Rotation and Translation coordinates

  • #1
I am currently reading Goldstein's Classical mechanics and come on to this problem. Let q1,q2,...,qn be generalized coordinates of a holonomic system and T its kinetic energy. qk correspondes to a translation of the entire system and qj 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
 

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  • #2
BvU
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T does not depend on ##\theta## but on ##\dot\theta## which is a different generalized coordinate...
idem ##r## and ##\dot r##
 
  • #3
wrobel
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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.
 
  • #4
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).
 
  • #5
wrobel
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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
 
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  • #6
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##, nθ, depends on time. The coordinate φ is a rotation coordinate since its rotation axis nφ correspondes to the fixed polar axis.
 

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