Physics:Moser regularization

Short description: Method to avoid singularities in classical mechanics

In mathematical physics, the Moser regularization is a method to transform the flow of the Kepler problem, meaning the motion of celestial bodies, into a geodesic flow on the sphere using the Moser (regularization) map. Both are named after Jürgen Moser, who introduced them in 1970.

Moser map

Similar to the stereographic projection, the Moser map $\mathrm{\Phi }:T^{*}{\mathbb{ℝ}}^n\to T^{*}(S^n\setminus \{0\})$ is given by:

${\mathrm{\Phi }}_1(\mathbf{𝐪},\mathbf{𝐩}):=(\frac{2\mathbf{𝐩}}{p^2+1},\frac{2p^2}{p^2+1}-1);$

${\mathrm{\Phi }}_2(\mathbf{𝐪},\mathbf{𝐩}):=(-\frac{1}{2}(p^2+1)\mathbf{𝐪}+(\mathbf{𝐪}\cdot \mathbf{𝐩})\mathbf{𝐩},-\mathbf{𝐪}\cdot \mathbf{𝐩}).$

Properties

• The Moser map $\mathrm{\Phi }$ is a symplectomorphism.

See also

• Levi-Civita regularization, based on the real Hopf fibration
• Kustaanheimo–Stiefel regularization, based on the complex Hopf fibration
• Ligon–Schaaf regularization

Literature

• Moser, Jürgen (1970). "Regularization of Kepler's Problem and the Averaging Method on a Manifold" (in en). Comm. Pure Appl. Math. 23 (4): 609–636. doi:10.1002/cpa.3160230406. 
• Heckman, Gert; de Laat, Tim (2012). "On the Regularization of the Kepler Problem" (in en). J. Symplectic Geom. 10 (3): 463–473. doi:10.4310/JSG.2012.v10.n3.a5. 
• Frauenfelder, Urs; van Koert, Otto (2018). "Chapter 4.1 Moser regularization" (in en). The Restricted Three-Body Problem and Holomorphic Curves. doi:10.1007/978-3-319-72278-8. ISBN 978-3-319-72278-8.