Noether's theorem: Difference between revisions

Revision as of 09:01, 29 December 2024

Revision as of 09:09, 29 December 2024

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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>.

| WhenA theretransformations by a aretiny time transformations<math>T \ll \tau</math>, <math>t \mapsto t + T</math>, they causecauses the "buffering" segments to contribute the two following terms to <math>\Delta S</math>:

<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 is due to stretching inthe temporalchanging dimensionsizes of the "buffering" segment. 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., Assumingwhich <math>Tchanges \llall \tautime derivatives </math>,by the dilation factor. theseThese 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,(+) respectivelysegment. Together they add to the consered quantity (theconserved action S) a summandterm <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.

| 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,