Editing Noether's theorem (section)

===Simple form using perturbations===

The essence of Noether's theorem is generalizing the notion of ignorable coordinates.

One can assume that the Lagrangian ''L'' defined above is invariant under small perturbations (warpings) of the time variable ''t'' and the [[generalized coordinate]]s '''q'''. One may write

:<math>\begin{align}
           t &\rightarrow t^{\prime} = t + \delta t \\
  \mathbf{q} &\rightarrow \mathbf{q}^{\prime} = \mathbf{q} + \delta \mathbf{q} ~,
\end{align}</math>

where the perturbations ''δt'' and ''δ'''''q''' are both small, but variable. For generality, assume there are (say) ''N'' such [[symmetry transformations]] of the action, i.e. transformations leaving the action unchanged; labelled by an index ''r''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;...,&nbsp;''N''.

Then the resultant perturbation can be written as a linear sum of the individual types of perturbations,
:<math>\begin{align}
           \delta t &= \sum_r \varepsilon_r T_r \\
  \delta \mathbf{q} &= \sum_r \varepsilon_r \mathbf{Q}_r ~,
\end{align}</math>

where ''ε''<sub>''r''</sub> are [[infinitesimal]] parameter coefficients corresponding to each: 
* [[Lie group#The exponential map|generator]] ''T<sub>r</sub>'' of [[time evolution]], and
* [[Lie group#The exponential map|generator]] '''Q'''<sub>''r''</sub> of the generalized coordinates.
For translations, '''Q'''<sub>''r''</sub> is a constant with units of [[length]]; for rotations, it is an expression linear in the components of '''q''', and the parameters make up an [[angle]].

Using these definitions, [[Emmy Noether|Noether]] showed that the ''N'' quantities

:<math>\left(\frac{\partial L}{\partial \dot{\mathbf{q}}} \cdot \dot{\mathbf{q}} - L \right) T_r - \frac{\partial L}{\partial \dot{\mathbf{q}}} \cdot \mathbf{Q}_r</math>

are conserved ([[constants of motion]]).

==== Examples ====

'''I. Time invariance'''

For illustration, consider a Lagrangian that does not depend on time, i.e., that is invariant (symmetric) under changes ''t'' → ''t'' + δ''t'', without any change in the coordinates '''q'''. In this case, ''N''&nbsp;=&nbsp;1, ''T''&nbsp;=&nbsp;1 and '''Q'''&nbsp;=&nbsp;0; the corresponding conserved quantity is the total [[energy]] ''H''<ref name=Lanczos1970>{{cite book | author-link= Cornelius Lanczos |last=Lanczos |first=C. | year = 1970 | title = The Variational Principles of Mechanics | edition = 4th | publisher = Dover Publications | location = New York | isbn = 0-486-65067-7}}</ref>{{rp|401}}

:<math>H = \frac{\partial L}{\partial \dot{\mathbf{q}}} \cdot \dot{\mathbf{q}} - L. </math>

'''II. Translational invariance'''

Consider a Lagrangian which does not depend on an ("ignorable", as above) coordinate ''q''<sub>''k''</sub>; so  it is invariant (symmetric) under changes ''q''<sub>''k''</sub> → ''q''<sub>''k''</sub> + ''δq''<sub>''k''</sub>. In that case, ''N''&nbsp;=&nbsp;1, ''T''&nbsp;=&nbsp;0, and ''Q''<sub>''k''</sub>&nbsp;=&nbsp;1; the conserved quantity is the corresponding linear [[momentum]] ''p''<sub>''k''</sub><ref name=Lanczos1970/>{{rp|403–404}}

:<math>p_k = \frac{\partial L}{\partial \dot{q_k}}.</math>

In [[special relativity|special]] and [[general relativity]], these two conservation laws can be expressed either ''globally'' (as it is done above), or ''locally'' as a continuity equation. The global versions can be united into a single global conservation law: the conservation of the energy-momentum 4-vector. The local versions of energy and momentum conservation (at any point in space-time) can also be united, into the conservation of a quantity defined ''locally'' at the space-time point: the [[stress–energy tensor]]<ref name="Goldstein1980">{{cite book |last=Goldstein |first=Herbert |author-link=Herbert Goldstein |year=1980 |title= [[Classical Mechanics (Goldstein)|Classical Mechanics]] |edition=2nd |publisher=Addison-Wesley |location=Reading, MA |isbn= 0-201-02918-9 }}</ref>{{rp|592}}(this will be derived in the next section).

'''III. Rotational invariance'''

The conservation of the [[angular momentum]] '''L''' = '''r''' × '''p''' is  analogous to its linear momentum counterpart.<ref name=Lanczos1970/>{{rp|404–405}} It is assumed that the symmetry of the Lagrangian is rotational, i.e., that the Lagrangian does not depend on the absolute orientation of the physical system in space. For concreteness, assume that the Lagrangian does not change under small rotations of an angle ''δθ'' about an axis '''n'''; such a rotation transforms the [[Cartesian coordinate system|Cartesian coordinates]] by the equation

:<math>\mathbf{r} \rightarrow \mathbf{r} + \delta\theta \, \mathbf{n} \times \mathbf{r}.</math>

Since time is not being transformed, ''T'' = 0, and ''N'' = 1. Taking ''δθ'' as the ''ε'' parameter and the Cartesian coordinates '''r''' as the generalized coordinates '''q''', the corresponding '''Q''' variables are given by

:<math>\mathbf{Q} = \mathbf{n} \times \mathbf{r}.</math>

Then Noether's theorem states that the following quantity is conserved,
:<math>
\frac{\partial L}{\partial \dot{\mathbf{q}}} \cdot \mathbf{Q} = 
\mathbf{p} \cdot \left( \mathbf{n} \times \mathbf{r} \right) = 
\mathbf{n} \cdot \left( \mathbf{r} \times \mathbf{p} \right) = 
\mathbf{n} \cdot \mathbf{L}.
</math>

In other words, the component of the angular momentum '''L''' along the '''n''' axis is conserved. And if '''n''' is arbitrary, i.e., if the system is insensitive to any rotation, then every component of '''L''' is conserved; in short, [[angular momentum]] is conserved.