\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" segmentsegments. 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\right) - 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,
