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

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The discussion centers on the validity of the equation \(\frac{\partial T}{\partial q} = 0\) in classical mechanics. It argues that kinetic energy \(T\) can vary with changes in generalized coordinates \(q\), particularly when considering angular velocity. An example is provided using polar coordinates, where the kinetic energy depends on both radial and angular components, demonstrating that the derivative is not zero. This contradicts the claim made by the professor regarding the general applicability of the equation. The conclusion emphasizes that the derivative can indeed be non-zero under certain conditions.
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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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