Geometric Origin of the Tennis Racket Effect
P. Mardesic, L. Van Damme, G. J. Gutierrez Guillen, D. Sugny
GGeometric Origin of the Tennis Racket Effect
P. Mardeˇsi´c ∗ , L. Van Damme, G. J. Gutierrez Guillen, D. Sugny † August 11, 2020
Abstract
The tennis racket effect is a geometric phenomenon which occurs in afree rotation of a three-dimensional rigid body. In a complex phase space,we show that this effect originates from a pole of a Riemann surface andcan be viewed as a result of the Picard-Lefschetz formula. We prove thata perfect twist of the racket is achieved in the limit of an ideal asymmet-ric object. We give upper and lower bounds to the twist defect for anyrigid body, which reveals the robustness of the effect. A similar approachdescribes the Dzhanibekov effect in which a wing nut, spinning around itscentral axis, suddenly makes a half-turn flip around a perpendicular axisand the Monster flip, an almost impossible skate board trick.
Consider an experiment that every tennis player has already made. Thetennis racket is held by the handle and thrown in the air so that the handlemakes a full turn before catching it. Assume that the two faces of the headcan be distinguished. It is then observed, once the racket is caught, that thetwo faces have been exchanged. The racket did not perform a simple rotationaround its axis, but also an extra half-turn. This twist is called the tennis racketeffect (TRE). An intuitive understanding of TRE is given in [1]. It is also knownas Dzhanibekov’s effect (DE), named after the Russian cosmonaut who madea similar experiment in 1985 with a wing nut in zero gravity [2, 3]. The wingnut spins rapidly around its central axis and flips suddenly after many rotationsaround a perpendicular axis [3]. The Monster Flip Effect (MFE) is a free styleskate board trick. It consists in jumping with the skateboard and making it turnaround its transverse axis with the wheels falling back to the ground. This trickis very difficult to execute since TRE predicts precisely the opposite, turningabout this axis should produce a π - flip and the wheels should end up in theair. The video [4] shows that this trick can be made with success after severalattempts.We propose in this letter to describe these phenomena. The results are es-tablished for a tennis racket and then extended to the two other systems. Themotion is modeled as a free rotation of an asymmetric rigid body, which hasthree different moments of inertia along its three inertia axes [5]. The axes ∗ Institut de Math´ematiques de Bourgogne - UMR 5584 CNRS, Universit´e de Bourgogne-Franche Comt´e, 9 avenue Alain Savary, BP 47870, 21078 DIJON, France, [email protected] † Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 6303 CNRS-Universit´eBourgogne-Franche Comt´e, 9 Av. A. Savary, BP 47 870, F-21078 Dijon Cedex, France,[email protected] a r X i v : . [ phy s i c s . c l a ss - ph ] A ug ith the smallest and largest moments of inertia are stable, while the interme-diate one is unstable. It is precisely this instability which is at the origin ofTRE [6]. A more detailed description can be obtained from Euler’s equations.The three-dimensional rotation is an example of Hamiltonian integrable sys-tems [7] in which the trajectories can be expressed analytically. The dynamicsof the rigid body in the space-fixed frame are given by elliptic integrals of thefirst and third kinds, which lead to a very accurate description of TRE [6, 8, 9].However, this analysis does not reveal its geometric character. A geometricpoint of view provides valuable physical insights, in particular with respect tothe robustness of the corresponding physical phenomenon. Different geomet-ric structures have been studied recently in the context of mechanical systemswith a small number of degrees of freedom. Among others, we can mention theBerry phase [10], Hamiltonian monodromy [11, 12, 13], singular tori [14] and theChern number [15] which found applications in classical and quantum physics.In this letter, we show that the geometric origin of TRE is a pole of a Riemannsurface defined in a complex phase space. This effect can be interpreted asthe result of the Picard-Lefschetz formula which describes the possible defor-mation of an integration contour in a complex space after pushing it around asingular fiber [16, 17]. The geometric character of DE and MFE can also bededuced from this approach and helps understanding in which conditions theycan be realized. Note that similar complex methods have been used to describeHamiltonian monodromy [18, 19, 20].The position of the body-fixed frame ( x, y, z ) with respect to the space-fixedframe ( X, Y, Z ) defines the free rotation of a rigid body [2, 5, 7]. Three Eulerangles ( θ, φ, ψ ) characterize the relative motion of the body-fixed frame. Theangle θ is the angle between the axis z and the space-fixed axis Z . The rotationof the body about the axes Z and z is respectively described by the angles φ and ψ (see Sup. Sec. II). The moments of inertia I x , I y and I z are the elementsof the diagonal inertia matrix in the body-fixed frame, with the convention I z < I y < I x . As displayed in Fig. 1, a tennis racket is a standard example ofan asymmetric rigid body in which the z -axis is along the handle of the racket, y lies in the plane of the head of the racket and x is orthogonal to the head(See Sup. Sec. I). TRE consists in a 2 π -rotation of the body around the y -axis.The precession of the handle is measured by the angle φ . TRE then manifestsby a twist of the head about the z - axis, i.e. by a variation ∆ ψ = π , along atrajectory such that ∆ φ = 2 π [9]. The Tennis Racket Effect. - TRE is a geometric phenomenon which does notdepend on time. From Euler’s equation, it can be described by the evolution of ψ with respect to φ (See Sup. Sec. II): dψdφ = ± (cid:112) ( a + b cos ψ )( c + b cos ψ )1 − b cos ψ , (1)where we introduce the parameters a = I y I z − b = 1 − I y I x and c = I y HJ − − b < c < a , a > < b < H and J denoterespectively the rotational Hamiltonian and the angular momentum of the rigidbody defined in Sup. Sec. II [5]. In the limit of a perfect asymmetric body, I z (cid:28) I y (cid:28) I x , we deduce that b → a → + ∞ . We consider only thepositive values of dψdφ defined in Eq. (16), the same analysis can be done forthe negative sign. Equation (16) defines a two-dimensional reduced phase space2igure 1: (Color online) A Tennis Racket with the three inertia axes ( x, y, z ).The angles φ and ψ used to define TRE describe respectively the rotation ofthe body around the y - and z - axes. TRE is a phenomenon in which a full turnin φ - direction produces an almost perfect half-turn in ψ - direction.with respect to ψ and dψ/dφ , as displayed in Fig. 2. Note the similarity of thisphase space with the one of a planar pendulum, except that two consecutiveunstable fixed points are separated by π instead of 2 π . The separatrix for which c = 0 is the trajectory connecting these points [5]. We extend below the studyto the complex domain and continue analytically all the functions.TRE is associated with a trajectory for which ∆ ψ (cid:39) π when ∆ φ = 2 π . Wedenote by ψ and ψ f the initial and final values of the angle ψ . To simplify thestudy of TRE, we consider a symmetric configuration for which ψ = − π + (cid:15) and ψ f = π − (cid:15) . A perfect TRE is thus achieved in the limit (cid:15) →
0. Notethat this symmetry hypothesis is not restrictive as shown numerically in Sup.Sec. VI. Using Eq. (16), we obtain that the variation of φ is given by:∆ φ = (cid:90) π − (cid:15) − π + (cid:15) − b cos ψ (cid:112) ( a + b cos ψ )( c + b cos ψ ) dψ. (2)For oscillating trajectories, the condition c + b cos ψ ≥ (cid:15) ≥ | cb | .From the parity of the integral and the change of variables x = cos ψ , ∆ φ canbe expressed as an incomplete elliptic integral, ∆ φ ( (cid:15) ) = (cid:82) (cid:15) ω , with ω = 1 b − bx (cid:112) x ( x − β )(1 − x )( x − α ) dx, (3)where α = − ab and β = − cb . As explained in Sup. Sec. III, we introduce afunction M defined by M ( u ) = √ √ − u + 2 ln(2) for u ∈ ]0 , m =3 ψ / π -101 d ψ / d φ Figure 2: (Color online) Reduced phase space describing the dynamics of therigid body in the space ( ψ, dψ/dφ ). The black and blue (dark gray) lines depictrespectively the rotating and oscillating trajectories of the angular momentum.The solid red line (light gray) represents the separatrix. The parameters a and b are set respectively to 12 and 0.05. M ( ) (cid:39) . u ∈ ]0 , m by M ( u ). Weput u = 1 / Theorem 1.
For all c such that: | c | < b exp( − π √ ab − m ) , for ab large enough, the equation ∆ φ a,b,c ( (cid:15) ) = 2 π has a unique solution (cid:15) S ( a, b, c ) which verifies: arcsin[ (cid:114) | cb | ] < (cid:15) S < arcsin[exp( − π √ ab − m . (4) This leads to: lim ab (cid:55)→ + ∞ (cid:15) S ( a, b, c ) = 0 . Several questions about its existence, uniqueness and robustness are raisedby the observation of TRE, all find a rigorous answer in Th. 1. A first fun-damental comment concerns a perfect TRE which occurs only in the limit of4 very asymmetric body. Such limits are common enough in physics to revealspecific phenomena. An example is given by the adiabatic evolution in mechan-ics [7] which is also based mathematically on an asymptotic analysis. The mainstatement of Th. 1 describes the asymptotic behavior of the twist defect whichapproximately evolves as (cid:15) (cid:39) e −√ ab ∆ φ for a sufficiently asymmetric body (with ab (cid:29) c goes to 0). Theorem 1 also establishes therobustness of TRE with respect to the shape of the body. Lower and upperbounds to the twist defect are given by Eq. (4) as a function of the differentparameters.These results have a geometric origin in the complex domain. We study thesolution (cid:15) of ∆ φ a,b,c ( (cid:15) ) = 2 π , where ∆ φ = ∆ φ a,b,c is given by Eq. (2). The originof TRE is revealed by a complexification of the problem in which ∆ φ can beinterpreted as an Abelian integral over the Riemann surface of the form ω [17].As displayed in Fig. 3, this surface has two sheets with four branch points in x = 0, 1, β and α . Branch cuts are introduced to define a single-valued function.In the limit c →
0, the two branch points x = 0 and x = β coincide, leadingto a pole whose integral is the logarithmic function. For large values of a , notethat there is no confluence of the branch point x = α with x = β or 0.Let F be the function defined by: F a,b,c ( u ) = (cid:90) u ω = (cid:90) γ ω, where γ is the integration path with 0 < u <
1. We have ∆ φ a,b,c ( (cid:15) ) = F a,b,c (sin (cid:15) ). The multi-valued character of F a,b,c is different for u < | β | and u > | β | . In the case | β | < u <
1, we consider in the upper sheet of the Riemannsurface the cycle δ passing by x = u and encircling the two branch points β and 0, as displayed in Fig. 3. By the Picard-Lefschetz formula [16, 17], theintegration contour γ is deformed to itself plus δ when the point x = u performsa loop along δ . The integral (cid:82) δ ω adds to F a,b,c , which reveals the multi-valuedcharacter of F a,b,c as a complex function. A single-valued function can be ob-tained by adding a convenient multiple of ln u = − (cid:82) u dxx , the factor being givenby πi (cid:82) δ ω . In the limit c → ω has a pole in x = 0 and this integral can becomputed from a residue formula.We present a heuristic proof of Th. 1, while a rigorous demonstration isprovided in Sup. Sec. III. We consider a simplified version of the problem whereonly two branch points are accounted for. We have: (cid:90) u (cid:112) x ( x − β ) dx = 2 ln( (cid:112) x − β + √ x ) (cid:12)(cid:12) x = u . Using the pole at infinity, we deduce that πi (cid:82) δ dxx = 1 and: (cid:90) u [ 1 (cid:112) x ( x − β ) − x ] dx = 2 ln (cid:0) √ − β (cid:113) − βu (cid:1) , u for | β | < u <
1. As shown inSup. Sec. III, this argument can be generalized to F a,b,c which can be expressedas: F a,b,c ( u ) = 1 √ ab h a,b,c ( u ) − √ ab ln u, (5)where h a,b,c is an analytic and bounded function in ] | β | , u [ with 0 < u < h a,b,c is the function M introduced in Th. 1. For ab large enough,the equation F a,b,c ( u ) = 2 π has a unique solution which proves Th. 1. In thesecond region in which u < | β | , the geometric situation is completely differentas can be seen in Fig. 3. The cycle ˜ δ encircles only the branch point x = 0 andno pole occurs when c →
0. Turning twice around x = 0 to get a closed path,we obtain (cid:82) ˜ δ ω = 0. This result stems from integrating the complex function x (cid:55)→ √ x along ˜ δ . The function F a,b,c is bounded with no logarithmic divergence.No information is gained about the existence, the uniqueness and the value of (cid:15) , i.e. the possibility to realize TRE. The Dzhanibekov effect. - A similar analysis can be used to describe DE [3].As represented in Sup. Sec. I, the z - and x - inertia axes of this rigid bodyare respectively along the wings and orthogonal to the wings, while the y - onecorresponds to the central axis of the rotation. The video [3] clearly shows thatthe motion of the wing nut is first guided by a screw which induces an almostperfect rotation around the central axis. In terms of Euler’s angles, this leadsto a very large angular velocity ˙ φ and a speed ˙ ψ approximatively equal to 0(i.e. dψdφ (cid:39) ψ is initially of the order of ± π . We deduce that theinitial point of the dynamics is very close to one of the unstable fixed pointsrepresented in Fig. 2, with a parameter c (cid:39)
0. Using Eq. (16), DE is describedby: ∆ φ = (cid:90) π − π − b cos ψ (cid:112) ( a + b cos ψ )( c + b cos ψ ) dψ, with c >
0, where ∆ φ represents the angle increment before the flip of thesystem. We assume that the wing nut performs a perfect twist for which ψ goesfrom − π to − π . We show in Sup. Sec. IV that:∆ φ = 1 √ ab [ h a,b ( c ) − ln( c )] , (6)where h a,b is a bounded function when c →
0. In this limit, the logarithmic di-vergence of ∆ φ occurs with the confluence of the two branch points in x = β and x = 0, which gives a pole as in TRE. Consequently, the speed dφ/dψ increasestremendously in the neighborhood of this point. Note that the parameter c forDE plays the same role as (cid:15) for TRE as can be seen in Eq. (5) and (6). DE withmany rotations around the intermediate axis can be observed for a sufficientsmall positive value of c . We stress that the number of turns does not need tobe complete. The Monster Flip. - This approach can be used for a skate board where the z - and y - inertia axes are respectively orthogonal and parallel to the wheel axis,while the x - axis is orthogonal to the board (see Sup. Sec. I). MFE correspondsto a complete turn around the transverse axis together with a small variationof ψ . It can be realized in a neighborhood of the unstable point where dψdφ = 06i.e. dφdψ = ∞ ). We search for a solution (cid:15) close to zero of ˜∆ φ ( (cid:15) ) = 2 π where˜∆ φ ( (cid:15) ) = 2 (cid:90) π + (cid:15)ψ i − b cos ψ (cid:112) a + b cos ψ (cid:112) c + b cos ψ dψ, (7)with ψ i = π/ ψ i = π/ (cid:112) | β | ] for rotating and oscillating trajec-tories, respectively. As in TRE, we get ˜∆ φ ( (cid:15) ) = (cid:82) sin (cid:15) cos ψ i ω , where ω is definedby Eq. (3). Introducing ˜ F a,b,c ( u ) = (cid:82) u cos ψ i ω , it can be shown in the region | β | < u < F a,b,c ( u ) = 1 √ ab ˜ h a,b,c ( u ) + 1 √ ab ln( u ) , where ˜ h a,b,c is a bounded and single-valued function. Note the change of signin front of the logarithmic term with respect to Eq. (5). The solution of˜∆ φ a,b,c = ˜ F a,b,c ( u ) can be approximated as (cid:15) (cid:39) √ | β | e π √ ab . The accuracy of thisapproximation is shown numerically in Sup. Sec. VI. For a body with ab ≥ | β | (cid:28) (cid:15) ≥ (cid:112) | β | . This result quantifies the difficulty of performing MFE.For an angle (cid:15) of 30 degrees, this leads for a standard skate board to c (cid:39) − ,while the maximum value of c is of the order of 10. Finally, as illustrated inSup. Sec. V, MFE cannot be realized in the second region u < | β | . Conclusion. - TRE originates from a pole of a Riemann surface and a perfecttwist of the head of the racket occurs in the limit of an ideal asymmetric body.Different properties such as the robustness of the effect have been derived fromthis geometric analysis. As a byproduct, we have described DE and establishedwhy the MFE is so difficult to perform. This study paves the way for the anal-ysis of other classical integrable systems and strongly suggests the importanceof complex geometry beyond the cases studied in this paper. An intriguingquestion is to transpose this effect to the quantum world. Different molecularsystems could show traces of this effect [21, 22]. Another field of applicationsis the control of quantum systems by external electromagnetic fields [23] using,e.g., the analogy between Bloch and Euler equations [24].
Acknowledgment
This work was supported by the EUR-EIPHI Graduate School (Grant No. 17-EURE-0002)
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This paragraph gives some details about the different model systems used in thispaper. We recall, in particular, how the moments of inertia can be estimatedfor different rigid bodies.The direction of the inertia axes of a standard tennis racket is representedin Fig. 1 of the main text. Figures 5 and 6 display the inertia axes of a wingnut and a skate board.The Tennis Racket Effect can also be observed with a book or a mobilephone. If the object of mass m is a homogeneous rectangular cuboid of height h , length L and width l , with h < l < L then the moments of inertia are givenby: I x = m ( L + l ) I y = m ( L + h ) I z = m ( h + l )We deduce that: a = L − l h + l b = l − h L + l If the object is almost flat, the height will be very small with respect to the otherdimensions. In this case, the intermediate axis is in the plane of the object andperpendicular to the largest side. Numerical values are given in Tab. 1. Thecomputation is more involved for a skate board since the wheels and the truckhave to be accounted for. We consider the mass repartition given in Fig. 6 of askate of length L = 80 cm, width l = 20 cm and height h = 5 cm. The massesof the board, a wheel and a truck are estimated to be respectively of the orderof 500 g, 200 g and 350 g, which leads to a total mass of 2 kg. We obtain I x = 0 .
123 kg.m , I y = 0 .
113 kg.m and I z = 0 .
012 kg.m .9able 1: Numerical values of the parameters a and b for different objects. Thebook is the book of mechanics by Goldstein. The mobile phone is a SamsungJS. The moments of inertia of the wing nut and a tennis racket are given in [1]and [2].Object a b ab Racket 12.54 0.06 0.75Book 1.11 0.31 0.34Mobile phone 2.97 0.198 0.59Wing nut 2.92 0.0972 0.28Skate board 8.82 0.078 0.69
As mentioned in the main text, the position of the body-fixed frame ( x, y, z )with respect to the space-fixed frame (
X, Y, Z ) can be described by three Eulerangles ( θ, φ, ψ ), which characterize the motion of the rigid body. The definitionof the Euler angles is shown in Fig. 7. Rotational motion is described by inte-grable dynamics which have two constants of the motion, namely the angularmomentum J and the Hamiltonian H . The Z - axis of the space-fixed frame isusually chosen along the direction of J . In the body-fixed frame, the componentsof J can be expressed as J x = − J sin θ cos ψ , J y = J sin θ sin ψ and J z = J cos θ ,where J is the modulus of J , while H is given by H = J x I x + J y I y + J z I z . Inthe ( J x , J y , J z )- space, the energy of the intermediate axis J I y defines the posi-tion of the separatrix which connects the unstable fixed points and is also theboundary between the rotating and oscillating trajectories for 2 I y H > J and2 I y H < J , respectively. Using the angular velocities Ω k = J k /I k , k = x, y, z ,it follows that the Euler angles satisfy the Euler differential system:˙ θ = J ( 1 I y − I x ) sin θ sin ψ cos ψ ˙ φ = J ( sin ψI y + cos ψI x )˙ ψ = J ( 1 I z − sin ψI y − cos ψI x ) cos θ (8)We introduce the parameters a = I y I z − b = 1 − I y I x and c = I y HJ −
1, withthe constraint − b < c < a . Note that c measures the signed distance to theseparatrix. For a standard tennis racket, we have a = 12 .
53 and b = 0 .
063 [2]while for a skate board the parameters are a = 8 .
82 and b = 0 .
078 (see Sec. 1).Using Eq. (8), we can describe the dynamics and the TRE in terms of theevolution of ψ with respect to φ . We have: dψdφ = ( a + b cos ψ ) cos θ − b cos ψ . (9)From c = a − sin θ ( a + b cos ψ ), we arrive at:cos θ = ± (cid:115) c + b cos ψa + b cos ψ , (10)10hich leads to: dψdφ = ± (cid:112) ( a + b cos ψ )( c + b cos ψ )1 − b cos ψ . (11)Equation (11) is the starting point of the analysis of the main text. We show rigorously in this section the different statements about the TennisRacket Effect. | β | < | u | < We study in this paragraph the function F a,b,c defined in Eq. (5) of the maintext. From the geometric analysis in the complex space, the Picard-Lefschetzformula states that the function F a,b,c can be expressed in the complex domain A = { u ∈ C : | β | < | u | < } as: F a,b,c ( u ) = g a,b,c ( u ) − k a,b,c ( u ) ln u, where g and k are two holomorphic functions on the annulus A , which areuniformly bounded and continuous on the closed annulus ¯ A . Note that k a,b,c ,which is given by πi (cid:82) δ ω , can be determined in the limit c → g a,b,c and k a,b,c by considering realintegrals. Analysis of F a,b,c in the real case: The function F a,b,c can be expressed as: F a,b,c ( u ) = 1 √ ab ( h ( u ) + h ( u )) − ln( u ) √ ab , with h ( u ) = (cid:90) u ( 1 (cid:112) x ( x − β ) − x ) 1 − bx (cid:112) (1 − x )(1 − xα ) dx, and h ( u ) = (cid:90) u dxx [ 1 − bx (cid:112) (1 − x )(1 − x/α ) − . We first determine a bound in the real domain of the h - function. Let 0
1. Since1 − bx (cid:112) (1 − x )(1 + xa/b ) ≤ √ − u , for x ∈ ]0 , u ], we deduce that: | h ( u ) | ≤ √ − u (cid:90) u | (cid:112) x ( x − β ) − x | dx. which gives | h ( u ) | ≤ √ − u | (cid:90) u (cid:0) (cid:112) x ( x − β ) − x (cid:1) dx | . | β | , u ]. As in the simplifiedcase of the main text, we use the fact that: (cid:90) u (cid:0) (cid:112) x ( x − β ) − x (cid:1) dx = 2 ln (cid:0) √ − β (cid:112) − β/u (cid:1) , and we arrive at | (cid:90) u (cid:0) (cid:112) x ( x − β ) − x (cid:1) dx | ≤ (cid:0) (cid:112) | β | (cid:112) − | β | (cid:1) ≤ √ | h ( u ) | ≤ √ √ − u . In a second step, we analyze the h - function. We have: | h ( u ) | ≤ (cid:90) u dxx | − bx − (cid:112) (1 − x )(1 − x/α ) (cid:112) (1 − x )(1 − x/α ) | ≤ (cid:90) u dxx − √ − x √ − x , which is valid for ab large enough. The upper bound of h can be exactlyintegrated: | h ( u ) | ≤ (cid:12)(cid:12) [ − √ − x )] u (cid:12)(cid:12) ≤ F a,b,c ( u ) = h a,b,c ( u ) √ ab − ln u √ ab , where h a,b,c = h + h is a bounded function with | h a,b,c ( u ) | ≤ √ √ − u + 2 ln(2) , which is the bound used in the main text. Note that the bound on h a,b,c does notdepend on a , b and c but only on a fixed parameter u which can be chosen at willin ]0 , M the function defined by M ( u ) = √ √ − u +2 ln(2) for u ∈ ]0 , h (cid:48) a,b,c of h a,b,c is given by the corresponding integrandsof h and h , leading to: h (cid:48) a,b,c ( u ) = 1 u − (cid:112) u ( u + c/b ) 1 − bu (cid:112) (1 − u )(1 + ub/a ) . Proposition 1.
For all u ∈ ]0 , , for all c such that ≤ | c | < be − π √ ab − M ( u ) , (12) for ab large enough, the equation F a,b,c ( u ) = 2 π (13) has a unique solution u = u S ( a, b, c ) in ] | cb | , u [ , which verifies: | cb | < u S < e − π √ ab + M ( u ) and, in particular, lim ab (cid:55)→ + ∞ u S ( a, b, c ) = 0 . roof. Equation (13) becomes:1 √ ab h a,b,c ( u ) − √ ab ln u = 2 π (14)Equation (14) can be expressed in terms of a fixed point problem u = f ( u ),with f ( u ) = e − π √ ab + h a,b,c ( u ) . We arrive at: e − π √ ab − M ( u ) < f ( u ) < e − π √ ab + M ( u ) . (15)We show by continuity the existence of a solution to the fixed point problem if f ( | β | ) > | β | and f ( u ) < u . The first condition is given by Eq. (12) while thesecond inequality is trivially verified from Eq. (15), for ab large enough. Theuniqueness of the solution is verified if the function g : u (cid:55)→ f ( u ) − u is strictlydecreasing. We show this statement for c ≤
0, while for c >
0, we prove that g is increasing on [ | β | , u m [, it reaches its maximum in u = u m and is strictlydecreasing on ] u m , u [.Let us first consider the case c ≤
0. The function h (cid:48) a,b,c can be bounded for u ∈ ] | β | , u ] by: h (cid:48) a,b,c ( u ) ≤ u (1 − − bu (cid:112) (1 − u )(1 + ub/a ) ) ≤ t ( u )where t ( u ) = 1 u (1 − (1 − u ) − / ) . Since lim u → t ( u ) = − and t is a strictly decreasing function, we deduce that h (cid:48) ( u ) ≤ − for u ∈ ] | c/b | , u [. g is therefore also strictly decreasing.We then study the case c >
0. A zero u m of g (cid:48) fulfills: h (cid:48) a,b,c ( u m ) e h a,b,c ( u m ) = e π √ ab , then: e π √ ab − M ( u ) < h (cid:48) a,b,c ( u m ) < e π √ ab + M ( u ) (16)For ab large enough, Eq. (16) shows that u m belongs to a small neighborhoodof u = | c/b | when | c/b | (cid:28)
1. Moreover, the function h (cid:48) a,b,c can be bounded by: h (cid:48) a,b,c ( u ) ≤ r ( u ) , where r ( u ) = u − √ u ( u + | c/b | ) √ − u . We have: r ( u ) ≤ ⇔ u + | c/b | u − | c/b | > . We obtain that r ( u ) ≤ h (cid:48) a,b,c ≤ u ≥ (cid:112) | c/b | when | c/b | →
0. In theinterval [ | c/b | , (cid:112) | c/b | ], h (cid:48) a,b,c is equivalent for | c/b | (cid:28) h (cid:48) a,b,c ( u ) (cid:39) u − (cid:112) u ( u + | c/b | ) , which is a strictly decreasing function tending to + ∞ when u and c go to 0. Wededuce that there exists a unique u m such that g (cid:48) ( u m ) = 0. We finally obtainthat g ( u m ) > g (cid:48) ( u ) < u m , u ], which leads to the uniqueness of thesolution u S . 13sing Proposition 1, we can deduce Theorem 1 of the main text. Proof.
The proof follows directly from Proposition 1 and the relation ∆ φ a,b,c ( (cid:15) ) = F a,b,c (sin (cid:15) ), since the change of variables u = sin (cid:15) is a bijection from [0 , π/ , ab large enough, the condition (15) gives that f :] | β | , u [ → ] | β | , u [. The above proof was used because it gives in additionan interval in which the respective fixed points u S and (cid:15) S of f and f ◦ arcsinbelong, showing thus the corresponding limits for u S and (cid:15) S , when ab → + ∞ .On the other hand, the fixed point theorem also shows the robustness of thephenomenon. u < | β | We consider now the function F a,b,c in the region | u | < | β | . We recall that thisanalysis only concerns the case with c > Lemma 1.
There exists a holomorphic function k defined on D = { v ∈ C : | v | < (cid:112) | β |} such that F a,b,c ( u ) = k ( √ u ) i.e. F ( v ) = k ( v ) .Proof. Turning around the origin in u , we do not catch the cycle δ as in theTRE, but a non-closed path. Turning twice around x = 0, we catch a closedcycle ˜ δ winding twice around the branch point x = 0 only. Note that here (cid:82) ˜ δ ω = 0. The result is equivalent to integrate x (cid:55)→ √ x on a loop winding twicearound zero. Let k be k ( v ) = F a,b,c ( v ). Then, we deduce that: k ( ve πi ) = F a,b,c ( v e πi ) = F a,b,c ( v ) + (cid:90) ˜˜ δ ω = F a,b,c ( v ) = k ( v ) . Moreover, k (0) = (cid:82) ω < ∞ is a complete elliptic integral. Hence, k has aremovable singularity at the origin and extends to a holomorphic function on D . Equation ∆ φ a,b,c ( (cid:15) ) = F a,b,c (sin (cid:15) ) becomes h a,b,c (sin (cid:15) ) = 2 π, where h a,b,c is a bounded and analytic function. Note that the nature of thisequation, valid in the small region D is completely different from Eq. (14).14 Analysis of the Dzhanibekov effect
After a very large number of rotations around its central axis, the wing nut flipssuddenly around a perpendicular axis. This phenomenon occurs for rotatingtrajectories for which c >
0. We have:∆ φ = (cid:90) π − π − b cos ψ (cid:112) ( a + b cos ψ )( c + b cos ψ ) dψ. With the same change of coordinates as for TRE, we arrive at:∆ φ = (cid:90) b − bx (cid:112) x ( x − β )(1 − x )( x − α ) dx with α = − ab and β = − cb . This integral can be viewed as an Abelian integralfor a cycle connecting the two branch points 0 and 1. It starts on one sheet ofthe Riemann surface and ends on the other. We now estimate the integral inthe real domain. We have:∆ φ = 1 √ ab (cid:90) − bx (cid:112) x ( x − β )(1 − x )(1 − xα ) . The variation ∆ φ can be written as the sum of two terms:∆ φ = 1 √ ab [ h ( c ) + g ( c )] , with g ( c ) = (cid:90) (cid:112) x ( x − β ) dx. Straightforward computations lead to: g ( c ) = 2 ln( √ b + √ b + c √ c ) . We show in a second step that the function h is bounded. We have: h ( c ) = 1 √ ab (cid:90) (cid:112) x ( x − β ) ( 1 − bx (cid:112) (1 − x )(1 − xα ) − dx. We can derive an upper bound as follows: | h ( c ) | ≤ √ ab (cid:90) (cid:112) x ( x − β ) | − bx − (cid:112) (1 − x )(1 − xα ) (cid:112) (1 − x )(1 − xα ) | dx We obtain: | h ( c ) | ≤ √ ab (cid:90) x [ 1 − √ − x √ − x ] dx ≤ , which is valid for ab large enough. It is then straightforward to derive Eq. (7)of the main text: ∆ φ = 1 √ ab [ h a,b ( c ) − ln( c )] . c →
0, we can estimate the variation ∆ φ . Direct computations lead to:∆ φ (cid:39) √ ab [ln( 4 bc ) + 2 ln 2] . The accuracy of this approximation is investigated in Fig. 10 for a standardwing nut.
We study in this section the Monster Flip for a standard skate board. As in themain text, we introduce the function ˜ F a,b,c = (cid:82) sin (cid:15) cos ψ i and search for solutions of˜ F a,b,c ( u ) = 2 π, ≤ u ≤ , (17)in the rotating case or ˜ F a,b,c ( u ) = 2 π, | β | ≤ u ≤ , (18)for oscillating trajectories. Note that cos ψ i is equal to 0 or to | β | in the rotatingand oscillating cases, respectively. Following the study used in TRE, we considerthe two regions 0 < u < | β | and | β | < u <
1. In the case 0 < u < | β | , whichonly concerns rotating trajectories, it can be shown that:˜ F a,b,c ( u ) = ˜ h a,b,c ( √ u ) , where ˜ h a,b,c is a holomorphic function vanishing at the origin. For the region | β | < | u | <
1, we get: ˜ F a,b,c ( u ) − ln( u ) (cid:90) δ ω = 1 √ ab ˜ h a,b,c ( u ) , where ˜ h a,b,c is a single-valued function.We consider now the different integrals in the real domain. Starting fromthe equation ∆ φ a,b,c = ˜ F a,b,c ( u ), we deduce that˜ F a,b,c ( u ) = e π √ ab +˜ h a,b,c ( u ) . (19)Approximate expressions of the variation (cid:15) can be obtained as follows. When u (cid:28)
1, we have:˜ F a,b,c ( u ) = 1 b (cid:90) u cos ψ i − bx (cid:112) x ( x − β )(1 − x )( x − α ) dx (cid:39) √ ab (cid:90) u dx (cid:112) x ( x − β ) , where we have replaced x by 0 except in the factor (cid:112) x ( x − β ). A standardintegration leads to: ˜ F a,b,c ( u ) (cid:39) √ ab ln (cid:0)(cid:114) u | β | + (cid:114) u | β | (cid:1) The equation ∆ φ a,b,c = 2 π = ˜ F a,b,c ( u ) can then be approximated as: (cid:114) u | β | + (cid:114) u | β | = e π √ ab .
16n the case 0 < u < | β | , we have (cid:113) u | β + (cid:113) u | β | ≤ √ h a,b,c - function is bounded. This also gives a strong constraint onthe parameters a and b : ab ≤ [ln(1 + √ π The bound on the product ab is of the order of 0.079 which means that thissituation is not very interesting in practice since the rigid body has to be slightlyasymmetric. The variation (cid:15) of MFE can be estimated as: (cid:15) (cid:39) π √ ac. (20)In the region u > | β | , a simple formula can be derived in the limit u/ | β | (cid:29) (cid:15) (cid:39) (cid:112) | β | e π √ ab , (21)which allows to estimate the bounded function ˜ h a,b,c . These different approxi-mations will be illustrated numerically in Sec. 6. The goal of this paragraph is to illustrate numerically the different results es-tablished in this work. Figure 8 gives a general overview of the twist | ∆ ψ | of thehead of the racket when the handle makes a 2 π - rotation. The twist is plottedas a function of the initial conditions ψ and dψ/dφ | . In addition, this numer-ical result shows that the TRE and the MFE are not limited to the symmetricconfiguration analyzed in this study. We observe that TRE can be achieved in alarge area around the separatrix. MFE occurs only in a very small band aroundthe separatrix, which shows the difficulty to realize the Monster flip.Figure 9 displays the evolution of (cid:15) in the TRE case. We observe that (cid:15) goesto zero when a increases. For the chosen value of the c parameter, it can beverified that (cid:15) > (cid:15) for a ≤ (cid:15) = arcsin[ (cid:112) | c/b | ].We study in Fig. 10 the evolution of ∆ φ with respect to the parameter c fora standard wing nut. We observe the divergence of ∆ φ when c goes to 0. Usingthe analysis of Sec. 4, we approximate with a good accuracy ∆ φ as follows:∆ φ (cid:39) √ ab [2 ln 2 + ln( 4 bc )] . (22)The behavior of (cid:15) in the MFE case is represented respectively in Fig. 11 and12 in the region where (cid:15) < (cid:15) and (cid:15) > (cid:15) . It can be seen that Eq. (20) and (21)give a very good estimate of (cid:15) . As could be expected, small values of (cid:15) are onlyobtained when c is sufficiently small. The parameters a and b have been chosenso that ab ≤ [ln(1+ √ π in the first situation. References [1] H. Murakami, O. Rios and T. J. Impelluso, J. App. Mech. , 111006(2016) 172] H. Brody, Phys. Teach. 213 (1985)18igure 3: (Color online) Riemann surface of the form ω with the four branchpoints (black dots) in x = α , β , 0 and 1 (from bottom to top). When c → x = β and x = 0 coincide and give birth to a pole. The top andbottom panels represent the cases where the TRE can or cannot be observed.The solid straight lines represent the branch cuts of the surface. The cycles δ and ˜ δ are depicted by solid red (dark gray) lines. The form ω is integrated alongthe path γ between the point u (black cross) and the ramification point x = 1(green or light gray solid line). 19 Figure 4: (Color online) Illustration of the Tennis Racket Effect: The head ofthe racket performs a π - flip when the handle makes a 2 π - rotation. yxz Figure 5: (Color online) A wing nut with the definition of the inertia axes ( x , y , z ). The intermediate axis is the central axis of the wing nut, while the axeswith the smallest and largest moments of inertia are respectively along andorthogonal to the wings. 20 yz Figure 6: (Color online) Schematic representation at the scale of a skate boardwith the definition of the inertia axes ( x , y , z ). The big dots indicate theposition of the wheels and the blue rectangles the trucks.21igure 7: (Color online) Definition of the Euler angles used to describe theposition of the body-fixed frame ( x, y, z ) with respect to the space-fixed frame( X, Y, Z ). The direction of the conserved angular momentum J is also indicated.22igure 8: (Color online) (upper panel) Contour plot of the twist | ∆ ψ | , with∆ φ = 2 π , as a function of the initial conditions ψ and dψ/dφ | . (lower panel)Zoom of the upper panel corresponding to the black rectangle.23
30 60 90 120 a ǫ Figure 9: (Color online) Evolution of (cid:15) as a function of a in the TRE case.Parameters are set to c = 10 − and b = 0 . (cid:15) = 1 . × − . Figure 10: (Color online) Evolution of the variation ∆ φ for the DE case as afunction of c (solid black line). The approximation of ∆ φ given in Eq. (22) isplotted in red (or dark gray). Parameters are set to a = 2 .
92 and b = 0 . c × -3 ǫ Figure 11: (Color online) Evolution of (cid:15) (black and red lines) and (cid:15) (blue line)as a function of c in the MFE case. The black and red curves depict respectivelythe numerical solution and the approximate expression of (cid:15) given by Eq. (20).Parameters are set to a = 12 .
65 and b = 0 . ab < . c × -3 ǫ Figure 12: (Color online) Same as Fig. 11 but for the region (cid:15) > (cid:15) . Theapproximate expression of (cid:15) is given by Eq. (21). Parameters are set to a = 8 . b = 0 ..