Embedding the Kepler Problem as a Surface of Revolution
Tóm tắt
Solutions of the planar Kepler problem with fixed energy h determine geodesics of the corresponding Jacobi–Maupertuis metric. This is a Riemannian metric on ℝ2 if h ⩾ 0 or on a disk D ⊂ ℝ2 if h < 0. The metric is singular at the origin (the collision singularity) and also on the boundary of the disk when h < 0. The Kepler problem and the corresponding metric are invariant under rotations of the plane and it is natural to wonder whether the metric can be realized as a surface of revolution in ℝ3 or some other simple space. In this note, we use elementary methods to study the geometry of the Kepler metric and the embedding problem. Embeddings of the metrics with h ⩾ 0 as surfaces of revolution in ℝ3 are constructed explicitly but no such embedding exists for h < 0 due to a problem near the boundary of the disk. We prove a theorem showing that the same problem occurs for every analytic central force potential. Returning to the Kepler metric, we rule out embeddings in the three-sphere or hyperbolic space, but succeed in constructing an embedding in four-dimensional Minkowski spacetime. Indeed, there are many such embeddings.
Tài liệu tham khảo
Abraham, R. and Marsden, J.E., Foundations of Mechanics, 2nd ed., rev. and enlarg., Reading,Mass.: Benjamin/Cummings, 1978.
Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1997.
Besse, A. L., Manifolds All of Whose Geodesics Are Closed, Ergeb. Math. Grenzgeb. (3), vol. 93, Berlin: Springer, 1978.
Engman, M., A Note on Isometric Embeddings of Surfaces of Revolution, Amer. Math. Monthly, 2004, vol. 111, no. 3, pp. 251–255.
Hwang, A.D., A Symplectic Look at Surfaces of Revolution, Enseign. Math. (2), 2003, vol. 49, nos. 1–2, pp. 157–172.
Lawden, D. F., Elliptic Functions and Applications, Appl. Math. Sci., vol. 80, New York: Springer, 1989.
Montgomery, R., The Kepler Cone and Weak KAM, Preprint, 2018.
Pin, O. Ch., Curvature and Mechanics, Adv. Math., 1975, vol. 15, pp. 269–311.
Whittaker, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies, Cambridge: Cambridge Univ. Press, 1988.
Zoll, O., Über Flächen mit Scharen geschlossener geodätisher Linien, Math. Ann., 1903, vol. 57, pp. 108–133.