Abstract
We consider the motion of a particle on the surface generated by a small perturbation of the standard sphere. The key observation is that a trajectory of the particle has the shape of a coil, and one may qualitatively describe the turns of the latter as a precessing great circle of the sphere. Thus, we change the configuration space of the initial problem for the space of great circles on the sphere. The construction enables us to derive a subsidiary Hamiltonian system having the shape of equations for the top with a 4th order Hamiltonian. The subsidiary system provides a detailed asymptotic description of the particle’s motion in terms of graphs on the standard sphere.
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References
E. T. Whittaker, A Treatise on the Analytical Dynamics (Cambridge Univ., Cambridge, 1927), Chs. III, IV, and XIII.
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Golo, V.L., Sinitsyn, D.O. Asymptotic Hamiltonian reduction for the dynamics of a particle on a surface. Phys. Part. Nuclei Lett. 5, 278–281 (2008). https://doi.org/10.1134/S1547477108030291
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DOI: https://doi.org/10.1134/S1547477108030291