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Most expressions of Noether's Theorem I have come across do not use lowercase delta, but a couple sites do. I am confused....... Check out page 21 of the June 23 issue of 'Science News' magazine. $$ \frac{d}{dt} \left(\sum_a \frac{δL}{δdq_a/dt}δq_a\right) = 0. $$

References:

  1. Science News Vol. 193, No. 11, June 23, 2018, p. 20.
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    $\begingroup$ $\uparrow$ Which sites? Link to sites? $\endgroup$ – Qmechanic Jul 19 '18 at 9:00
  • $\begingroup$ I just wrote the equation that appeared in the June 23, 2018 issue of 'Science News' as best I could...... $\endgroup$ – Kurt Hikes Jul 20 '18 at 2:13
  • $\begingroup$ This is standard notation in the calculus of variations. Check out Goldstein's Classical Mechanics for instance. $\endgroup$ – Ryan Thorngren Jul 20 '18 at 17:45
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There is really too little context to be sure what the author meant but the formula looks superficially like the Noether conservation law for vertical variations in point mechanics (rather than field theory). The $\delta$ usually means functional derivatives, cf. e.g. this Phys.SE post. However functional derivatives wrt. velocity does not make sense without further explanation, cf. e.g. this & this Phys.SE posts.

Let us from now on restrict to the case where the Lagrangian $L(q,\dot{q},t)$ only depends on up to first time derivatives. Then the vertical Noether current is a product of a vertical generator and the momemtum $p = \partial L / \partial \dot{q}$. Note in particular that the momentum is constructed via partial differentiation not functional differentiation.

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