I Kinetic energy depends on ##\theta## but this argument says otherwise

[Kashmir]

A free particle with coordinates as shown has kinetic energy $T = \frac{1}{2}m\left(\dot r^2 + r^2\dot\theta^2 + r^2\sin^2\theta\dot\phi^2\right)$
So we see $T$ depends on $\theta$.

Now suppose we rotate our coordinate system such that only one coordinate $\theta$ changes from $\theta$ to
$\theta'$ and fix it there as is shown in figure below. In this coordinate system the kinetic energy should be the same as before since the kinetic energy should be same in all inertial frames.

However if we substitue values of $r, \theta',\phi$ in the formula for $T$ we will get a different value of $T$ hence a contradiction.

Can anyone please tell me what is wrong. Thank you.

 

[vanhees71]

I do not understand what your specific change of coordinates is. Can you give a formula?

Of course, the Hamilton principle is covariant under arbitrary changes of coordinates (diffeomorphisms). So no matter how you choose your coordinates you always describe the same physical system.

 

[PeroK]

However if we substitue values of $r, \theta',\phi$ in the formula for $T$ we will get a different value of $T$ hence a contradiction.

Can anyone please tell me what is wrong. Thank you.

You didn't actually do any calculations to back up your claim! Let's see the coordinate transformation you have in mind.

 

[Kashmir]

I do not understand what your specific change of coordinates is. Can you give a formula?

Of course, the Hamilton principle is covariant under arbitrary changes of coordinates (diffeomorphisms). So no matter how you choose your coordinates you always describe the same physical system.

I don't have a formula. I'm just imagining that we rotate our coordinate system by a finite rotation with the origin fixed where it was initially in such a manner that all other variables except $\theta$ remain the same.

I hope I'm clear?

 

[Kashmir]

You didn't actually do any calculations to back up your claim! Let's see the coordinate transformation you have in mind.

I am not able to write them in mathematics.

 

[Lnewqban]

I don't have a formula. I'm just imagining that we rotate our coordinate system by a finite rotation with the origin fixed where it was initially in such a manner that all other variables except $\theta$ remain the same.

I hope I'm clear?

When you rotate the coordinate system, keeping angle theta the same, the projection on x-y plane changes; therefore, at least one of the two other parameters must change.

 

[PeroK]

I am not able to write them in mathematics.

If you let $\theta = \theta' + \alpha$, then the formula for kinetic energy becomes: $$T = \frac 1 2 m(\dot r^2 + r^2\dot\theta'^2 + r^2\sin^2(\theta' + \alpha)\dot\phi^2)$$ You can't change $\theta$ and keep the same formula as before, if that's what you are thinking.

 

[Kashmir]

If you let $\theta = \theta' + \alpha$, then the formula for kinetic energy becomes: $$T = \frac 1 2 m(\dot r^2 + r^2\dot\theta^2 + r^2\sin^2(\theta' + \alpha)\dot\phi^2)$$ You can't change $\theta$ and keep the same formula as before, if that's what you are thinking.

If you let $\theta = \theta' + \alpha$, then the formula for kinetic energy becomes: $$T = \frac 1 2 m(\dot r^2 + r^2\dot\theta^2 + r^2\sin^2(\theta' + \alpha)\dot\phi^2)$$ You can't change $\theta$ and keep the same formula as before, if that's what you are thinking.

Do you mean the correct formula is $T = \frac 1 2 m(\dot r^2 + r^2\dot\theta^2 + r^2\sin^2(\theta' + \alpha)\dot\phi^2)$ ?

 

[weirdoguy]

since the kinetic energy should be same in all inertial frames.

... frames that are at rest with respect to each other. Without this condition your statement is not true.

 

[Kashmir]

... frames that are at rest with respect to each other. Without this condition your statement is not true.

Yes the two are at rest

 

[PeroK]

Do you mean the correct formula is $T = \frac 1 2 m(\dot r^2 + r^2\dot\theta^2 + r^2\sin^2(\theta' + \alpha)\dot\phi^2)$ ?

Yes, if you change one of your coordinates, you must do a proper substitution in the formula. Note that with the original $\theta$ we have: $$x = r\sin \theta \cos \phi, \ y = r \sin \theta \sin \phi, \ z = r \cos \theta$$ And plugging those into $$T = \frac 1 2 m(\dot x^2 + \dot y^2 + \dot z^2)$$ gives you the formula for $T$ in terms of $r, \theta, \phi$. That's where that formula comes from. If you choose your angles $\theta$ and/or $\phi$ differently, then you get a different formula for $T$ in terms of your new coordinates.

 

[Kashmir]

Yes, if you change one of your coordinates, you must do a proper substitution in the formula. Note that with the original $\theta$ we have: $$x = r\sin \theta \cos \phi, \ y = r \sin \theta \sin \phi, \ z = r \cos \theta$$ And plugging those into $$T = \frac 1 2 m(\dot x^2 + \dot y^2 + \dot z^2)$$ gives you the formula for $T$ in terms of $r, \theta, \phi$. That's where that formula comes from. If you choose your angles $\theta$ and/or $\phi$ differently, then you get a different formula for $T$ in terms of your new coordinates.

Suppose I'm in the new system.
I'll still write $$x = r\sin \theta' \cos \phi, \ y = r \sin \theta' \sin \phi, \ z = r \cos \theta'$$ leading to
$$T = \frac{1}{2}m\left(\dot r^2 + r^2\dot\theta'^2 + r^2\sin^2\theta'\dot\phi^2\right)$$

---

 

[PeroK]

Suppose I'm in the new system.
I'll still write $$x = r\sin \theta' \cos \phi, \ y = r \sin \theta' \sin \phi, \ z = r \cos \theta'$$ leading to
$$T = \frac{1}{2}m\left(\dot r^2 + r^2\dot\theta'^2 + r^2\sin^2\theta'\dot\phi^2\right)$$

---

Okay, but then $\theta' = \theta$. That's just regular spherical coordinates.

 

[Kashmir]

Okay, but then $\theta' = \theta$. That's just regular spherical coordinates.

I'll add a figure, maybe I'm not clear. Please bear with me. Thank you.

 

[PeroK]

Suppose I'm in the new system.
I'll still write $$x = r\sin \theta' \cos \phi, \ y = r \sin \theta' \sin \phi, \ z = r \cos \theta'$$

---

The only way those three equations can hold is if you have the usual spherical coordinates, with $\theta$ measured from the positive x-axis. If you do anything different, you can't demand the same equations.

 

[Kashmir]

The only way those three equations can hold is if you have the usual spherical coordinates, with $\theta$ measured from the positive x-axis. If you do anything different, you can't demand the same equations.

Suppose I've rotated my coordinate system such that only $\theta$ changed to $\theta'$. This will look like the figure shown

Isn't it correct till now?

 

[PeroK]

Suppose I've rotated my coordinate system such that only $\theta$ changed to $\theta'$. This will look like the figure shown View attachment 290785
Isn't it correct till now?

I don't see any rotation there. I just see spherical coordinates.

 

[Kashmir]

I don't see any rotation there. I just see spherical coordinates.

I've just placed it upright,

 

[Kashmir]

I don't see any rotation there. I just see spherical coordinates.

 

[PeroK]

View attachment 290787

So, you have a new $z$ axis?

 

[Kashmir]

So, you have a new $z$ axis?

Yes ,even x, y will be different. I've shown the two systems in this figure below. S2 is the rotated one

 

[kuruman]

Perhaps this might help. What you can always write is $$T = \frac{1}{2}m\left(\dot r^2 + r^2\dot\theta^2 + \rho^2\dot\phi^2\right)$$where $\rho$ is the projection of $r$ on the $xy$-plane. In this form, you can see the "decomposition" of the kinetic energy into the sum of three terms, radial, meridional (if that's what it's called), and azimuthal.

 

[PeroK]

With your new $z$ axis, you no longer have $$T = \frac 1 2 m (\dot x^2 + \dot y^2 + \dot z^2)$$That is only valid for mutually orthogonal Cartesian axes. You can't change something like the z-axis without changing the formulas that involve the z-coordinate.

 

[Kashmir]

With your new $z$ axis, you no longer have $$T = \frac 1 2 m (\dot x^2 + \dot y^2 + \dot z^2)$$That is only valid for mutually orthogonal Cartesian axes. You can't change something like the z-axis without changing the formulas that involve the z-coordinate.

I've rotated the whole set of three axis, keeping them orthogonal during the rotation. The rotation is such that only theta changes. The mutually perpendicular axis are shown here with the only change in the coordinates of the particle being in $\theta$

 

[PeroK]

I've rotated the whole set of three axis, keeping them orthogonal during the rotation. The rotation is such that only theta changes. The mutually perpendicular axis are shown here with the only change in the coordinates of the particle being in $\theta$View attachment 290790

That's still simply the usual spherical coordinates. Except you've chosen to use $x, y, z, r, \phi$ but used $\theta'$ instead of $\theta$.

If you want to keep both coordinate systems (perhaps on the same diagram), then you need to use $x', y', z', r, \phi', \theta'$ as only $r$ is the same coordinate in both systems.

 

[Lnewqban]

only theta changesI've rotated the whole set of three axis, keeping them orthogonal during the rotation. The rotation is such that only theta changes. The mutually perpendicular axis are shown here with the only change in the coordinates of the particle being in $\theta$

In that case, the value of the projection on each axis of r must change.
In case you have missed it, please see post #6 above.

 

[mjc123]

OP, what you may not realize is that when you make that coordinate rotation, although φ does not change, dφ/dt does.
Consider two objects on the surface of the Earth (assumed a perfect sphere), both on the Greenwich meridian (same r and φ) but one near the north pole and one near the equator. Suppose they both move at V m/s eastward along their respective lines of latitude. What is dφ/dt in each case?
So in your equation for KE, it's not just θ that changes, but also dφ/dt.

 

[Kashmir]

OP, what you may not realize is that when you make that coordinate rotation, although φ does not change, dφ/dt does.
Consider two objects on the surface of the Earth (assumed a perfect sphere), both on the Greenwich meridian (same r and φ) but one near the north pole and one near the equator. Suppose they both move at V m/s eastward along their respective lines of latitude. What is dφ/dt in each case?
So in your equation for KE, it's not just θ that changes, but also dφ/dt.

dφ/dt is more near the pole.

 

[Kashmir]

In that case, the value of the projection on each axis of r must change.
In case you have missed it, please see post #6 above.

So we can't rotate in such a way that changes theta but keeps r and phi the same?

 

[Kashmir]

When you rotate the coordinate system, keeping angle theta the same, the projection on x-y plane changes; therefore, at least one of the two other parameters must change.

I am not keeping theta the same. I'm keeping the other two same and vary theta

 

[PeroK]

So we can't rotate in such a way that changes theta but keeps r and phi the same?

No. You can keep $\phi$ fixed on a single line of longitude, but $\phi$ will changes for points not on that line.

 

[wrobel]

I don't have a formula. I'm just imagining that we rotate our coordinate system by a finite rotation with the origin fixed where it was initially in such a manner that all other variables except remain the same.

I hope I'm clear?

Yes you are
but this is not a rotation of the space
$\varphi\mapsto \varphi+c$ is a rotation
$\theta\mapsto\theta+c$ is not

 

[Kashmir]

Yes you are
but this is not a rotation of the space
$\varphi\mapsto \varphi+c$ is a rotation
$\theta\mapsto\theta+c$ is not

That means a change of coordinate system such that only theta changes is not a rotation?

 

[wrobel]

That means a change of coordinate system such that only theta changes is not a rotation?

exactly, just take two different points and trace how they move as $\theta$ changes

 

[PeroK]

That means a change of coordinate system such that only theta changes is not a rotation?

It's a change to the z-axis only. It would leave the x and y axes unchanged. Note that $\tan \phi = \frac y x$.

 

[vanhees71]

Okay, but then $\theta' = \theta$. That's just regular spherical coordinates.

No! If you direct the polar axis in another direction, it's another angle.

 

[vanhees71]

It's a change to the z-axis only. It would leave the x and y axes unchanged. Note that $\tan \phi = \frac y x$.

Be careful with this "sloppy formula". In the programming language you have the function atan2 for what you want (in Fortran it's atan2(y,x) in C atan2(x,y) ;-))).

A formula with the cut of the polar angle at $\pm \pi$ is
$$\phi=\text{sign} \, y \arccos \left (\frac{x}{\sqrt{x^2+y^2}} \right), \quad \phi \in (-\pi,\pi).$$

 

[PeroK]

No! If you direct the polar axis in another direction, it's another angle.

The OP's diagram indicated a rotation of all three axes, but with the false assumption that $\phi$ was unchanged for all points.

 

[Lnewqban]

I am not keeping theta the same. I'm keeping the other two same and vary theta

That is not what I wrote, sorry.
If you increase theta by 90 degrees, for example, the value of the other angle respect to vector r must change.

As you tilt the z axis, you are simultaneously rotating it around the x-axis and around the y axis.
Because of that, the x-y plane tilts about an axis that crosses the origin and is perpendicular to vector r.

When all the above happens, the length of the projection of vector r on x-y plane changes, making change the projections of vector r on the x-axis and on the y axis, even when the spatial position of vector r and its absolute magnitude remain the same.

An example would be to assign x, y and z axes to three edges of a carboard box.
Looking directly from above, you will see and angle of 90 degrees between axes x and y.
Now, as you tilt edge z back diagonally to edges x and y, you could see how the angle between edges x and y increases until reaching 180 degrees when edge z reaches the ground: your phi angle has naturally changed.

 

[Lnewqban]

So we can't rotate in such a way that changes theta but keeps r and phi the same?

The value of r remains unchanged, but its projections on the three planes change (please, see attached animation).
The value of phi is affected by the rotation of the three axes, as well as it happened to theta.

Perhaps I am wrong, but I believe that:
1) The particle with mass m is located at the origin of coordinates x-y-z.
2) Vector r is the spatial velocity of mass m.
3) Theta and phi are angles that can define the spatial location of r because those angles are contained in planes that are perpendicular to each other.

Please, look for "6.4 Choosing Generalized Coordinates" in the following link:
https://courses.physics.ucsd.edu/2010/Fall/physics200a/LECTURES/CH06.pdf