The simple system under study possesses a rich and varied range of dynamical behaviour. For large amplitudes the motion is chaotic. Breitenberger and Mueller (1981) remark that `this simple system looks like a toy at best, but its behaviour is astonishingly complex, with many facets of more than academic lustre'. However, the concern here is the range of amplitudes where the motion is regular so that classical perturbation techniques yield meaningful results.
This work is the earliest comprehensive analysis of the elastic pendulum. Although the paper is frequently referenced by later authors, it is clear that, in some cases, they have not studied this work. Van der Burgh (1968) inaccurately describes the paper as `a mainly qualitative description'; in fact, his own paper contains little that is not already contained in Vitt and Gorelik [Indeed, the incorrect reference given by Van der Burgh to the Vitt and Gorelik paper is identical to that in Minorsky (1962, p.506), suggesting that he took the reference from there and not from the original paper.]. Breitenberger and Mueller (loc. cit.) note that this important paper has often been misquoted. Davidovic, et al. (1996) give a brief but accurate synopsis of its contents, and state even more strongly that the paper has been `more frequently quoted and misquoted than read by other authors'. I think this is a fair point; it is time the work was available in English.
The contents of the paper will now be summarised. Vitt and Gorelik (1933) consider the motion of an elastic pendulum confined to a plane, thus having two degrees of freedom. The authors set up the Lagrangian equations for the system, assuming the amplitude is sufficiently small that terms beyond cubic order can be ignored. They identify the linear vertical (springing) and horizontal (swinging) modes of the system. They concentrate on the special case where the vertical frequency is twice the horizontal frequency; in this case, each type of linear oscillation can induce the other through nonlinear interactions. Vertical oscillations can induce horizontal motion through parametric resonance, whereas horizontal or swinging motion can lead to vertical springing oscillations through direct resonant forcing.
In §2, periodic solutions are sought using the technique of secular perturbations. Two distinct solutions are found in which the trajectory of the bob is a parabola. For these particular solutions, the effect of the nonlinear interactions is to modify the frequency of the oscillations, but preserving the 2:1 ratio. The cup-like solutions, with concave-upward parabola, have frequency slightly depressed, the cap-like ones with a concave-downward trajectory have a somewhat augmented frequency. There is no energy transfer between the springing and swinging motion. These solutions are easily demonstrated in the physical system.
In §3, solutions which transfer energy back and forth between the swinging and springing motion are considered. A perturbed Hamiltonian is constructed, action-angle variables are introduced, and the Hamiltonian is averaged with respect to the fast variations, so that the lowest-order solution is immediate. An equation (equation (20) in the paper) is derived for the slowly-varying amplitude of the horizontal oscillation. The integral curves of the equation are illustrated, and the patterns of the trajectories in phase-space are depicted, clearly illustrating both the generic behaviour and important limiting cases. Curiously, although Eq. (20) is easily solved in terms of Jacobian elliptic functions, the authors make no mention of this. A qualitative description of the energy transfer follows, and an explicit formula for the modulation period is derived (equation [21] in the translation, un-numbered in the original). Again, this may be expressed as a complete elliptic integral of the first kind, though the authors do not say this.
In §4, the authors describe a series of experiments, and show that the theoretically calculated results are in good agreement with the observed behaviour of the physical system. They make no reference to its three-dimensional motion. This is surprising because, in their experiments, they cannot have failed to have noticed the remarkable propensity of the bob to deviate from the original swing plane, either in a precessing elliptical orbit, or in successive horizontal excursions with different azimuthal directions. The three-dimensional motion is discussed in Lynch (1999). Probably, Vitt and Gorelik did notice the interesting behaviour, but found it not directly relevant to their goal of providing a classical analogue for quantum resonance.
In the concluding section, the nonlinear interaction of the elastic pendulum is compared and contrasted to modal interactions in linear systems. One of the crucial differences is the dependence of the non-linear interactions on the initial conditions. The authors then discuss the original motivation for the work, the phenomenon of Fermi resonance, seen in the line spectrum of CO2 and in other molecules for which there is a frequency ratio close to 2:1. Although this is a quantum-mechanical effect, it is closely analogous to the classical phenomenon of nonlinear resonance seen in the swinging spring.
Current interest in the swinging spring arises from the rich variety of its solutions. For very small amplitudes, the motion is regular, and classical perturbation theory yields valid results. As the amplitude is increased, the regular motion breaks down into a chaotic regime which occupies more and more of phase space as the energy grows. However, for very large energies, a regular and predictable regime is re-established (Núñez-Yépez, et al., 1990). This can easily be understood: for very high energies, the system rotates rapidly around the point of suspension and is no longer libratory.
Of course, the chaotic regime was not considered by Vitt and Gorelik, as the relevant concepts were unavailable to them. However, recent studies have examined this behaviour in some detail. A large list of references may be found in Lynch (2000). That paper considers the elastic pendulum as a simple model for balance in the atmosphere. The concepts of filtering, initialization and the slow manifold, so important for atmospheric dynamics, can be introduced and lucidly illustrated in the context of the simple system. The swinging and springing oscillations act as analogues of the Rossby and gravity waves in the atmosphere.
Finally we may remark that Jin, et al., (1994) have modelled the El Niño phenomenon using arguments based on transition to chaos through a series of frequency-locked steps induced by non-linear resonance with the Earth's annual cycle. Their model produces results consistent with currently available data. Thus, the non-linear resonance observed in our simple mechanical system may provide the basis for a paradigm of the most important interannual variation in the ocean-atmosphere climate system.
Breitenberger, E and R D Mueller, 1981: The elastic pendulum: a nonlinear paradigm. J. Maths. Phys., 22, 1196-1210.
Burgh, A van der, 1968: On the asymptotic solutions of the differential equations of the elastic pendulum. J. Mécan, 7, 507-520.
Davidovic, D , B A Anicin and V M Babovic, 1996: The libration limits of the elastic pendulum. Am. J. Phys., 64, 338-342.
Jin, F-F, J.D. Neelin and M. Ghil, 1994: El Niño on the devil's staircase: annual subharmonic steps to chaos. Science, 264, 70-72.
Lynch, Peter, 1996: The Elastic Pendulum: a Simple Mechanical Model of Atmospheric Balance. Tech. Note No. 54, Met Éireann, Dublin, Ireland.
Lynch, Peter, 1999: Resonant Motions of the Swinging Spring. Tech. Note No. 56, Met Éireann, Dublin, Ireland. (PostScript version available from Peter.Lynch@met.ie )
Lynch, Peter, 2000: The Swinging Spring: a Simple Model for Atmospheric Balance. To appear in Proceedings of the Symposium on the Mathematics of Atmosphere-Ocean Dynamics. Isaac Newton Institute, June-December, 1996. Cambridge University Press.
Minorsky, N, 1962: Nonlinear Oscillations. Van Nostrand, Princeton. p506.
Núñez-Yépez, H N, A L Salas-Brito, C A Vargas and L Vicente, 1990: Onset of chaos in an extensible pendulum. Phys. Lett., A145, 101-105.
Vitt, A and G Gorelik, 1933: Oscillations of an Elastic Pendulum as an Example of the Oscillations of Two Parametrically Coupled Linear Systems. Zh. Tekh. Fiz. (J. Tech. Phys.) 3(2-3), 294-307.