Spherical pendulum

The spherical pendulum, which may be used to model the bending states of HCP [7] and the pendular states of dipolar molecules in strong electric fields [8], is of... 

By the argument in Section IIB, the presence of a locally quadratic cylindrically symmetric barrier leads one to expect a characteristic distortion to the quantum lattice, similar to that in Fig. 1, which is confirmed in Fig. 7. The heavy lower lines show the relative equilibria and the point (0,1) is the critical point. The small points indicate the eigenvalues. The lower part of the diagram differs from that in Fig. 1, because the small amplitude oscillations of a spherical pendulum approximate those of a degenerate harmonic oscillator, rather than the fl-axis rotations of a bent molecule. Hence the good quantum number is... 

Figure 7. Eigenvalues of the spherical pendulum (points) joined by continuous lines of constant Vb and dashed lines of constant j.

The simplest way to model isomerization is to add a quadratic term to the spherical pendulum Hamiltonian [10, 24]. Thus... 

Figure 8. Evidence of quantum monodromy in the the spherical pendulum eigenvalue lattice. The heavy continuous lines are the relative equihbria, and the large dot indicates the critical point.

Elliptic Functions of the Worst Kind Non-linear Quantisation of the Classical Spherical Pendulum... 

In general, the 3D motion of the spherical pendulum is very complex, but for fixed initial angular displacements, values of the kinetic energy can be found (by trial and error) for which this motion is periodic. The approximation discussed above leads to the approximate description of the horizontal motion in terms of Mathieu functions, for which Flocquet analysis determines periodic solutions in terms of two integers k and n, which can be thought of as quantum numbers. 

Approximate frequency dependence of the spherical pendulum on the kinetic energy I2I... 

The spherical pendulum, which consists of a mass attached by a massless rigid rod to a frictionless universal joint, exhibits complicated motion combining vertical oscillations similar to those of the simple pendulum, whose motion is constrained to a vertical plane, with rotation in a horizontal plane. Chaos in this system was first observed over 100 years ago by Webster [2] and the details of the motion discussed at length by Whittaker [3] and Pars [4]. All aspects of its possible motion are covered by the case, when the mass is projected with a horizontal speed V in a horizontal direction perpendicular to the vertical plane containing the initial position of the pendulum when it makes some acute angle with the downward vertical direction. In many respects, the motion is similar to that of the symmetric top with one point fixed, which has been studied ad nauseum by many of the early heroes of quantum mechanics [5]. 

So how is the spherical pendulum quantised The answer is that its motion is generally chaotic, except for discrete values of the initial projection speed, for which it is periodic. The precise details of this phenomenon are difficult to get a handle on, because, although the vertical motion is always described by periodic elliptic functions, the horizontal motion is described in terms of Lame functions, which are very difficult to study and for which periodicity is difficult to diagnose. 

The motion of the spherical pendulum for general values of its initial vertical coordinate and horizontal velocity can be quite complex, as can be seen in Fig. 2 which shows the projection of three trajectories onto the horizontal plane, where... 

Fig. 2. The horizontal projection of three trajectories of the spherical pendulum with different initial horizontal velocities, increasing from left to right.

Eor the case of spherical pendulum, it is relevant to ask how the frequency of vertical oscillations changes as the initial horizontal velocity V increases from zero, in terms of the dimensionless kinetic energy K defined by... 

Fig. 4. The acceleration associated with the vertical coordinate for the simple pendulum (the lowest curve), the slow spherical pendulum (the middle curve) and the fast spherical pendulum (the upper curve).

To our knowledge, this is the first analytical estimate of the frequency of the spherical pendulum to have been published. 

There is actually a considerable literature on the approximate amplitude dependence of the simple pendulum [9-11], although this is the only one we know of which is based on approximating the physics rather than the mathematics. The formula is remarkably accurate even for initial angular displacements of 90° from the downward vertical. The corresponding equations for the spherical pendulum in generalised coordinates are altogether more complicated, very... 

APPROXIMATE FREQUENCY DEPENDENCE OF THE SPHERICAL PENDULUM ON THE KINETIC ENERGY... 

Figure 6 shows the approximate frequency of the spherical pendulum relative to the ideal value for the simple pendulum, plotted as a function of the dimensionless kinetic energy. 

Whittaker [3] has given an expression for the azimuthal angle of the spherical pendulum in terms of elliptic functions of the third kind, so that, not surprisingly, there has been very little numerical discussion of its motion. Instead, we see if our approximate theory can be used to obtain a simpler picture of the motion. 

Numerical simulations of the spherical pendulum for arbitrary values of K and W will usually reveal a very complicated, a periodic motion of the type shown in Fig. 2, but in some cases the motion is periodic. The theory can be found in Refs. [9,11], but is summarised here. Let r be any integer or simple fraction (such as 3/2, etc.). Then solutions of Mathieu s equation of the form... 

A viscous term predominates when the ratio of wavelength to particle size is so high that particles do not follow the liquid movement. The viscous losses term can be derived from Stokes equation for the effect of viscosity on a spherical pendulum swinging in a viscous liquid. 

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