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

  • Thread starter Thread starter pardesi
  • Start date Start date
  • Tags Tags
    Classical Doubt
AI Thread Summary
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.
pardesi
Messages
337
Reaction score
0
is it necessarily true that we have
\frac{\partial T}{\partial q}=0?
 
Physics news on Phys.org
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
 

Thread 'The Effect of Linear and Rotational Motion on Measured Weight'

Consider an extremely long and perfectly calibrated scale. A car with a mass of 1000 kg is placed on it, and the scale registers this weight accurately. Now, suppose the car begins to move, reaching very high speeds. Neglecting air resistance and rolling friction, if the car attains, for example, a velocity of 500 km/h, will the scale still indicate a weight corresponding to 1000 kg, or will the measured value decrease as a result of the motion? In a second scenario, imagine a person with a...
View full post »

Thread 'Coulomb gauge implies instantaneous radial electric field?'

Scalar and vector potentials in Coulomb gauge Assume Coulomb gauge so that $$\nabla \cdot \mathbf{A}=0.\tag{1}$$ The scalar potential ##\phi## is described by Poisson's equation $$\nabla^2 \phi = -\frac{\rho}{\varepsilon_0}\tag{2}$$ which has the instantaneous general solution given by $$\phi(\mathbf{r},t)=\frac{1}{4\pi\varepsilon_0}\int \frac{\rho(\mathbf{r}',t)}{|\mathbf{r}-\mathbf{r}'|}d^3r'.\tag{3}$$ In Coulomb gauge the vector potential ##\mathbf{A}## is given by...
View full post »
Thread 'Griffith, Electrodynamics, 4th Edition, Example 4.8. (First part)'

Thread 'Griffith, Electrodynamics, 4th Edition, Example 4.8. (First part)'

I am reading the Griffith, Electrodynamics book, 4th edition, Example 4.8 and stuck at some statements. It's little bit confused. > Example 4.8. Suppose the entire region below the plane ##z=0## in Fig. 4.28 is filled with uniform linear dielectric material of susceptibility ##\chi_e##. Calculate the force on a point charge ##q## situated a distance ##d## above the origin. Solution : The surface bound charge on the ##xy## plane is of opposite sign to ##q##, so the force will be...
View full post »

Similar threads

I Canonical transformation from canonical to kinetic momentum
Replies
7
Views
2K
I Question on derivation of a property of Poisson brackets
Replies
0
Views
1K
I Taylor expansion about lagrangian in noether
Replies
1
Views
1K
I Coulomb gauge implies instantaneous radial electric field?
Replies
5
Views
145
I Does Poisson's equation hold due to vector potential cancellation?
Replies
3
Views
145
I Lagrangian doesn't change when adding total time derivative
Replies
19
Views
4K
I Is ##p^k = \partial L / \partial \dot{x}^k## true for all ##L##'s?
Replies
2
Views
392
A Treating Velocity as Independent of position till the end in Lagrangian Mechanics
Replies
6
Views
2K
I Landau's inertial frame logic
Replies
1
Views
1K
I Momentum and Action: Understanding Lagrangian Mechanics
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top