Monodromy in the resonant swing spring

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Abstract

In this paper, it is shown that an integrable approximation of the spring pendulum, when tuned to be in 1:1:2 resonance, has monodromy. The stepwise precession angle of the swing plane of the resonant spring pendulum is shown to be a rotation number of the integrable approximation. Due to the monodromy, this rotation number is not a globally defined function of the integrals. In fact at lowest order it is given by arg(χ+iλ), where χ and λ are functions of the integrals. The resonant swing spring is therefore a system where monodromy has easily observed physical consequences.

Introduction

The spring pendulum, or swing spring, is one of the simplest possible mechanical systems. It is a spring with one end fixed, a mass attached at the other end, and a constant vertical gravitation field acting upon it. The name swing spring comes from the fact that, for appropriate initial conditions, the mass can either swing like a pendulum or bounce up and down like a spring. However, if in linear approximation near the equilibrium, the frequencies of the swinging and springing motion are in resonance, then these two types of motions are intricately intertwined. In particular, the following motion is easily observed: starting with a weakly unstable vertical springing motion, the system evolves into a planar swinging motion. This swinging motion is transient and the system returns to its original springing motion. This cycle then repeats. Lynch [6] observed that the orientation of the swing plane typically changes from one swinging phase to the next. Moreover, the angle between the swing planes of any two successive swinging phases is constant. However, the angle between the swing planes depends on initial conditions. He called this phenomenon the stepwise precession of the swing plane of the swing spring. It is this phenomenon that we are going to explain both qualitatively and quantitatively.

The swing spring has a long history that is well described in [7]. The earliest comprehensive work on the planar spring pendulum is [12]. This paper gives a classical treatment of the 1:2 resonance using action-angle variables. It is written in the spirit of the old quantum mechanics and was actually motivated by the Fermi 1:2 resonance in CO2. The advent of modern quantum mechanics seems to have made this type of analysis old-fashioned if not forgotten. As Lynch [7] points out most of the previous work is only concerned with the planar spring pendulum, so that the stepwise precession of the swing plane cannot be found in the old literature. Some progress on the three-dimensional system was made in [6]. After that Holm and Lynch [5] found that the system can be approximated by the 3-wave system and derived a differential equation for the angle of the swing plane. This was done using “pattern evocation in shape space” [8]. We show that the equation found by Holm and Lynch is exact and is nothing but the equation for the evolution of one of the angles of the action-angle coordinates of an integrable approximation to the resonant swing spring. We then trace the origin of the stepwise precession to the existence of Hamiltonian monodromy in this integrable approximation.

Hamiltonian monodromy is an obstruction to the existence of global action variables, which was first described in [3] (see also [2]). It generically appears around an equilibrium point of an integrable two degree of freedom Hamiltonian system whose linearization has a complex quartuple of eigenvalues [9], [13]. Such an equilibrium point is called a focus–focus point. The integrable approximation of the resonant swing spring has three degrees of freedom. But after reduction of a symmetry, one obtains a two degree of freedom system with a focus–focus point as a relative equilibrium. Physically, this equilibrium corresponds to the pure springing motion of the system.

For the purpose of the present paper the most important consequence of monodromy is that the rotation number of invariant tori, that is, the ratio of their frequencies, near the singularity is not a single valued function. Our main result is that the stepwise precession of the swing plane is given by such a rotation number W, which explicitly has the form: 2πW=arg(χ+iλ)+O(χ22).Here χ is a scaled nonlinear energy and λ a scaled angular momentum. Hence both are simple functions of the integrals of the system, and χ, λ→0 near the equilibrium of the swing spring. The amazing feature of (1) is that it is multi-valued and thus not differentiable at the origin. The multi-valuedness means that no matter how small the initial perturbation from the equilibrium is, one can always obtain all possible values for W. In general, such multi-valuedness has been described for integrable foliations near focus–focus points by Vũ Ngoc [11]. In some sense our result is a special case of his. However, he did not study the influence of the Hamiltonian, but only the foliation.

The paper is organized as follows. We briefly recall the physics of the swing spring in Section 2, and some basic facts about the harmonic oscillator in Section 3. Then we derive an integrable approximation in Section 4 which is valid near the resonant equilibrium point. This integrable system is reduced to a one degree of freedom system in Section 5, and the geometry of its energy momentum map is described in Section 6. The dynamics of the swing angle ϑ is analyzed in Section 7. In Section 8 we show that there is monodromy. Finally we obtain Eq. (1) for the rotation number W, by approximating an elliptic integral in Section 9.

Section snippets

The physics of the swing spring

The spring pendulum is a point particle r=(x,y,z) in R3 of mass m attached to a spring which moves in a constant vertical gravitation field. Its potential energy is Ṽ(r)=mgz+12k(ℓ0−∥r∥)2,where ∥r∥=x2+y2+z2. The unstretched length of the spring with spring constant k is ℓ0 and g is the constant of gravity. The motion of the swing spring is governed by Newton’s equations: mr̈=−gradṼ(r).The system is in equilibrium when the forces of gravity and the spring balance, that is, when gradṼ(r)=0.Thus

Prelude: the 1:1 resonance

In order to understand the swing plane angle and to motivate a coordinate change needed in the following sections, we recall some facts about the isotropic harmonic oscillator with two degrees of freedom. Its Hamiltonian is the xy-part of H2 (10), namely, Hxy=12(px2+py2+x2+y2).The Hamiltonian Hxy is an action, because it has a periodic flow with minimal period 2π for every initial condition with positive energy.1 The system (11) is

An approximating integrable system

The first approximation to the Hamiltonian (6) of the swing spring is the truncation of its Taylor expansion at the equilibrium, namely, (10) with ν=2. The Hamiltonian of the resonant swing spring in its cubic approximation is Ĥ=H2+V3=12(p2x+p2y+p2z)+12(x2+y2+4z2)−μ̃(x2+y2)z.By rescaling the coordinates and changing the time scale, we may consider μ̃ to be a small parameter, which measures the distance to the origin.

The second approximation is the first order normal form of Ĥ (17). In our

Reduction to one degree of freedom

The truncated averaged resonant swing spring (H,TR3) is Liouville integrable, and has two S1 symmetries. In the classical approach, a “coordinate system” is introduced that has the corresponding conserved quantities as momenta, together with conjugate angles. This is very efficient, but this “coordinate system” has a singularity exactly at the pure springing motion we want to study. This is why we use singular reduction [2] in order to obtain a one degree of freedom system that properly

Critical values of the energy momentum map

The truncated and averaged swing spring (H,TR3) is Liouville integrable and has three degrees of freedom. The Liouville–Arnold theorem implies that almost all initial conditions lead to dynamics on three-tori. At special points where the constants of motion are not independent, the motion takes place on a lower-dimensional space. An example of this occurs for relative equilibria in which only rotation about the z-axis takes place, or equilibria, with no motion at all. All this information is

Reconstruction: the swing plane angle

The most striking feature of the resonant swing spring is the phenomenon of stepwise precession of the swing plane. In Section 3, we have seen that the angle ϑ (15) gives the orientation of the ellipse in the xy-plane of the isotropic harmonic oscillator. For the full Hamiltonian H (21c) of the resonant swing spring we expect this ellipse to change orientation and eccentricity. However, its area is still constant, because the angular momentum L is conserved. When the solutions of the swing

Monodromy and rotation numbers

Let us now compute the monodromy matrix for the swing spring system in a way similar to that used in [2]. As we have shown, the reduced system is everywhere periodic except for the homoclinic orbits (whose points are mapped onto the thread by the energy momentum map). To obtain an expression of the missing action function we will focus on the Θi, the change of θi over one period of the motion of ρ3. To do so we first express such change as the time integral of the time derivative of θi and

Analysis of the swing angle

In this section, we examine more closely the change of the swing angle ϑ (54) over one period of the reduced motion. This is the stepwise precession angle: 2πW=−Θ=−122−Θ1)=−12γτ2−τ1.We prove the following proposition.

Proposition 5

The rotation number describing the angle of stepwise precession for the integrable approximation of the resonant swing spring is given by2πW=−arg(χ+iλ)+O(χ22).

Eq. (72) shows that the rotation number W is a multi-valued function of the parameters h, j1 and j2. This therefore

Acknowledgements

This work was supported by the EU network HPRN-CT-2000-0113 MASIE—Mechanics and Symmetry in Europe. A first draft of this paper was written at University of Warwick during the Symposium on Geometric Mechanics and Symmetry 2002. The authors would like to thank the Math Research Center at Warwick for its hospitality. HRD was partially supported by EPSRC grant GR/R44911/01.

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