The double pendulum

The purpose of this section is to illustrate the methods of Lagrangian and Hamiltonian mechanics with the help of a simple mechanical system the double pendulum. It is shown that although the equations of motion for this system look very simple, the double pendulum is a chaotic system. 

the pendulum phase space is four-dimensional. For the special case considered here, the equations of motion (3.2.6) agree with those derived by Shinbrot et al. (1992). 

For a given set of initial conditions Zh t = 0), k = 1,. ..,4, the solution of (3.2.6) defines a system trajectory. Since the energy is conserved, this trajectory winds in a three-dimensional sub-space of the four-dimensional phase space of the pendulum. It is not easy to imagine the motion in such a high-dimensional space, and we need to devise a visualization method that gives some clues as to the qualitative nature of the system trajectories. One particularly useful method, the method of the surface of 

The trajectory in Fig. 3.2 is obviously very compUcated. This demonstrates that the double pendulum, even in its simpUfied version studied here, is capable of exhibiting very complicated motion. 

We will now formulate the dynamics of the double pendulum within the Hamiltonian approach. The generaUzed momenta of the double pendulum are given by 

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Consider a hamionic oscillator connected to another hamionic oscillator (Fig. 5-13). Write the sum of forces on each mass, mi and m2. This is a classic problem in mechanics, closely related to the double pendulum (one pendulum suspended from another pendulum). 

For simplicity, take the specific case where ki = k2 = k. Write the matrix of force constants analogous to matrix (5-29). Diagonalize this matrix. What are the roots Discuss the motion of the double pendulum in contrast to two coupled, tethered masses (Fig. 5-1). 

Fig. 3.2. Projection of a phase-space trajectory of the double pendulum on the plane.

We construct the surface of section for the double pendulum in the following way. 

We are now ready to plot a Poincare surface of section for the double pendulum. We choose E = -2 and 19 different initial conditions defined... 

Fig. 3.3. Poincare section for the double pendulum, (a) E = -2, (b) E = 2. The full lines indicate the dynamically accessible regions.

Besides the one little regular island at 0 and — 2 there are undoubtedly more regular islands in the phase space of the double pendulum at E = 2. We missed them by our rather coarse choice of initial conditions. As indicated in Fig. 3.3(b), their total area in phase space is probably very small. Nevertheless, Fig. 3.3(b) illustrates an important feature of the phase space of most physical systems the phase space contains an intricate mixture of regular and chaotic regions. The system is said to exhibit a mixed phase space. 

As illustrated by Figs. 3.3(a) and (b), Poincare sections are a very powerful tool for the visual inspection and classification of the dynamics of a given Hamiltonian. The double pendulum illustrates that for autonomous systems with two degrees of fireedom a Poincare section can immediately suggest whether a given Hamiltonian allows for the existence of chaos or not. Moreover, it tells us the locations of chaotic and regular regions in phase space. 

The salient features of the dynamics of our model molecule are best exhibited with the help of Poincare sections that have already proved useful in the analysis of the double pendulum presented in Section 3.2. Fig. 4.7 shows the rpp projection of an x = 0 surface of section of a trajectory for a = 0.1, uq = 10 and E = 4 started at 0 = 0.957T, x = sin(0), y = 0, z = cos(0) and tj = 1.42. The resulting y-p Poincare section clearly shows chaotic features. This indicates that the classical dynamics of the skeleton of the model molecule is chaotic. But the most striking feature of the model molecule is its fully chaotic quantum dynamics. This is proved by Fig. 4.8, which shows the chaotic quantum fiow of the molecule on the southern hemisphere of the Bloch sphere. Fig. 4.8 was produced in the following way. First we defined the Poincar6 section by p = 0, dp/dt > 0. Then, we ran 40 trajectories in x,y,z,r],p) space for a = 0.1, Uo = 10 and E = Q starting at the 40 different initial conditions... 

The helium atom is an atomic physics example of a three-body problem. On the basis of Poincare s result we have to expect that the helium atom is classically chaotic. Richter and Wintgen (1990b) showed that this is indeed the case the helium atom exhibits a mixed phase space with intermingled regular and chaotic regions (see also Wintgen et al. (1993)). Thus, conceptually, the helium atom is a close relative of the double pendulum studied in Section 3.2. Given the classical chaoticity of the helium atom we are confronted with an important question How does chaos manifest itself in the helium atom ... 

The Double Pendulum An object of mass M is connected to two massless rods of length and as shown in the figure below (Fig. 1.11). The motion of the system is restricted to the vertical plane containing the two rods. By writing the lagrangian in terms of the two angles 6 and (p, determine the equations of motion for the object. 

H. Rippel and R. Jaacks, Performance data of the double pendulum interferometer, in Recent Aspects of Fourier Transform Spectroscopy, special issue of Mikrochim. Acta [Wien], G. GuelachviU, R. Kellner, and G. Zerbi, Eds., Springer-Verlag, Vienna, Austria 1988, Vol. m, p. 303. 

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