Applications of Noether’s Theorem

Abstract

If the action S = ∫ t 1 t 2 L ( q , q ˙ t ) d t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003055174/2504eff5-148a-48bb-bb28-78512f02fa67/content/ieq0004.tif"/> is invariant under the infinitesimal transformation t ˜ = t + ε τ ( q , t ) , q ˜ r = q r + ε ξ r ( q , t ) , r = 1 , … , n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003055174/2504eff5-148a-48bb-bb28-78512f02fa67/content/ieq0005.tif"/> , with ε= constant ≪ 1,, then the Noether’s theorem permits to construct the corresponding conserved quantity. The Lanczos approach employs to ε =qN+1 as a new degree of freedom, thus the Euler-Lagrange equation for this new variable gives the Noether’s constant of motion. Torres del Castillo and Rubalcava-García showed that each variational symmetry implies the existence of an ignorable coordinate; here we apply the Lanczos approach to the Noether’s theorem to motivate the principal relations of these authors. The Maxwell equations without sources are invariant under duality rotations, then we show that this invariance implies, via the Noether’s theorem, the continuity equation for the electromagnetic energy. Besides, we demonstrate that if we know one solution of P(x)y”+q(x)y’+r(x)y=0, then this Lanczos technique allows obtain the other solution of this homogeneous linear differential equation.