Editing Noether's theorem (section)

==Brief illustration and overview of the concept==
[[File:Noether theorem scheme.png|thumb|upright=2|Plot illustrating Noether's theorem for a coordinate-wise symmetry]]
The main idea behind Noether's theorem is most easily illustrated by a system with one coordinate <math>q</math> and a continuous symmetry <math> \varphi: q \mapsto q + \delta q </math> (gray arrows on the diagram). 

Consider any trajectory <math>q(t)</math> (bold on the diagram) that satisfies the system's [[Euler-Lagrange equation|laws of motion]]. That is, the [[Action (physics)|action]] <math>S</math> governing this system is [[stationary point|stationary]] on this trajectory, i.e. does not change under any local [[Calculus of variations|variation]] of the trajectory. In particular it would not change under a variation that applies the symmetry flow <math>\varphi</math> on a time segment {{closed-closed|''t''<sub>0</sub>, ''t''<sub>1</sub>}} and is motionless outside that segment. To keep the trajectory continuous, we use "buffering" periods of small time <math>\tau</math> to transition between the segments gradually.

The total change in the action <math>S</math> now comprises changes brought by every interval in play. Parts where variation itself vanishes, i.e outside <math>[t_0,t_1]</math>, bring no <math>\Delta S</math>. The middle part does not change the action either, because its transformation <math>\varphi</math> is a symmetry and thus preserves the Lagrangian <math>L</math> and the action <math display="inline"> S = \int L </math>. The only remaining parts are the "buffering" pieces. In these regions both the coordinate <math>q</math> and velocity <math>\dot{q}</math> change, but <math>\dot{q}</math> changes by <math>\delta q / \tau</math>, and the change <math>\delta q</math> in the coordinate is negligible by comparison since the time span <math>\tau</math> of the buffering is small (taken to the limit of 0), so <math>\delta q / \tau\gg \delta q</math>. So the regions contribute mostly through their "slanting" <math>\dot{q}\rightarrow \dot{q}\pm \delta q / \tau</math>. 

That changes the Lagrangian by <math>\Delta L \approx \bigl(\partial L/\partial \dot{q}\bigr)\Delta \dot{q} </math>, which integrates to
<math display="block">\Delta S =
  \int \Delta L \approx \int \frac{\partial L}{\partial \dot{q}}\Delta \dot{q} \approx
  \int \frac{\partial L}{\partial \dot{q}}\left(\pm \frac{\delta q}{\tau}\right) \approx
  \ \pm\frac{\partial L}{\partial \dot{q}} \delta q =
    \pm\frac{\partial L}{\partial \dot{q}} \varphi.
</math>

These last terms, evaluated around the endpoints <math>t_0</math> and <math>t_1</math>, should cancel each other in order to make the total change in the action <math>\Delta S</math> be zero, as would be expected if the trajectory is a solution. That is
<math display="block">
  \left(\frac{\partial L}{\partial \dot{q}} \varphi\right)(t_0) =
  \left(\frac{\partial L}{\partial \dot{q}} \varphi\right)(t_1),
</math>
meaning the quantity <math>\left(\partial L /\partial \dot{q}\right)\varphi</math> is conserved, which is the conclusion of Noether's theorem. For instance if pure translations of <math>q</math> by a constant are the symmetry, then the conserved quantity becomes just <math>\left(\partial L/\partial \dot{q}\right) = p</math>, the canonical momentum.

More general cases follow the same idea:{{bulleted list
| When more coordinates <math>q_r</math> undergo a symmetry transformation <math>q_r \mapsto q_r + \varphi_r</math>, their effects add up by linearity to a conserved quantity <math display="inline">\sum_r \left(\partial L/\partial \dot{q}_r\right)\varphi_r</math>.
| ''Time invariance'' implies conservation of energy: Suppose the Lagrangian is invariant to time transformations, <math>t \mapsto t + T</math>. We effect such a transformation with a very small time shift <math>T \ll \tau</math> in the time between <math>t_0+\tau</math> and <math>t_1-\tau</math>, by stretching the first buffering segment <math>(t_0,t_0+\tau)</math> to  <math>(t_0,t_0+\tau+T)</math> and compressing the second buffering segment <math>(t_1-\tau,t_1)</math> to  <math>(t_1-\tau+T,t_1)</math>. Again, the action outside the interval <math>(t_0,t_1)</math> and between the buffering segments remains the same. However, the buffering segments each contribute two terms to the change of the action:
<math display="block">\Delta S \approx
  \pm \left(TL + \int \sum_r \frac{\partial L}{\partial \dot{q}_r}\Delta \dot{q}_r\right) \approx
  \pm T \left(L - \sum_r \frac{\partial L}{\partial \dot{q}_r}\dot{q}_r\right).
</math>
The first term <math>TL</math> is due to the changing sizes of the "buffering" segments. The first segment changes its size from <math>\tau</math> to <math>\tau + T</math>, and the second segment form <math>\tau</math> to <math>\tau - T</math>. Therefore, the integral over the first segment changes by <math> +T L(t_0)</math> and the integral over the second segment changes by <math> -T L(t_1)</math>. The second term is due to the time dilation by a factor <math>(\tau+T)/\tau</math> in the first segment and by <math>(\tau-T)/\tau</math> in the second segment, which changes all time derivatives by the dilation factor. These time dilations change <math>\dot{q}_r</math> to <math>\dot{q}_r \mp (T/\tau) \dot{q}_r</math> (to first order in <math>T/\tau</math>) in the first (-) and second (+) segment. Together they add to the conserved action S a term <math display="inline"> \pm T \left(L - \sum_r \left(\partial L/\partial \dot{q}_r\right)\dot{q}_r\right)</math> for the first (+) and second (-) segment. Since the change of action must be zero, <math>\Delta S = 0</math>, we conclude that the total energy <math>\sum_r \frac{\partial L}{\partial \dot{q}_r}\dot{q}_r - L</math> must be equal at times <math>t_0</math> and <math>t_1</math>, so total energy is conserved. 
| Finally, when instead of a trajectory <math>q(t)</math> entire fields <math>\psi(q_r,t)</math> are considered, the argument replaces 
* the interval <math>[t_0,t_1]</math> with a bounded region <math>U</math> of the <math>(q_r,t)</math>-domain, 
* the endpoints <math>t_0</math> and <math>t_1</math> with the boundary <math>\partial U</math> of the region, 
* and its contribution to <math>\Delta S</math> is interpreted as a flux of a [[conserved current]] <math>j_r</math>, that is built in a way analogous to the prior definition of a conserved quantity.

Now, the zero contribution of the "buffering" <math>\partial U</math> to <math>\Delta S</math> is interpreted as vanishing of the total flux of the current <math>j_r</math> through the <math>\partial U</math>. That is the sense in which it is conserved: how much is "flowing" in, just as much is "flowing" out.
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