- #31
Kashmir
- 466
- 74
Done.vanhees71 said:
Cyclic coordinates in a two body central force problem
- Context: Graduate
- Thread starter Kashmir
- Start date
Click For Summary
Discussion Overview
The discussion revolves around the concept of cyclic coordinates in the context of a two-body central force problem, particularly focusing on the implications of spherical symmetry and the Lagrangian formulation. Participants explore the definitions and properties of cyclic coordinates, angular momentum, and the effects of coordinate choice on symmetry and conservation laws.
Discussion Character
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants note that the potential energy in a central force problem depends only on the radial distance, leading to spherical symmetry.
- It is argued that the angle coordinate ##\phi## is not included in the Lagrangian, thus it is a cyclic coordinate that generates angular momentum.
- Conversely, participants assert that the angle coordinate ##\theta## is included in the Lagrangian, making it non-cyclic.
- Some participants question the interpretation of Goldstein's statement regarding cyclic coordinates and suggest that ##\theta## should be cyclic due to its representation of rotation.
- There is a discussion about the fixed axes for rotations associated with ##\theta## and ##\phi##, with differing opinions on whether ##\theta## has a fixed axis.
- One participant expresses dissatisfaction with the 3rd edition of Goldstein, claiming it distorts the previous edition's content and affects the understanding of cyclic coordinates.
- Another participant argues that all components of angular momentum are conserved, not just one, and relates this to Noether's theorem and the invariance of the Lagrangian under arbitrary rotations.
- Some participants seek clarification on how to prove the conservation of total angular momentum based on the cyclic nature of coordinates.
Areas of Agreement / Disagreement
Participants generally disagree on the status of ##\theta## as a cyclic coordinate, with some asserting it is non-cyclic due to its presence in the Lagrangian, while others argue it should be cyclic based on symmetry arguments. The discussion remains unresolved regarding the implications of coordinate choice on angular momentum conservation.
Contextual Notes
The discussion highlights limitations in understanding due to the dependence on the definitions of cyclic coordinates and the implications of spherical symmetry. There are unresolved mathematical steps regarding the proof of total angular momentum conservation.
Similar threads
High School
Clarification of coordinate fictitious forces
- · Replies 29 ·
- · Replies 2 ·
- · Replies 39 ·
- · Replies 3 ·
Undergrad
Lagrangian for pendulum
- · Replies 11 ·
- · Replies 40 ·
- · Replies 11 ·
- · Replies 2 ·
- · Replies 17 ·
- · Replies 3 ·