I Rotation and Translation coordinates

[FrederikPhysics]

I am currently reading Goldstein's Classical mechanics and come on to this problem. Let q₁,q₂,...,q_n be generalized coordinates of a holonomic system and T its kinetic energy. q_k correspondes to a translation of the entire system and q_j a rotation of the entire system around some axis, then ∂_qjT=∂_qkT=0 since velocities are not affected by a translation of the origin or a rotation of the coordinate axes.
Now take as an example the kinetic energy of a single particle in spherical coordiantes, here the polar angle θ correspondes to a rotation of the system, but T is dependent on θ. r correspondes to a translation of the particle and therefore the system but T depends on r. How should I undersand this?
Mvh. Frederik

 

[BvU]

T does not depend on $\theta$ but on $\dot\theta$ which is a different generalized coordinate...
idem $r$ and $\dot r$

 

[wrobel]

Actually it is a definition:
"the system admits translation along the generalized coordinate $q^j$"$\Longleftrightarrow$ $"\frac{\partial T}{\partial q^j}=0"$. Or equivalently iff $T$ is invariant under the group $q^j\mapsto q^j+s,\quad s\in \mathbb{R}$
It doesn't matter what is $q^j$. If $q^j$ is an angle then we say that the system admits a rotation.

 

[FrederikPhysics]

T does not depend on $\theta$ but on $\dot\theta$ which is a different generalized coordinate...
idem $r$ and $\dot r$

For a system consisting of one particle with mass m one writes T=½m(dr/dt²+r² dθ/dt²+r²sin²θ dφ/dt²) in spherical coordinates, θ being the polar angle and φ the azimutal angle, that makes T=T(r, θ, dr/dt, dθ/dt, dφ/dt).

 

[wrobel]

O now I see what you want. This is a good question indeed. It is because $\theta\mapsto \theta+s$ is not a rotation of the space about a fixed axis. Look how all the points of the space conduct under this transformation

 

[FrederikPhysics]

Okay, so one might say:
Let $T=\frac{1}{2}m\dot{\textbf{r}}\cdot\dot{\textbf{r}}$ be the kinetic energy of a one particle system, $\textbf{r}$ the position vector in some frame and q a set of generalized coordinates. $q_{k}$ is called a translation coordinate when there exists a vector $\textbf{n}$ such that $\frac{\partial \textbf{r}}{\partial q_{k}}=\textbf{n}$ fulfilling $\dot{\textbf{n}}=\textbf{0}$. Then $\frac{\partial T}{\partial q_{k}}=m\dot{\textbf{r}}\cdot\frac{d}{dt}\frac{\partial\textbf{r}}{\partial q_{k}}=0$.
When this is true $q_{k}$is said to translate the system in the direction of $\textbf{n}$.

$q_{j}$ is called a rotation coordinate when there exists a vector $\textbf{n}$ such that $\frac{\partial \textbf{r}}{\partial q_{j}}=\textbf{n}\times\textbf{r}$ fulfilling $\dot{\textbf{n}}=\textbf{0}$. Then $\frac{\partial T}{\partial q_{j}}=m\dot{\textbf{r}}\cdot\frac{d}{dt}\frac{\partial\textbf{r}}{\partial q_{j}}=0$.
When this is true $q_{k}$ is said to rotate the system around the direction of $\textbf{n}$.
The generalazation to more particles follows naturally.

Within this definition the coordinates r and $\theta$ of the spherical coordinate system, in general neither translates nor rotates the system since the r unitvector, $\hat{\textbf{r}}$, depends on time and the rotation axis of $\theta$, n_θ, depends on time. The coordinate φ is a rotation coordinate since its rotation axis n_φ correspondes to the fixed polar axis.