ, (91) comparingwiththecorrespondingperiodassociatedwiththeISPEq . (41), wenotethattheangularmomentumassociated with the spin Pψof the PSP, introduces two corrections: (1)InP1by means of the definition (85a) of P2[compare with Eqs. (31a, 34a)], and (2)In the quadratic correction of the denominator of (91). From (70c) and (81) we infer that ˙Φ and˙ψare periodic functions with the same period Tgiven by (91). Therefore, we can obtain linear approximations for the angles Φ and ψby using the following definitions ˜Φ(τ)≡Φ0+/bracketleftigg 1 T/integraldisplayT 0˙Φ(s)ds/bracketrightigg τ= Φ0+/bracketleftbiggΦ(T)−Φ(0) T/bracketrightbigg τ, ˜Φ(τ) =π 2+1 8/parenleftbig 8−P1+3P2+4Pψ+P2 ψ/parenrightbig τ, (92) ˜ψ(τ)≡ψ0+/bracketleftigg 1 T/integraldisplayT 0˙ψ(s)ds/bracketrightigg τ=ψ0+/bracketleftbiggψ(T)−ψ(0) T/bracketrightbigg τ, ˜ψ(τ) =1 8/parenleftbig −8+P1+P2−4Pψ−P2 ψ+2P1Pψ+8αP2 ψ/parenrightbig τ. (93) The mean angular velocity of apsidal precession (that we call Allais’ p recession), can be obtained analogously to the procedure in (43) ΩAllais≡Φ(T)−Φ(0)−2π T(94) ΩAllais=3P2 8+Pψ 2=1 8(3PΦ+Pψ), (95) where we have used Eqs. (90, 91) and Eq. (85a). By comparing Eq. (95) with Eq. (43) we see clearly the contribution of the spin to the Allais’ precession, through the canonical moment umPψ[compare also the expressions (31a) and (85a) ofP2for the ISP and PSP respectively]. We also note that the correction introduced by the spin is of the same order of theZ−component of the angular momentum P2. Now, introducing the numerical values I¯x=I¯y= 1Kg·m2, I¯z= 10−2Kg·m2, l= 1m , m= 1Kg θ0= 0.1,Φ0=π/2, ψ0= 0,( rad) ˙θ0= 0,˙Φ0= 1,˙ψ0= 0,(rad/seg), (96) Apsidal precession of the Physical Pendulum 15 we obtain P1= 0.0054868, P2= 0.00316787, Pψ= 0.00317842 k= 0.00447991, Q10=π 4, Q20= 0.306786,¯ψ0=π/2. (97a) the period of motion reads T=16π 8−P1+P2 ψ1 Ω0= 2.00846 s, (98) which is in excellent agreement with the exact solution [16]. The apsidal Allais’ precession in this approximation gives ΩAllais= 0.00871254 rad/s. (99) In Fig. 6 we compare the exact solution for the nutation angle θwith the approximate one Eq. (87), for 500 ≤t≤504 s. It is observedthat both solutions aresuperposed and cannot b e distinguished, showing the excellent agreement between them 501 502 503 504t/LParen1s/RParen10.020.040.060.080.10Θ Figure 6: Plot of the approximate solution (87) for the nutation angle ¯θ(sinθ)asafunctionof time, for 500≤t≤504 s. The exact solution is superposed to the approximate one. In Fig. 7 we compare the exact and approximate solutions for the az imuthal angle Φ (90) and the spin angle ψ(81), for 500≤t≤504 s. Once again, the solutions are indistinguishable showing the exc ellent agreement between them. The straight lines shown in these figures are the linear approximations (9 2) and (93). On the other hand, the interval of time chosen shows the global character of the approximate solutions. 501502503504t/LParen1s/RParen1157215741576157815801582/CapPhi/LParen1rad/RParen1A , 500501502503504t/LParen1s/RParen1/Minus1076/Minus1074/Minus1072/Minus1070Ψ/LParen1rad/RParen1B Figure 7: (A) Approximate solutions for the azimuthal angle Φ,for500≤t≤504s, (B) Approximate solutions of the spin angleψ, for500≤t≤504s. The straight lines correspond to the linear approximation s for each one of these angles. The exact solutions cannot be distinguished from the approx imate ones. 16 H´ ector R. Maya, Rodolfo A. Diaz, W. J. Herrera /Minus0.05 0.05Y/LParen1m/RParen1 /Minus0.10/Minus0.050.050.10X/LParen1m/RParen1 T 30T 60T Figure 8: Projections of the trajectory of the CM in the XY−plane,obtained with the approximate solutions, for time intervals of a period T,(A) for0≤t≤T: (T).(B)For29T≤t≤30T: (30T).(C)for59T≤t≤60T: (60T).In all cases the exact solutions are superposed to the approximate ones. Finally, we compare in Fig. 8 the projections of the trajectory in the XY−plane obtained from the exact solution and from the approximate solution given by (see appendix): /bracketleftbiggq1 q2/bracketrightbigg =/bracketleftbiggcos(ΩAllaisτ)−sin(ΩAllaisτ) sin(ΩAllaisτ) cos(Ω Allaisτ)/bracketrightbigg/bracketleftbigg¯θ0cos(ωτ)/parenleftbig P2/¯θ0/parenrightbig sin(ωτ)/bracketrightbigg (100) where q1=x l=¯θsinΦ, q2=y l=−¯θcosΦ, ω=1 8/parenleftbig 8−P1+P2 ψ/parenrightbig . (101) once again the solutions are practically identical. The dotted straigh t lines show the evolution of the apsidal axis for the intervals of time plotted, which obey the linear equation y=xtan(Ω AllaisnT),withn= 1,30,60 (102) 4 Concluding remarks We have obtained approximate solutions for the equations of motion of the Ideal Spherical Pendulum (ISP) and the Physical Symmetrical Pendulum (PSP), by using a Hamilton-Jacobi a pproach along with the technics of variation of parameters and the averaging method. The success of the avera ging method is due to the presence of variables that vary very slowly within a period of the polar angular coordinate. In particular, we found approximate expressions for the apsidal p recession that undergoes an ISP and a PSP in their elliptical modes, when the pendula are initiated with small initial amplitud es and small angular momenta of precession with respect to an inertial reference frame. In addition, we obtain ed approximate analytical expressions that describe the projections of the trajectories on the horizontal plane for both types of pendula. The method implemented in this paper to obtain these expressions is o peratively and conceptually simpler than the standard procedure used in obtaining the exact expressions in ter ms of elliptical functions of Jacobi and Weierstrass. In particular, our method permits to show with clarity the origin of the p recession of the ISP and of the PSP. Moreover, in the case of the PSP, our approximate expressions introduce in a na tural way the correction due to the spin for its apsidal precession. Up to our knowledge, this is a new approximation for an e ffect already observed in the paraconical pendulum with symmetrical support. Apsidal precession of the Physical Pendulum 17 On the other hand, this method also outlines the way to obtain appro ximate solutions for the Physical Assymetrical Pendulum (paraconical pendulum), which is still an open problem. It is because canonical transformations to slowly varying coordinates are also possible for the Asymmetrical Pendulu m, and in terms of such kind of coordinates the new averaged Hamiltonian leads to trivial (though approximate) equatio ns of motion. The examples presented here show an excellent numerical agreeme nt of our approximations with respect to the exact solutions. Further, such an agreementis preservedeven for long periods of time, so that they are globallyvalid. Finally, as a consequence of the geometrical interpretation of the approxim ate trajectories, we have associated the period of elliptical motionTwith the period of oscillations of the coordinate ¯θ, this association permits to propose Tas the period of motion, which coincides with the exact period up to the order of microsecond s. 5 Acknowledgments We acknowledge to Divisi´ on de Investigaci´ on de Bogot´ a (DIB) of U niversidad Nacional de Colombia, for its financial support. 6 Appendix We present in this appendix the deduction of Eqs. (46) and (100), w hich expresses the geometrical interpretation that we have done for the approximate solutions obtained in this paper. Fur ther, we prove the canonical character of the Linear Transformation introduced in (72). 1. In order to deduce Eq. (46) /bracketleftbigg q1 q2/bracketrightbigg =/bracketleftbigg cos(ωaτ)−sin(ωaτ) sin(ωaτ) cos(ωaτ)/bracketrightbigg/bracketleftbigg¯θ0cos(ωτ)/parenleftbig α2/¯θ0/parenrightbig sin(ωτ)/bracketrightbigg . (103) we start with the following definitions q1=x l= sinθcosϕ=¯θcosϕ, (104a) q2=y l= sinθsinϕ=¯θsinϕ, (104b) and from the approximate solution for the angle of precession ϕEqs. (34)-(36b) we get Q1=−α1 8τ+Q10, Q2=3α2 8τ+Q20, (105a) α1=¯θ2 0 2+α2 2 2¯θ2 0, k=¯θ2 0 2−α2 2 2¯θ2 0, (105b) k2=α2 1−α2 2 (105c) Q10=π 4, Q20=−arctan/parenleftbiggk+α1 α2/parenrightbigg (105d) ϕ=Q2+arctan/braceleftbigg1 α2/bracketleftig k+α1tan/parenleftig ωτ+π 4/parenrightig/bracketrightig/bracerightbigg (106) ϕ=ωaτ+Q20+η (107) where η≡arctan/braceleftbigg1 α2/bracketleftig k+α1tan/parenleftig ωτ+π 4/parenrightig/bracketrightig/bracerightbigg , (108) ωa=3α2 8, ω= 1−α1 8. (109) 18 H´ ector R. Maya, Rodolfo A. Diaz, W. J. Herrera From the definitions (108) and (105c) it can be proved that cosη=α2cos/parenleftbig ω τ+π 4/parenrightbig √α1¯θ, (110a) sinη=kcos/parenleftbig ω τ+π 4/parenrightbig +α1sin/parenleftbig ω τ+π 4/parenrightbig √α1¯θ(110b) and without any kind of approximation we find cos(Q20+η) =√α1+kcos(ωτ) ¯θ, (111) sin(Q20+η) =√α1−ksin(ωτ) ¯θ, (112) and from the initial conditions associated with the elliptical mode (105 b) it follows that /radicalbig α1+k=¯θ0,/radicalbig α1−k=α2 ¯θ0 Picking up all these results in Eq. (104a) we see that q1=¯θcosϕ=¯θ0cos(ωτ)cos(ωaτ)−α2 ¯θ0sin(ωτ)sin(ωaτ), (113) with a similar result for q2. 2. In order to deduce Eq. (100) /bracketleftbigg q1 q2/bracketrightbigg =/bracketleftbigg cos(Ω Aτ)−sin(ΩAτ) sin(ΩAτ) cos(Ω Aτ)/bracketrightbigg/bracketleftbigg¯θ0cos(ωτ)/parenleftbig P2/¯θ0/parenrightbig sin(ωτ)/bracketrightbigg (114) We take into account that Φ = ϕ+π/2,whereϕis the spherical coordinate and Φ is the Euler angle. The definitions (104) are converted into q1=x l= sinθsinΦ = ¯θsinΦ, (115a) q2=y l=−sinθcosΦ =−¯θcosΦ. (115b) On the other hand, the approximate solution for the azimuthal ang le Eqs. (70c), (80b) and (86) are given by Φ =/parenleftbigg3P2 8+Pψ 2/parenrightbigg τ+Q20+arctan/braceleftbigg1 P2[k+P1tan(Q1+τ)]/bracerightbigg , (116a) Q1=1 8/parenleftbig P2 ψ−P1/parenrightbig τ+Q10, (116b) Q10=π 4, Q20=π 2−arctan/parenleftbiggP1+k P2/parenrightbigg , (116c) P1=P2 2 2¯θ2 0+¯θ2 0 2, P2=PΦ−Pψ, (116d) k=P2 1−P2 2=¯θ2 0 2−P2 2 2¯θ2 0. (116e) which can be expressed as Φ = Ω Aτ+π 2−λ+η, (117) where we have used the auxiliary functions ΩA≡3P2 8+Pψ 2, ω≡1−P1 8+P2 ψ 8(118) λ≡arctan/parenleftbiggP1+k P2/parenrightbigg , η≡arctan/braceleftbigg1 P2/bracketleftig k+P1tan/parenleftig ωτ+π 4/parenrightig/bracketrightig/bracerightbigg (119) Apsidal precession of the Physical Pendulum 19 from these definitions, it is easy to prove that cosλ=P2√2P1√P1+k,sinλ=√P1+k√2P1(120) cosη=P2(cosωτ−sinωτ) ¯θ√2P1, (121) sinη=(P1+k)cosωτ+(P1−k)sinωτ ¯θ√2P1. (122) using these results and Eq. (117) in Eqs. (115a, 115b), we have q1=¯θsinΦ =/radicalbig P1+kcos(Ω Aτ)cosωτ−/radicalbig P1−ksin(ΩAτ)sinωτ, q2=−¯θcosΦ =/radicalbig P1+ksin(ΩAτ)cosωτ+/radicalbig P1−kcos(ΩAτ)sinωτ, and considering finally the conditions (116d) and (116e) it is easy to s ee that /radicalbig P1+k=¯θ0,/radicalbig P1−k=P2 ¯θ0, (123) which proves Eq. (114). 3. Now, to prove the canonical character of the linear transform ation in (72), we introduce a more general linear transformation ( Q, P)→(Γ,Π) whose definition is Qi=fi(Π)τ+Γi, Pi= Πi, (124) wherefiis any polynomic function of the momenta Π k. The canonical character of this family of transformations follows from the Poisson brackets of the variables Qi,Piwith respect to the new variables Γ k,Πk [Qi, Pj]Γk,Πk=∂Qi ∂Γk∂Pj ∂Πk−∂Qi ∂Πk∂Pj ∂Γk=δij (125) References [1] Lagrange, J. L.: M´ echanicque Analitique . Veuve Desaint, Paris (1788) [2] Poisson, S. D.: Sur un cas particulier du mouvement de rotation des corps pes ans. Journal de I’ ´Ecole Polytechnique. 16, 247-267 (1813) [3] Golubev, V. V.: Lectures on Integration of Equations of Motion of a Rigid Bod y about a Fixed Point . Israeli Pro- gram for Scientific Translations, Israel (1960) [4] Leimanis, E.: The General Problem of the Motion of Coupled Rigid Bodies abo ut a Fixed Point . Springer Tracts in Natural Philosophy Vol. 7. 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