Proving that the free particle lagrangian is rotationally symmetric

[stormyweathers]

Homework Statement

Show that the free particle lagrangian is invariant to rotations in $$\Re^{3}$$, but I assume this means invariant up to a gauge term.
$$L=m/2 [\dot{R^{2}} + R^{2}\dot{θ^{2}} +R^{2}Sin^{2}(θ)\dot{\phi^{2}}$$

Homework Equations

I consider an aribtrary infinitesimal rotation:
$$ \theta(t,\epsilon)=\theta(t,0)+\epsilon \Delta \theta $$
$$ \phi(t,\epsilon)=\phi(t,0)+\delta \Delta \phi $$

The Attempt at a Solution

The new angle derivitives are identical to the first, since we evaluate them by taking the time partial of the transformed coordinates.
I am running into issues with the $$Sin^{2}(\theta)$$ term.
$$Sin^{2}(\theta) \rightarrow Sin^{2}(\theta)+2Cos(\theta)Sin(\theta) \epsilon \Delta \theta + O(\epsilon^{2})$$

The epsilon term is throwing me off, because I can't get it to disappear or rewrite it as a gauge term.

 

[George Jones]

Why do you have to use infinitesimal rotations and a Lagrangian written in terms of spherical coordinates?

Whey not apply a finite rotation to

L = \frac{1}{2} m \dot{\vec{r}} \cdot \dot{\vec{r}} ?

 

[stormyweathers]

It seemed more straightforward to apply displacements in the angular directions. To apply a general rotation matrix would be a ton more algebra, no?

I need to use infinitesimal rotations because I am trying to prove a continuous symmetry for noether's theorem