Abstract
It is customary to describe the behaviour and stability of oscillators with the help of phase space representation. However, two-dimensional (2D) oscillators like the Foucault pendulum call for a 4D phase space that is not simple to visualize. Applying celestial body perturbation theory to the Foucault pendulum in his doctor dissertation, Nobel laureate Kamerlingh Onnes showed that the essential features of a Foucault pendulum are its inherent circular and linear anisotropies. A spherical differential 2D sub-space can be defined, where the group of the points of a spherical surface with respect to the operation rotation about a diametral axis is isomorphic with the group of sequential states of oscillation of a 2D pendulum with respect to the operation translation in time. Any Foucault pendulum is then characterized by two elliptical eigenstates which are represented by the poles of that rotation axis on the so-called anisosphere. Such poles play the role of attractor/repellor when “dichroic” damping is present. Moreover, they move drastically within a meridian plane when nonlinear restoring torque giving rise to Airy precession occurs. The concept of anisosphere constitutes a very powerful tool for analysing and optimizing actual Foucault pendulum implementations. That feature is illustrated by a numerical model.
Highlights
Textbooks and research articles usually represent the behaviour and stability features of onedimensional (1D) pendulums with the help of θ-θphase space [1,2]
Assuming that an experiment such as A-II could be pursued without time restrictions, the anisosphere model states that the trace of the pendulum representing point describing the phase curve would spiral around the anisosphere from the slow elliptic eigenstate N until the fast elliptic eigenstate M, as would the trace of a knife peeling an apple in one shot from the pedicel all around until the remnants of calyx
Linear oscillators In the above sections, the anisosphere model has been applied to a linearly anisotropic Foucault pendulum operating in the Northern earth hemisphere
Summary
Textbooks and research articles usually represent the behaviour and stability features of onedimensional (1D) pendulums with the help of θ-θphase space [1,2]. Considering that linear anisotropy due to suspension asymmetry results in slightly different effective pendulum lengths for two orthogonal azimuths, and that the effective length for a given azimuth is the radius of curvature of the potential well cross-section containing that azimuth, linear anisotropy must be associated with a slight hyperboloidal deformation of the pre-existing rotation symmetric gravitational potential well. Airy first mentions that when the elliptical orbit of the pendulum is almost a circle (b ≈ a) [25], the orbit major axis undergoes a rotation (precession) in the same sense as angular velocity ω of the bob travel along the ellipse, at the particular rate ψ H. The same result was obtained by Deakin [28] from dimensional analysis and symmetry considerations
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