Is \(\frac{\partial T}{\partial q} = 0\) Always True in Classical Mechanics?

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Discussion Overview

The discussion centers around the question of whether the partial derivative of kinetic energy \( T \) with respect to a generalized coordinate \( q \) is always zero in classical mechanics. Participants explore the conditions under which this derivative may or may not hold true, considering various scenarios and examples.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions the generality of the statement \(\frac{\partial T}{\partial q} = 0\), suggesting that kinetic energy can vary with changes in generalized coordinates.
  • Another participant provides an example involving angular velocity, arguing that if the generalized coordinate \( q \) represents angular velocity and varies while other coordinates remain constant, then the kinetic energy \( T \) changes, indicating that the derivative is not zero.
  • A third participant references a specific case in polar coordinates, asserting that the derivative is not zero for a single particle's motion, supporting the argument against the claim.
  • One participant expresses agreement with the previous points, indicating that they contradict a professor's assertion regarding the derivative being zero.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are multiple competing views regarding the validity of the statement \(\frac{\partial T}{\partial q} = 0\). Some argue that it is not generally true, while others may imply that it could be under specific conditions.

Contextual Notes

The discussion highlights the dependence on the definitions of generalized coordinates and the specific scenarios being considered, as well as the potential for different interpretations in classical mechanics.

pardesi
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is it necessarily true that we have
\frac{\partial T}{\partial q}=0?
 
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I imagine kinetic energy often varies if a generalized coordinate of the system varies. I don't see why that derivative would be 0 in general.

For instance, if the generalized coordinate q describes the angular velocity of a body about some axis, and q varies while holding all other generalized coordinates constant, then the kinetic energy T of the system varies, and that derivative is non-0... right?
 
It's trivially not true for motion of one particle using polar coordinates (Goldstein, p. 26).

T=\frac{1}{2}m (\dot{r}^{2} + (r\dot{\theta})^{2})
 
exactly that was my point of contradiction to my profs claim
 

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