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The Motion of the Spherical Pendulum Subjected to a Dn Symmetric Perturbation∗

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The Motion of the Spherical Pendulum Subjected to a Dn Symmetric Perturbation∗ SIAM J. APPLIED DYNAMICAL SYSTEMS c 2005 Society for Industrial and Applied Mathematics Vol. 4, No. 4, pp. 1140–1158 The Motion of the Spherical Pendulum Subjected to a Dn Symmetric Perturbation∗ † ‡ Pascal Chossat and Nawaf M. Bou-Rabee Abstract. The motion of a spherical pendulum is characterized by the fact that all trajectories are relative periodic orbits with respect to its circle group of symmetry (invariance by rotations around the vertical axis). When the rotational symmetry is broken by some mechanical effect, more complicated, possibly chaotic behavior is expected. When, in particular, the symmetry reduces to the dihedral group Dn of symmetries of a regular n-gon, n>2, the motion itself undergoes dramatic changes even when the amplitude of oscillations is small, which we intend to explain in this paper. Numerical simulations confirm the validity of the theory and show evidence of new interesting effects when the amplitude of the oscillations is larger (symmetric chaos). Key words. spherical pendulum, elasticity, dihedral symmetry, reduction, nonlinear oscillations, chaos AMS subject classifications. 34Cxx, 37Jxx, 70Kxx, 74Hxx DOI. 10.1137/040616681 1. Introduction. The spherical pendulum is a pendulum with a fixed length and an extremity consisting of a massive bob, which is allowed to move freely on the sphere of radius . In the subsequent analysis we are interested in the case when the angle of deviation of the pendulum with respect to its stable state of rest is small. We can therefore replace the spherical coordinates which define the position of the pendulum (the configuration space of 2 this system is S ) by the Cartesian coordinates on the horizontal (x, y) plane. We therefore set z = − 2 − x2 − y2 for the vertical coordinate (Figure 1). This mechanical system with two degrees of freedom is a paradigm for integrable systems with symmetry [13]. It is very well-known that this system has two kinds of periodic motions: planar oscillations (in any vertical plane) and relative equilibria, which assume the form of circular motions in either one or the other direction of rotation around the vertical axis. These two kinds of periodic motion are “limiting” cases of quasi-periodic motions with two frequencies, where one frequency is related to the “oscillatory” behavior and the other one characterizes a “drift” around the vertical axis (precession). As is well known, when the ratio of these two frequencies is irrational, the motion is purely quasi-periodic (it never closes up), while if it is rational, the motion is in fact periodic (with, however, a period which can be very long). In experiments, it can be virtually impossible to distinguish between the truly quasi- periodic motions and the periodic ones. This behavior is a fundamental consequence of the ∗Received by the editors October 10, 2004; accepted for publication (in revised form) by P. Holmes June 22, 2005; published electronically November 22, 2005. http://www.siam.org/journals/siads/4-4/61668.html †Institut Non Lin´eaire de Nice (CNRS UMR 6618), Sophia Antipolis, 06560 Valbonne, France, and Embassy of France in New Delhi, India ([email protected]). ‡Applied & Computational Mathematics, California Institute of Technology, Pasadena, CA 91125 (nawaf@acm. caltech.edu). This author was supported by a US DOE Computational Science Graduate Fellowship. 1140 PENDULUM WITH Dn SYMMETRY 1141 z y 0 x length l g m Figure 1. The spherical pendulum. invariance of the energy (Hamiltonian function) through any rotation around the vertical axis (and reflection through any plane containing that axis). The symmetry group of the system is isomorphic to O(2), the group of 2 × 2 orthogonal matrices. The presence of rotational symmetry implies the existence of a first integral (conservation of angular momentum), which allows us to reduce the system to an integrable one. This will be recalled in section 3. The following observation was made on a physical pendulum which can be found on the premises of the French embassy in New Delhi. It is part of a sculpture by Lalanne, and it consists of a long rod in bronze held (welded) in the hand of a centaur and terminated by a massive bob (Figure 2). The rod has hexagonal cross section. When this pendulum is moved away from its vertical position (state of rest), it oscillates in a vertical plane as expected, but after a while it starts rotating in one direction. Then after some other time, the rotational behavior decreases until the pendulum oscillates again in a vertical plane, different from the initial one. The whole sequence is then repeated, with the difference that now the pendulum rotates in the direction opposite to the previous one. This behavior repeats itself as long as there is enough energy to sustain a discernable motion. In term of coordinates in the (x, y) plane, the trajectories therefore follow the sequence which is schematized in Figure 3. This behavior is clearly not compatible with the classical model of a spherical pendulum. What is the origin of this discrepancy? In Lalanne’s pendulum, the rigid rod is clamped at its upper end (in the hand of the centaur); therefore the motion is due to elastic deformations. Nevertheless, if this system did satisfy the same O(2) invariance property as the spherical pendulum, we would expect that the dynamics would be similar and, in particular, that it would not exhibit alternating rotating oscillations. We mentioned that the rod had hexagonal cross section, which means that it has D6 symmetry, where D6 is the symmetry group of a regular hexagon. Our aim is to show how this fact can explain the unexpected motion of Lalanne’s pendulum. In the classical theory of rods it is assumed that (i) planar cross sections are transformed to planar cross sections (Bernoulli hypothesis) and (ii) Kirchhoff–Hooke linear constitutive equa- tions hold. The former assumption allows for a drastic reduction of the number of kinematic 1142 PASCAL CHOSSAT AND NAWAF M. BOU-RABEE Figure 2. The sculpture by Lalanne. The pendulum is held in the right hand of the centaur. Only small oscillations are permitted. Figure 3. The motion of the perturbed spherical pendulum projected on the horizontal plane. variables characterizing the deformations. In this case, however, since the principal moments of inertia of a regular n-gonal cross section (n ≥ 3) are equal, the stored (elastic) energy function W has the same form as if the cross sections were O(2) invariant (see [3] for details). Such models are therefore not suitable for studying Lalanne’s pendulum, and higher order effects should be taken into account. Moreover, it was shown in [6] that terms of order at least n in the Taylor expansion of W are required in order to distinguish between Dn and O(2) symmetry. The precise form of the Dn invariant terms will be recalled in section 4. Following [3], where the compression of an inextensible and unshearable rod with Dn PENDULUM WITH Dn SYMMETRY 1143 symmetric cross section was studied, we may assume that a nonlinear, higher order moment/ curvature (or stress/strain) relationship holds. Another possibility is to relax the hypothesis that planar cross sections are transformed to planar ones, thereby introducing a more compli- cated geometric description of the deformation (see also [7]). Whatever hypothesis is adopted to ensure Dn-symmetry of the model, we expect that it should lead to the same qualitative results. In this paper we do not intend to derive and solve an exact model of Lalanne’s pendulum. We shall instead exploit the fact that this system is close to being a spherical pendulum, thanks to the “large” aspect ratio of the rod (its length amounts to about 200 times its mean diameter) and to the presence of a massive bob at its lower end. We will then make the ansatz that the potential energy of the system can be written as the sum of the gravitational potential and of a D6 (or more generally, Dn, n ≥ 3) symmetric perturbation due to the elastic deformation, and we will show that such a system enjoys the same dynamics as the one observed with Lalanne’s pendulum. There are various ways in which an experiment can be mounted in order to precisely satisfy this hypothesis. For example, n identical springs can be attached to the rod at some distance 0 from its upper extremity and fixed to the “ceiling” at which the pendulum is suspended, and be equally distributed around the vertical axis. A similar idea was exposed in [7] for modeling elastic systems with discrete symmetry. Another possibility is to place the pendulum, the bob being a magnetic material (iron), into a magnetic field with Dn-symmetry (see [11] for a related study). In each of these experiments, the setup can be arranged so that its nonisotropic components act as a perturbation of the spherical pendulum. The potential energy V of the system can therefore be expressed as follows (under the above hypothesis on the amplitude of the oscillations): 2 2 2 (1) V (x, y)=mg − − x − y + V1(x, y), where g is the acceleration of gravity, m is the mass of the bob, and V1 is a Dn invariant smooth function. According to the above commentaries, this assumption implies that the Taylor expansion of V1 contains terms of order at least n (see section 4 below). The number characterizes the “strength” of the perturbation potential. It depends on the mechanical system which is considered; for example, in the simulations of section 5 below, = ke is proportional to the stiffness k of the springs. We can therefore write the Hamiltonian function of the system as 2 m 2 2 (xx˙ + yy˙) 2 2 2 (2) H (x, x,˙ y, y˙)= x˙ +˙y + + mg − − x − y + V1(x, y).
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