Chaotic Modeling and Simulation (CMSIM) 1: 57 67, 014 Dnamics of multiple pendula without gravit Wojciech Szumiński Institute of Phsics, Universit of Zielona Góra, Poland (E-mail: uz88szuminski@gmail.com) Abstract. We present a class of planar multiple pendula consisting of mathematical pendula and spring pendula in the absence of gravit. Among them there are sstems with one fied suspension point as well as freel floating joined masses. All these sstems depend on parameters (masses, arms lengths), and possess circular smmetr S 1. We illustrate the complicated behaviour of their trajectories using Poincaré sections. For some of them we prove their non-integrabilit analsing properties of the differential Galois group of variational equations along certain particular solutions of the sstems. Kewords: Hamiltonian sstems, Multiple pendula, Integrabilit, Non-integrabilit, Poincaré sections, Morales-Ramis theor, Differential Galois theor. 1 Introduction The complicated behaviour of various pendula is well known but still fascinating, see e.g. books [,3] and references therein as well as also man movies on outube portal. However, it seems that the problem of the integrabilit of these sstems did not attract sufficient attention. According to our knowledge, the last found integrable case is the swinging Atwood s machine without massive pulles [1] for appropriate values of parameters. Integrabilit analsis for such sstems is difficult because the depend on man parameters: masses m i, lengths of arms a i, Young modulus of the springs k i and unstretched lengths of the springs. In a case when the considered sstem has two degrees of freedom one can obtain man interesting information about their behaviour making Poincaré cross-sections for fied values φ 1 a 1 of the parameters. However, for finding new integrable cases m one needs a strong tool to distinguish values of 1 a parameters for which the sstem is suspected φ to be integrable. Recentl such effective and strong tool, the so-called Morales-Ramis theor [5] has appeared. It is based on analsis of differential Galois group of variational Fig. 1. Simple double pendulum. equations Received: 9 March 013 / Accepted: 3 October 013 c 014 CMSIM ISSN 41-0503
58 Szumiński obtained b linearisation of equations of motion along a non-equilibrium particular solution. The main theorem of this theor states that if the considered sstem is integrable in the Liouville sense, then the identit component of the differential Galois group of the variational equations is Abelian. For a precise definition of the differential Galois group and differential Galois theor, see, e.g. [6]. The idea of this work arose from an analsis of double pendulum, see Fig. 1. Its configuration space is T = S 1 S 1, and local coordinates are (φ 1, φ ) mod π. A double pendulum in a constant gravit field has regular as well as chaotic trajectories. However, a proof of its non-integrabilit for all values of parameters is still missing. Onl partial results are known, e.g., for small ratio of pendulums masses one can prove the non-integrabilit b means of Melnikov method [4]. On the other hand, a double pendulum without gravit is integrable. It has S 1 smmetr, and the Lagrange function depends on difference of angles onl. Introducing new variables θ 1 = φ 1 and θ = φ φ 1, we note that θ 1 is cclic variable, and the corresponding momentum is a missing first integral. The above eample suggests that it is reasonable to look for new integrable sstems among planar multiple-pendula in the absence of gravit when the S 1 smmetr is present. Solutions of such sstems give geodesic flows on product of S 1, or products of S 1 with R 1. For an analsis of such sstems we propose to use a combination of numerical and analtical methods. From the one side, Poincaré section give quickl insight into the dnamics. On the other hand, analtical methods allow to prove strictl the non-integrabilit. In this paper we consider: two joined pendula from which one is a spring pendulum, two spring pendula on a massless rod, triple flail pendulum and triple bar pendulum. All these sstems possess suspension points. One can also detach from the suspension point each of these sstems. In particular, one can consider freel moving chain of masses (detached multiple simple pendula), and free flail pendulum. We illustrate the behaviour of these sstems on Poicaré sections, and, for some of them, we prove their non-integrabilit. For the double spring pendulum the proof will be described in details. For others the main steps of the proofs are similar. In order to appl the Morales-Ramis method we need an effective tool which allows to determine the differential Galois group of linear equations. For considered sstems variational equations have two-dimensional subsstems of normal variational equations. The can be transformed into equivalent second order equations with rational coefficients. For such equations there eists an algorithm, the so-called the Kovacic algorithm θ 1 a 1 [7], determining its differential Galois groups effectivel. Double spring pendulum θ + 1 θ k( a) The geometr of this sstem is shown in Fig.. The mass is attached to on a spring Fig.. Double spring pendulum. with Young modulus k. Sstem has S 1 smmetr, and θ 1 is a cclic coordinate.
Chaotic Modeling and Simulation (CMSIM) 1: 57 67, 014 59 The corresponding momentum p 1 is a first integral. The reduced sstem has two degrees of freedom with coordinates (θ, ), and momenta (p, p 3 ). It depends on parameter c = p 1. The Poincaré cross sections of the reduced sstem shown in Fig. 3 suggest that the sstem is not integrable. The main problem is to prove that in fact (a) E=0.00005 (b) E=0.001 Fig. 3. The Poincaré sections for double spring pendulum. Parameters: = = a 1 = a = 1, k = 0.1, p 1 = c = 0 cross-plain = 1. the sstem is not integrable for a wide range of the parameters. In Appendi we prove the following theorem. Theore. Assume that a 1 0, and c = 0. Then the reduced sstem descended from double spring pendulum is non-integrable in the class of meromorphic functions of coordinates and momenta. 3 Two rigid spring pendula θ k ( a ) 1 1 1 k ( a) Fig. 4. Two rigid spring pendula. The geometr of the sstem is shown in Fig. 4. On a massless rod fied at one end we have two masses joined b a spring; the first mass is joined to fied point b another spring. As generalised coordinates angle θ and distances 1 and are used. Coordinate θ is a cclic variable and one can consider the reduced sstem depending on parameter c - value of momentum p 3 corresponding to θ. The Poincaré cross sections in Fig. 5 and in Fig. 6 show the compleit of the sstem. We are able to prove non-integrabilit onl under assumption k = 0. Theore. If k 1 c 0, and k = 0, then the reduced two rigid spring pendula sstem is non-integrable in the class of meromorphic functions of coordinates and momenta.
60 Szumiński Moreover, we can identif two integrable cases. For c = 0 the reduced Hamilton equations become linear equations with constant coefficients and the are solvable. For k 1 = k = 0 original Hamiltonian simplifies to H = 1 ( p 1 + p p ) 3 + 1 + and is integrable with two additional first integrals F 1 = p 3, F = p 1 p 1. (a) E=0.1, (b) E=0.. Fig. 5. The Poincaré sections for two rigid spring pendula. Parameters: = = a 1 = a = 1, k 1 = k = 1/10, p 3 = c = 1/10, cross-plain 1 = 0, p 1 > 0. (a) E=0.15, (b) E=4 Fig. 6. The Poincaré sections two rigid spring pendula. Parameters: = 1, = 3, k 1 = 0.1, k = 1.5, a 1 = a = 0, p 3 = c = 0.1, cross-plain 1 = 0, p 1 > 0
Chaotic Modeling and Simulation (CMSIM) 1: 57 67, 014 61 4 Triple flail pendulum In Fig. 7 the geometr of the sstem is shown. Here angle θ 1 is a cclic coordinate. Fiing value of the corresponding momentum p 1 = c R, we consider the reduced sstem with two degrees of freedom. Eamples of Poincaré sections for this sstem are shown in Fig. 8 and 9. For more plots and its interpretations see [11]. One can also prove that this sstem is not integrable, see [9]. θ 1 a1 θ 1 + θ a 3 a m3 m Fig. 7. Triple flail pendulum. θ 1 + θ + θ 3 (a) E=0.01, (b) E=0.01. Fig. 8. The Poincaré sections for flail pendulum. Parameters: = 1, = 3, m 3 =, a 1 = 1, a =, a 3 = 3, p 1 = c = 1, cross-plain θ = 0, p > 0. (a) E=0.0035, (b) E=0.0363. Fig. 9. The Poincaré sections for flail pendulum. Parameters: = 1, = m 3 =, a 1 =, a = a 3 = 1, p 1 = c = 1, cross-plain θ = 0, p > 0.
6 Szumiński Theorem 3. Assume that l 1 l l 3 m 3 0, and l = m 3 l 3. If either (i) 0, c 0, l l 3, or (ii) l = l 3, and c = 0, then the reduced flail sstem is not integrable in the class of meromorphic functions of coordinates and momenta. 5 Triple bar pendulum Triple bar pendulum consists of simple pendulum of mas and length a 1 to which is attached a rigid weightless rod of length d = d 1 + d. At the ends of the rod there are attached two simple pendula with masses, m 3, respectivel, see Fig.10. Like in previous cases fiing value for the first integral p 1 = c corresponding to cclic variable θ 1, we obtain the reduced Hamiltonian θ 1 a1 d 1 θ 1 + θ d a 3 a m 3 θ 1 + θ Fig. 10. Triple bar pendulum. depending onl on four variables (θ, θ 3, p, p 3 ). Therefore we are able to make Poincaré cross sections, see Fig. 11, and also to prove the following theorem [10].: + θ 3 (a) E=0.008, (b) E=0.009. Fig. 11. The Poincaré sections for bar pendulum. Parameters: = = 1, m 3 =, a 1 = 1, a =, a 3 = 1, d 1 = d = 1, p 1 = c = 1, cross-plain θ = 0, p > 0. Theorem 4. Assume that l l 3 m 3 0, and l = m 3 l 3, d 1 = d. If either (i) c 0, l l 3 or (ii) l = l 3, and c = 0, then the reduced triple bar sstem governed b Hamiltonian is not integrable in the class of meromorphic functions of coordinates and momenta.
Chaotic Modeling and Simulation (CMSIM) 1: 57 67, 014 63 6 Simple triple pendulum Problem of dnamics of a simple triple pendulum in the absence of gravit field was numericall analsed in [8]. Despite the fact that θ 1 is again cclic variable, and the corresponding momentum p 1 is constant, the Poincaré sections suggest that this sstem is also nonintegrable, see Fig.13. One can think, that the approach applied to the previous pendula can be used for this sstem. However, for this pendulum we onl found particular solutions that after reductions become equilibria and then the Morales-Ramis theor does not give an obstructions to the integrabilit. θ 1 + θ 1 θ a 1 a θ + θ + 1 θ 3 Fig. 1. Simple triple pendulum. a 3 m 3 (a) E=0.0097 (b) E=0.011 Fig. 13. The Poincaré sections for simple triple pendulum: =, = 1, m 3 = 1, a 1 =, a = a 3 = 1, p 1 = c = 1, cross-plain θ = 0, p > 0. 7 Chain of mass points θ ( 1 ) 1, a We consider a chain of n mass points in a plane. C The sstem has n + 1 degrees of freedom. Let a3 θ m r 3 i denote radius vectors of points in the center 3 of mass frame. Coordinates of these vectors θ ( 4 i, i ) can be epressed in terms of ( 1, 1 ) a 4 m 4 and relative angles θ i, i =,..., n. In the Fig. 14. Chain of mass points centre of mass frame we have m i r i = 0, thus we can epressed ( 1, 1 ) as a function of angles θ i. Lagrange and Hamilton functions do not depend on ( 1, 1 ), (ẋ 1, ẏ 1 ), and θ is a cclic variable thus the corresponding momentum p is a first integral. The reduced sstem has n degrees of freedom. Thus the m n a n θ n
64 Szumiński chain of n = 3 masses is integrable. Eamples of Poincaré sections for reduced sstem of n = 4 masses are given in Fig. 15. In the case when m 3 a 4 = a a non-trivial particular solution is known and non-integrabilit analsis is in progress. (a) E=0.04, (b) E=0.045. Fig. 15. The Poincaré sections for chain of 4 masses. Parameters: = m 3 = 1, =, m 4 = 3, a = 1, a 3 = 1, a 4 = 3, p = c = 3, cross-plain θ3 = 0, p3 > 0. 8 Unfied triple flail pendulum θ ( 1, 1 ) One can also unfi triple flail pendulum described C θ 4 in Sec.4, and allow to move it freel. θ a4 3 As the generalised coordinates we choose coordinates a 3 m 4 ( 1, 1 ) of the first mass, and relative angles, see Fig. 16. In the center of masses m 3 frame coordinates ( 1, 1 ), and their derivatives Fig. 16. Chain of mass points (ẋ 1, ẏ 1 ) disappear in Lagrange function, and θ is a cclic variable. Thus we can also consider reduced sstem depending on the value of momentum p = c corresponding to θ. Its Poincaré sections are presented in Fig. 17. One can also find a non-trivial particular solution when a 3 = a 4. The non-integrabilit analsis is in progress. 9 Open problems We proved non-integrabilit for some sstems but usuall onl for parameters that belong to a certain hpersurface in the space of parameters. It is an open question about their integrabilit when parameters do not belong to these hpersurfaces. Another problem is that for some sstems we know onl ver simple particular solutions that after reduction b one degree of freedom transform into equilibrium. There is a question how to find another particular solution for them. a
Chaotic Modeling and Simulation (CMSIM) 1: 57 67, 014 65 (a) E=0.4, (b) E=0.3. Fig. 17. The Poincaré sections for unfied flail pendulum. Parameters: =, = 1, m 3 =, m 4 = 1, a = a 3 = a 4 = 1, p = c = 3, cross-plain θ3 = 0, p3 > 0. 10 Appendi: Proof of non-integrabilit of the double spring pendulum, Theore Proof. The Hamiltonian of the reduced sstem for p 1 = c = 0 is equal to H = [ p + a 1 p (p cos θ + p 3 sin θ ) + a 1( (p + (p 3 +k ( a ) )) + (p cos θ + p 3 sin θ ) ] /(a 1 ), (1) and its Hamilton equations have particular solutions given b θ = p = 0, ẋ = p 3, ṗ 3 = k(a ). () We chose a solution on the level H(0,, 0, p 3 ) = E. Let [Θ, X, P, P 3 ] T be variations of [θ,, p, p 3 ] T. Then the variational equations along this particular solution are following Θ Ẋ P = P 3 p 3(a 1+) a 1 0 a 1 m1+m(a1+) a 1 m1m 0 0 0 0 1 p 3 0 p3(a1+) a 1 0 0 k 0 0 Θ X P, (3) P 3 where and p 3 satisf (). Equations for Θ and P form a subsstem of normal variational equations and can be rewritten as one second-order differential equation Θ + P Θ + QΘ = 0, Θ Θ, P = a 1p 3 (a 1 ( + ) + ) (a 1 + (a 1 + ) ), Q = k(a 1 + )( a ) a 1 a 1p 3 a 1 (a 1 + (a 1 + ) ). (4)
66 Szumiński The following change of independent variable t z = (t) + a 1, and then a change of dependent variable [ Θ = w ep 1 z ] p(ζ) dζ (5) z 0 transforms this equation into an equation with rational coefficients where w = r(z)w, r(z) = q(z) + 1 p (z) + 1 4 p(z), (6) p = [a 1 ( 4E + k(a + 3a 1 (a 1 z) + 5a (a 1 z)) + 3kz ) + z(a 1 a k + a 1 ( 4E + a 1 k(a 1 3z)) + kz 3 + a k(a 1 z)(4a 1 + z))]/[(a 1 + z ) (z a 1 )( E + kz (a 1 + a )k(z a 1 a ))], q = (a 1 (4E k((a 1 + a ) 3z)(a 1 + a z)) + k (a 1 + a z)z 3 ) a 1 ( e + k(a 1 + a z) )(z a 1 )(a 1 + z. ) We underline that both transformations do not change identit component of the differential Galois group, i.e. the identit components of differential Galois groups of equation (4) and (6) are the same. Differential Galois group of (6) can be obtained b the Kovacic algorithm [7]. It determines the possible closed forms of solutions of (6) and simultanousl its differential Galois group G. It is organized in four cases: (I) Eq. (6) has an eponential solution w = P ep[ ω], P C[z], ω C(z) and G is a triangular group, (II) (6) has solution w = ep[ ω], where ω is algebraic function of degree and G is the dihedral group, (III) all solutions of (6) are algebraic and G is a finite group and (IV) (6) has no closed-form solution and G = SL(, C). In cases (II) and (III) G has alwas Abelian identit component, in case (I) this component can be Abelian and in case (IV) it is not Abelian. Equation (6) related with our sstem can onl fall into cases (I) or (IV) because its degree of inifinit is 1, for definition of degree of infinit, see [7]. Moreover, one can show that there is no algebraic function ω of degree such that w = ep[ ω] satisfies (6) thus G = SL(, C) with non-abelian identit component and the necessar integrabilit condition is not satisfied. References 1.O. Pujol, J. P. Pérez, J. P. Ramis, C. Simó, S. Simon, and J. A. Weil. Swinging Atwood machine: eperimental and numerical results, and a theoretical stud. Phs. D, 39(1):1067-1081, 010..G. L. Baker and J. A. Blackburn. The Pendulum. A Case Stud in Phsics. Oford Universit Press, Oford, 005. 3.M. Gitterman.The Chaotic Pendulum. World Scientific, 010. 4.A. V. Ivanov. Problem of non-integrabilit for the double mathematical pendulum. In Progress in nonlinear science, Vol. 1 (Nizhn Novgorod), 001, pages 51 56. RAS, Inst. Appl. Phs., Nizhni Novgorod, 00.
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