11.3 Steiner’s Theorem; Heavy Roller; Physical Pendulum 75 We should mention in this context that modifications of this problem fre- quently appear in written examination questions, in particular the problem of the transition from rolling motion down an inclined segment to a subsequent horizontal motion, i.e., on a horizontal plane segment, and we also mention the difference between gliding and rolling motions (see below). As an additional problem we should also mention the physical pendulum, as opposed to the mathematical pendulum (i.e., a point mass suspended from a weightless thread or rod), on which we concentrated previously. The physical pendulum (or compound pendulum) is represented by a rigid body, which is suspended from an axis around which it can rotate. The center of mass is at adistances from the axis of rotation, such that L≡T−V≡ 1 2 Θ ˙ ϑ 2 − Mgs·(1 −cos ϑ) . Here M is the mass of the body, and Θ := Θ (s) + Ms 2 is the moment of inertia of the rigid body w.r.t. the axis of rotation. The oscillation eigenfrequency for small-amplitude oscillations is therefore ω 0 = g l red , where the so-called reduced length of the pendulum, l red , is obtained from the following identity: l red = Θ Ms . At this point we shall mention a remarkable physical “non-event” occur- ring at the famous Gothic cathedral in Cologne. The towers of the medieval cathedral were completed in the nineteenth century and political representa- tives planned to highlight the completion by ringing the so-called Emperor’s Bell, which was a very large and powerful example of the best technology of the time. The gigantic bell, suspended from the interior of one of the towers, was supposed to provide acoustical proof of the inauguration ceremony for the towers. But at the ceremony, the bell failed to produce a sound, since the clapper, in spite of heavy activity of the ringing machinery, never actu- ally struck the outer mantle of the bell 3 . This disastrous misconstruction can of course be attributed to the fact that the angle of deflection ϑ 1 (t)ofthe clapper as a function of time was always practically identical with the angle of deflection of the outer mantle of the bell, i.e., ϑ 2 (t) ≈ ϑ 1 (t) (both angles measured from the vertical direction). This is again supported by a diagram (Fig. 11.1). 3 An exercise can be found on the internet, [2], winter 1992, file 8. 76 11 Rigid Bodies Fig. 11.1. Schematic diagram for a bell. The diagram shows the mantle of the bell (the outer double lines, instantaneously swinging to the right) and a clapper (instantaneously swinging to the left). It is obvious from this sketch that there are two characteristic angles for the bell, the first one characterizing the deviation of the center of mass (without clapper) from the vertical, the second one for the clapper itself. The reason for the failure, which is often used as a typical examination question on the so-called double compound pendulum, is that the geometrical and material parameters of the bell were such that l red|{bell with fastened clapper} ≈ l red|{clapper alone} . Another remarkable example is the reversion pendulum.Ifwestartfrom the above-mentioned equation for the reduced length of the pendulum, inser- tion of the relation Θ ! = Θ (s) + Ms 2 gives l red (s) ! = Θ (s) Ms + s. This yields a quadratic equation with two solutions satisfying s 1 + s 2 = l red . For a given system, i.e., with given center of mass, we thus obtain two solu- tions, with s 2 = l red − s 1 . If we let the pendulum oscillate about the first axis, i.e., corresponding to the first solution s 1 , then, for small oscillation amplitudes we obtain the oscillation eigenfrequency ω 0 = g l red . If, subsequently, after reversion, we let the pendulum oscillate about the par- allel second axis, i.e., corresponding to s 2 , then we obtain the same frequency. 11.4 Inertia Ellipsoids; Poinsot Construction 77 11.4 Inertia Ellipsoids; Poinsot Construction The positive-definite quadratic form 2T rot = 3 i,k=1 θ i,k ω i ω k describes an ellipsoid within the space R 3 (ω), the so-called inertia ellipsoid. This ellipsoid can be diagonalized by a suitable rotation, and has the normal form 2T rot = 3 α=1 Θ α · ω 2 α , with so-called principal moments of inertia Θ α and corresponding principal axes of lengths a α = 2T rot Θ α . (See Fig. 11.2.) Every rigid body, however arbitrarily complex, possesses such an inertia ellipsoid. This is a smooth fictitious body, in general possessing three principal axes of different length, and where the directions of the principal axes are not easily determined. However, if the body possesses a (discrete!) n-fold axis of rotation 4 ,withn ≥ 3, then the inertia ellipsoid is an ellipsoid of (continuous!) revolution around the symmetry axis. As a practical consequence, e.g., for a three-dimensional system with a square base, this implies the following property. If the system is rotated about an arbitrary axis lying in the square, then the rotational energy of the system does not depend on the direction of the rotation vector ω, e.g., whether it is parallel to one of the edges or parallel to one of the diagonals; it depends only on the magnitude of the rotational velocity. The angular momentum L is calculated by forming the gradient (w.r.t. ω)ofT rot (ω). Thus L(ω) = grad ω T rot (ω) , i.e., L α = ∂T rot ∂ω α = Θ α ω α . Since the gradient vector is perpendicular to the tangent plane of T rot (ω), one thus obtains the so-called Poinsot construction: For given vector of rotation the angular momentum L(ω) is by construc- tion perpendicular to the tangential plane of the inertial ellipsoid T rot (ω)= constant corresponding to the considered value of ω. This can be seen in Fig. 11.2. 4 This means that the mass distribution of the system is invariant under rotations about the symmetry axis by the angle 2π/n. 78 11 Rigid Bodies Fig. 11.2. Inertia ellipsoid and Poinsot con- struction. For the typical values a := 2 and b := 1, a (ω x ,ω y )-section of an inertia ellip- soid ω 2 1 Θ 1 + + ω 2 3 Θ 3 =2· T rot (with, e.g., a 2 ≡ 2·T rot Θ 1 ) and an initial straight line (here diagonal) through the origin is plot- ted as a function of ω x . The second (exter- nal) straight line is perpendicular to the tan- gential plane of the first line, i.e., it has the direction of the angular momentum L.This is the so-called Poinsot construction One can also define another ellipsoid, which is related to the inertia el- lipsoid, but which is complementary or dual to it, viz its Legendre transform. The transformed ellipsoid or so-called Binet ellipsoid is not defined in the space R 3 (ω), but in the dual space R 3 (L)bytherelation 2T rot = 3 α=1 L 2 α Θ α . Instead of the angular velocity ω,theangular momentum L is now central. In fact, L and ω are not parallel to each other unless ω has the direction of a principal axis of the inertia ellipsoid. We shall return to this point in Part II, while discussing crystal optics, when the difference between the directions of the electric vectors E and D is under debate. 11.5 The Spinning Top I: Torque-free Top A spinning top is by definition a rigid body supported at a canonical reference point r 0 (see above), where in general v 0 = 0. For example, r 0 is the point of rotation of the rigid body on a flat table. In this case, the gravitational forces, virtually concentrated at the center of mass, produce the following torques w.r.t. the point of rotation of the top on the table: D r 0 =(R s − r 0 ) ×(−Mgˆz) . A torque-free top is by definition supported at its center of mass; there- fore no gravitational torque does any work on the top. The total angular momentum L is thus conserved, as long as external torques are not applied, which is assumed. Hence the motion of the torque-free top can be described as a “rolling of the inertia ellipsoid along the Poinsot plane”, see above. In this way, ω-lines are generated on the surface of the inertia ellipsoid as described in the following section. 11.6 Euler’s Equations of Motion and the Stability Problem 79 11.6 Euler’s Equations of Motion and the Stability Problem We now wish to describe the stability of the orbits of a torque-free top.There- fore we consider as perturbations only very weak transient torques. Firstly, we have ω(t)= 3 α=1 ω α (t)e α (t)andL = 3 α=1 L α (t)e α (t) . Here the (time-dependent) unit vectors e α (t) describe the motion of the prin- cipal axes of the (moving) inertia ellipsoid; ω α (t)andL α (t) are therefore co-moving cartesian coefficients of the respective vectors ω and L; i.e., the co-moving cartesian axes e α (t), for α =1, 2, 3, are moving with the rigid body, to which they are fixed and they depend on t,whereastheaxese j ,for j = x, y, z, are fixed in space and do not depend on t: L α (t) ≡ L α (t) |co−moving , ω α (t) ≡ ω α (t) |co−moving . But d dt e α = ω × e α ; hence d dt L = 3 α=1 dL α dt |co−moving e α (t)+ω × L = δD(t) , (11.6) where the perturbation on the right describes the transient torque. We also have d dt ω = 3 α=1 dω α dt |co−moving e α (t) . Hence in the co-moving frame of the rigid body: dL α dt + ω β L γ − ω γ L β = δD α etc., and with L α = Θ α ω α we obtain the (non-linear) Euler equations for the calculation of the orbits ω(t) on the surface of the inertia ellipsoid: Θ α · dω α dt + ω β ω γ ·(Θ γ − Θ β )=δD α (t) , Θ β · dω β dt + ω γ ω α · (Θ α − Θ γ )=δD β (t) , 80 11 Rigid Bodies Θ γ · dω γ dt + ω α ω β ·(Θ β − Θ α )=δD γ (t) . (11.7) Let us now consider the case of an ellipsoid with three different principal axes and assume without lack of generality that Θ α <Θ β <Θ γ . In (11.7), of the three prefactors ∝ ω β ω γ , ∝ ω γ ω α and ∝ ω α ω β the first and third are positive, while only the second (corresponding to the so-called middle axis) is negative. As a consequence, small transient perturba- tions δD α and δD γ lead to elliptical motions around the respective principal axis, i.e., ∝ ε β ω 2 β + ε γ ω 2 γ and ∝ ε α ω 2 α + ε β ω 2 β , whereas perturbations corresponding to δD β may have fatal effects, since in the vicinity of the second axis the ω-lines are hyperbolas, ∝ ε α ω 2 α − ε γ ω 2 γ , i.e., along one diagonal axis they are attracted, but along the other diagonal axis there is repulsion, and the orbit runs far away from the axis where it started. The appearance of hyperbolic fixed points is typical for the transition to chaos, which is dicussed below. Motion about the middle axis is thus unstable. This statement can be shown to be plausible by plotting the orbits in the frame of the rotating body for infinitesimally small perturbations, δD → 0, The lines obtained in this way on the surface of the inertia ellipsoid by integration of the three coupled equations (11.7) describe the rolling motion of the inertia ellipsoid on the Poinsot plane. Representations of these lines can be found in almost all relevant textbooks. Another definition for the ω-lines on the inertia ellipsoid is obtained as follows: they are obtained by cutting the inertia ellipsoid 3 α=1 Θ α ω 2 i =2T rot = constant . with the so-called L 2 or swing ellipsoid , i.e., the set 3 α=1 Θ 2 α ω 2 i = L 2 = constant . We intentionally omit a proof of this property, which can be found in many textbooks. 11.7 The Three Euler Angles ϕ, ϑ and ψ; the Cardani Suspension 81 To calculate the Hamiltonian it is not sufficient to know d dt ω α|co−moving . Therefore we proceed to the next section. 11.7 The Three Euler Angles ϕ, ϑ and ψ; the Cardani Suspension Apart from the location of its center of mass, the position of a rigid body is characterized by the location in space of two orthogonal axes fixed within the body : e 3 and e 1 . For example, at t = 0 we may assume e 3 ≡ ˆz and e 1 (without prime) ≡ ˆx . For t>0 these vectors are moved to new directions: e 3 (0) → e 3 (t)(≡ e 3 )ande 1 → e 1 (now with prime) . This occurs by a rotation D R , which corresponds to the three Euler angles ϕ, χ and ψ, as follows (the description is now general): a) Firstly, the rigid body is rotated about the z-axis(whichisfixedinspace) by an angle of rotation ϕ, in such a way that the particular axis of the rigid body corresponding to the ˆx direction is moved to a so-called node direction e 1 . b) Next, the particular axis, which originally (e.g., for t =0)pointsinthe ˆz-direction, is tilted around the nodal axis e 1 in the e 3 direction by an angle ϑ = arccos(ˆz ·e 3 ). c) Finally, a rotation by an angle ψ around the e 3 axis follows in such a way that the nodal direction, e 1 , is rotated into the final direction e 1 . As a consequence we obtain D R ≡D 3) e 3 (ψ) ·D 2) e 1 (ϑ) ·D 1) ˆz (ϕ) , where one reads from right to left, and the correct order is important, since fi- nite rotations do not commute (only infinitesimal rotations would commute). A technical realization of the Euler angles is the well-known Cardani sus- pension , a construction with a sequence of three intertwined rotation axes in independent axle bearings (see the following text). In particular this is a realization of the relation ω = ˙ ψe 3 + ˙ ϑe 1 +˙ϕˆz , which corresponds to the special case of infinitesimal rotations; this relation will be used below. 82 11 Rigid Bodies In the following we shall add a verbal description of the Cardani suspen- sion, which (as will be seen) corresponds perfectly to the Euler angles. The lower end of a dissipationless vertical rotation axis, 1 ( ˆ= ± ˆz,cor- responding to the rotation angle ϕ), branches into two axle bearings, also dissipationless, supporting a horizontal rotation axis, 2 (= ±e 1 ), with corre- sponding rotation angle ϑ around this axis. This horizontal rotation axis corresponds in fact to the above-mentioned nodal line; in the middle, this axis branches again (for ϑ = 0: upwards and downwards) into dissipationless axle bearings, which support a third rotation axis, 3, related to the direction e 3 (i.e., the figure axis), with rotation angle ψ. For finite ϑ, this third axis is tilted with respect to the vertical plane through ˆz and e 1 . The actual top is attached to this third axis. Thus, as mentioned, for the so-called symmetrical top, i.e., where one is dealing with an inertia ellipsoid with rotational symmetry, e.g., Θ 1 ≡ Θ 2 , the (innermost) third axis of the Cardani suspension corresponds to the so- called figure axis of the top. The reader is strongly recommended to try to transfer this verbal descrip- tion into a corresponding sketch! The wit of the Cardani suspension is that the vertical axis 1 branches to form the axle bearings for the horizontal nodal axis, 2, and this axis provides the axle bearings for the figure axis 3. To aid the individual imagination we present our own sketch in Fig. 11.3. TheimportanceoftheEuler angles goes far beyond theoretical mechanics as demonstrated by the technical importance of the Cardani suspension. Fig. 11.3. The figure illustrates a so-called Cardani sus- pension, which is a construction involving three different axes of rotation with axle bearings, corresponding to the Euler angles ϕ, ϑ and ψ 11.8 The Spinning Top II: Heavy Symmetric Top 83 11.8 The Spinning Top II: Heavy Symmetric Top The “heavy symmetric top” is called heavy, because it is not supported at the center of mass, so that gravitational forces now exert a torque, and it is called symmetric, because it is assumed, e.g., that Θ 1 ≡ Θ 2 (= Θ 3 ) . The axis corresponding to the vector e 3 is called the figure axis. Additionally one uses the abbreviations Θ ⊥ := Θ 1 ≡ Θ 2 and Θ || := Θ 3 . Let us calculate the Lagrangian L(ϕ, ϑ, ψ, ˙ϕ, ˙ ϑ, ˙ ψ) . Even before performing the calculation we can expect that ψ and ϕ are cyclic coordinates, i.e., that ∂L ∂ϕ = ∂L ∂ψ =0, such that the corresponding generalized momenta p ϕ := ∂L ∂ ˙ϕ and p ψ := ∂L ∂ ˙ ψ are conserved quantities. Firstly, we express the kinetic energy T rot := 1 2 · Θ ⊥ · ω 2 1 + ω 2 2 + Θ || ω 2 3 in terms of the Euler angles by ω = ˙ ψe 3 + ˙ ϑe 1 +˙ϕˆz . Scalar multiplication with e 3 , e 1 and e 2 yields the result 5 : ω 3 = ˙ ψ +˙ϕ ·cos ϑ, ω 1 = ˙ ϑ and ω 2 =˙ϕ · sin ϑ. As a consequence, we have T rot = Θ ⊥ 2 · ˙ ϑ 2 +sin 2 ϑ ˙ϕ 2 + Θ || 2 · ˙ ψ +˙ϕ cos ϑ 2 . (11.8) 5 because e 3 , ˆz and e 2 are co-planar, such that e 3 · ˆz =cosϑ and e 2 · ˆz =sinϑ. 84 11 Rigid Bodies Additionally we have the potential energy V = Mgs· cos ϑ, where s ·cos ϑ is the z-component of the vector (R s − r 0 ). The Lagrangian is L =T rot −V. Obviously ψ and ϕ are cyclic, and the above-mentioned generalized momenta are conserved, viz the following two components of the angular momentum L: p ψ = ∂T rot ∂ ˙ ψ = Θ || · ( ˙ ψ +˙ϕ cos ϑ)=Θ || ω 3 ≡ L 3 ≡ L(t) ·e 3 (t) , and p ϕ = ∂T rot ∂ ˙ϕ = Θ ⊥ ·sin 2 ϑ ˙ϕ + Θ || ·( ˙ ψ +˙ϕ cos ϑ)cosϑ = Θ 2 ω 2 sin ϑ + Θω 3 cos ϑ = L 2 sin ϑ + L 3 cos ϑ ≡ L z ≡ L(t) · ˆz . The symmetric heavy top thus has exactly as many independent 6 con- served quantities as there are degrees of freedom. The number of degrees of freedom is f = 3, corresponding to the three Euler angles,andthethree independent conserved quantities are: a) The conservation law for the (co-moving!) component L 3 of the angu- lar momentum, i.e., the projection onto the (moving!) figure axis.(This conservation law is remarkable, in that the figure axis itself performs a complicated motion. On the one hand the figure axis precesses around the (fixed) z-axis; on the other hand, it simultaneously performs so-called nutations, i.e., the polar angle ϑ(t) does not remain constant during the precession, but performs a more-or-less rapid periodic motion between two extreme values.) For the asymmetric heavy top this conservation law would not be true. b) The component L z of the angular momentum (fixed in space!). c) As a third conservation law we have, of course, conservation of energy, since on the one hand the Lagrangian L does not depend explicitly on time, while on the other hand we have used skleronomous coordinates, i.e., the relation between the Euler angles and the cartesian coordinates x, y and z does not explicitly depend on time. 6 Two dependent conserved quantities are e.g., the Hamiltonian H of a conservative system and any function F (H). . so-called double compound pendulum, is that the geometrical and material parameters of the bell were such that l red|{bell with fastened clapper} ≈ l red|{clapper alone} . Another remarkable example. axis the ω-lines are hyperbolas, ∝ ε α ω 2 α − ε γ ω 2 γ , i.e., along one diagonal axis they are attracted, but along the other diagonal axis there is repulsion, and the orbit runs far away. 11.3. TheimportanceoftheEuler angles goes far beyond theoretical mechanics as demonstrated by the technical importance of the Cardani suspension. Fig. 11.3. The figure illustrates a so-called Cardani sus- pension,