Note that:formula_108This has the form of:formula_109(where we have performed a change of dummy indices) so set:formula_110Then,:formula_111:formula_112Noether's theorem states that formula_113 (as one may explicitly check by substituting the Euler–Lagrange equations into the left hand side).
Note that: formula_141This has the form of: formula_142(where we have performed a change of dummy indices) so we can set: formula_143Then,: formula_144: formula_145Noether’s theorem states that formula_146 (as one may explicitly check by substituting the Euler-Lagrange equations into the left hand side).