Discussion Overview
The discussion revolves around solving dynamical systems using polar coordinates, specifically focusing on deriving formulas for the angular velocity \(\dot{\theta}\) and the radial velocity \(\dot{r}\). Participants explore various mathematical approaches and expressions related to these derivatives.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant mentions using the formula \(r\dot{r} = x\dot{x} + y\dot{y}\) to solve for \(\dot{r}\) and seeks a similar formula for \(\dot{\theta}\).
- Another participant suggests starting from \(\tan \theta = y/x\) and hints at a relationship involving \(\sec^2 \theta\) for deriving \(\dot{\theta}\).
- A participant expresses difficulty in formulating a nice expression for \(\dot{\theta}\) despite working from the tangent relationship.
- One participant provides an expression for \(\dot{\theta}\) as \(\theta' = (xy' - yx')/r\) but later questions its correctness based on dimensional analysis.
- Another participant proposes differentiating \(r^2 = x^2 + y^2\) to derive \(\dot{r}\) and \(\ddot{r}\) correctly.
- A participant shares their exploration of deriving equations related to Kepler's laws and expresses uncertainty about their previous calculations.
- One participant concludes with a refined expression for \(\dot{\theta}\) as \(\theta' = (xy' - yx')/r^2\) after correcting earlier mistakes.
Areas of Agreement / Disagreement
Participants express varying degrees of confidence in their derived formulas, with some acknowledging potential mistakes. There is no consensus on the correctness of the expressions for \(\dot{\theta}\) and \(\dot{r}\), and multiple competing views remain regarding the derivations.
Contextual Notes
Some participants indicate that their derivations may contain errors, particularly in dimensional analysis and the relationships between variables. The discussion includes attempts to connect the mathematical expressions to physical laws, such as Newton's laws and Kepler's laws, but these connections remain unresolved.
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