A coordinate representing rotation about a variable axis and ##T##

[Kashmir]

If a system is represented by a set of generalized coordinates $q_i$ in which one coordinate say $\theta$ is such that $d \theta$ represents a rotation of the system about a fixed axis( an axis whose orientation remains fixed in space) then the kinetic energy $T$ shouldn't depend on it. I understand why this is true.

What if the rotation axis isn't fixed and it's orientation changes in space? What can we say about $T$? Will it depend on $\theta$ or not?

Thank you in advance :)

 

[PeroK]

As you saw from your previous post, the kinetic energy may depend on a coordinate such as $\theta$ or it may not. You have to take each coordinate system specifically.

 

[Kashmir]

As you saw from your previous post, the kinetic energy may depend on a coordinate such as $\theta$ or it may not. You have to take each coordinate system specifically.

Is there no general rule, like for a fixed axis it is independent, about a variable axis is it independent or not?

 

[PeroK]

Is there no general rule, like for a fixed axis it is independent, about a variable axis is it independent or not.

For any general rule, you need to state the rule and then prove it. "Fixed axis" and "variable axis" don't sound like well-defined terms to me.

 

[Kashmir]

For any general rule, you need to state the rule and then prove it. "Fixed axis" and "variable axis" don't sound like well-defined terms to me.

By fixed axis I mean an axis whose orientation is fixed in space.

 

[Kashmir]

As you saw from your previous post, the kinetic energy may depend on a coordinate such as $\theta$ or it may not. You have to take each coordinate system specifically.

I don't have a Cartesian or spherical system in mind. I'm thinking about a general set of generalized coordinates $q_i$ that somehow represent the system. In this set of $q_i$ I've a coordinate theta whose change means a rotation of the system in space.

 

[PeroK]

I don't have a Cartesian or spherical system in mind. I'm thinking about a general set of generalized coordinates $q_i$ that somehow represent the system. In this set of $q_i$ I've a coordinate theta whose change means a rotation of the system in space.

It seems plausible that you can write an expression for kinetic energy does not depend on $\theta$.

For example, you could choose Cartesian coordinates where $\theta$ represents the usual polar angle.

 

[Kashmir]

It seems plausible that you can write an expression for kinetic energy does not depend on $\theta$.

For example, you could choose Cartesian coordinates where $\theta$ represents the usual polar angle.

Can we make a general statement as to if $d\theta$ corresponds to a rotation of the system about an axis whose orientation isn't fixed in space then the kinetic energy will/will not depend on $\theta$ ?

(If the axis is fixed, I think I can prove that T will be independent of $\theta$ )

 

[PeroK]

Can we make a general statement as to if $d\theta$ corresponds to a rotation of the system about an axis whose orientation isn't fixed in space then the kinetic energy will/will not depend on $\theta$ ?

What does "an axis whose orientation isn't fixed in space" mean? You mean a time dependent axis?

PS The definition of "rotation" entails a "fixed" axis. That's what rotation means.

 

[Kashmir]

What does "an axis whose orientation isn't fixed in space" mean? You mean a time dependent axis?

Yes ,maybe it's orientation depends on time or other coordinates $q_j$ or both. What I'm trying to say is that the orientation isn't fixed. If the axis points this way right now the next instant it will point in other direction.

 

[PeroK]

Yes ,maybe it's orientation depends on time or other coordinates $q_j$ or both. What I'm trying to say is that the orientation isn't fixed. If the axis points this way right now the next instant it will point in other direction.

I'm struggling to see the purpose of this. If you allow time dependent coordinates, then I think anything is possible.

 

[Kashmir]

What does "an axis whose orientation isn't fixed in space" mean? You mean a time dependent axis?

PS The definition of "rotation" entails a "fixed" axis. That's what rotation means.

https://en.m.wikipedia.org/wiki/Rotation_around_a_fixed_axis maybe it clears what I mean by fixed and variable axis.

 

[PeroK]

https://en.m.wikipedia.org/wiki/Rotation_around_a_fixed_axis maybe it clears what I mean by fixed and variable axis.

Okay, but what are you trying to achieve by your question? If your coordinates are sufficiently generalised, then you can't say very much about them.

 

[PeroK]

Imagine, for example, that $\theta$ is a polar angle and you define a coordinate $\theta'$ as some (non-linear) function of $\theta$? Then is $d\theta'$ still a rotation? In this case, clearly the KE depends on $\theta'$.

This is why I'm reluctant to get drawn into general claims when the hypotheses are not clear.

 

[PeroK]

E.g. take the usual 2D polar coordinates and define: $$\theta = (\alpha t + 1)\theta'$$where $\alpha$ is a constant with dimensions of $1/t$. At each time $t$, $\theta'$ instantaneously looks like a polar angle, but$$\dot \theta = \alpha \theta' + (\alpha t +1 )\dot \theta'$$

 

[Kashmir]

Okay, but what are you trying to achieve by your question? If your coordinates are sufficiently generalised, then you can't say very much about them.

I'll write the passage from where my question emerged. Thank you for taking your time to write.

 

[Kashmir]

Okay, but what are you trying to achieve by your question? If your coordinates are sufficiently generalised, then you can't say very much about them.

Goldstein pg72 2nd Ed. while discussing a particle in a central force field says "... Since potential energy involves only the radial distance, the problem has spherical symmetry, i.e., any rotation, about any **fixed axis**, can have no effect on the solution. Hence an angle coordinate representing rotation about **fixed axis** must be cyclic".

Why restrict to fixed axis? What about an angle coordinate representing rotation about ** non fixed axis** ?

 

[PeroK]

Why restrict to fixed axis? What about an angle coordinate representing rotation about ** non fixed axis** ?

Because it's patently not true. If the axis is moving in time, then points at different angular coordinates may be moving at different speeds and that affects the KE.

 

[Kashmir]

Because it's patently not true. If the axis is moving in time, then points at different angular coordinates may be moving at different speeds and that affects the KE.

I couldn't understand that. Could you explain that please.

 

[PeroK]

I couldn't understand that. Could you explain that please.

If you are reading Goldsmith, you must be able to work it out for yourself.

 

[Kashmir]

If you are reading Goldsmith, you must be able to work it out for yourself.

OK Thank you :)

 

[Dale]

Is there no general rule, like for a fixed axis it is independent, about a variable axis is it independent or not?

I agree with @PeroK. I don’t think that any general statements can be proved based on your statements.

However, I would go a little further. While you could take an arbitrary such coordinate system and transform the usual Lagrangian into that coordinate system, it may not even be clear how to separate the resulting transformed Lagrangian into kinetic and potential terms.

 

[Kashmir]

I agree with @PeroK. I don’t think that any general statements can be proved based on your statements.

However, I would go a little further. While you could take an arbitrary such coordinate system and transform the usual Lagrangian into that coordinate system, it may not even be clear how to separate the resulting transformed Lagrangian into kinetic and potential terms.

Thank you. That did help. :)