Cyclic coordinates in a two body central force problem

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SUMMARY

The discussion centers on the concept of cyclic coordinates in the context of the two-body central force problem as presented in Goldstein's "Classical Mechanics" (3rd edition). It establishes that while the angle coordinate ##\phi## is cyclic and contributes to the conservation of angular momentum, the angle coordinate ##\theta## is not cyclic due to its presence in the Lagrangian. The conversation highlights the implications of spherical symmetry and the role of cyclic coordinates in determining conserved quantities, ultimately clarifying misconceptions about angular momentum conservation in spherical coordinates.

PREREQUISITES
  • Understanding of Lagrangian mechanics and the Euler-Lagrange equations.
  • Familiarity with the concepts of cyclic coordinates and conservation laws in classical mechanics.
  • Knowledge of spherical coordinates and their application in physics.
  • Basic grasp of angular momentum and its conservation in physical systems.
NEXT STEPS
  • Study the implications of Noether's theorem on conservation laws in classical mechanics.
  • Explore the derivation of angular momentum conservation in Cartesian coordinates.
  • Investigate the differences between Goldstein's 2nd and 3rd editions regarding cyclic coordinates.
  • Learn about integrable systems and the conditions under which complete sets of cyclic coordinates can be found.
USEFUL FOR

Students and professionals in physics, particularly those focusing on classical mechanics, theoretical physicists, and anyone interested in the mathematical foundations of motion under central forces.

  • #31
vanhees71 said:
By accident I posted a wrong (copyright-violating!) link from another thread first. I corrected my posting with the correct link pointing to the standard drawing of the spherical coordinates in Wikipedia. Please erase the wrong link in the quote and substitute it with the right one in order not to spread the bad link further!
Done.
 
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