Noether theorem

Noether’s theorem associates a conserved quantity to each continuous variational symmetry.

For independent coordinates with $L={\sum }_i{m}_i({\dot{x}}_i^2-{\omega }^2{x}_i^2)/2$, the variables $X_i=\sqrt{m_i}{x}_i$ give an isotropic oscillator even at unequal masses. Rotations of the $X_i$ are continuous symmetries. Their conserved Noether charges $X_i{P}_j-X_j{P}_i$ therefore predate any equality of physical masses. Equal masses alone cannot create new invariances of arbitrary additional interactions.

Invariance under a shift of the time coordinate, with other required background structures held fixed. A continuous time-translation symmetry gives an energy conservation law through Noether theorem. A generic expanding cosmological background is not stationary and need not supply a corresponding globally conserved total energy.

Suppose an action is invariant under a local transformation $\delta {\Phi }_a=R_a\alpha +R_a^{\mu }{\partial }_{\mu }\alpha$ for every compactly supported function $\alpha$. Variation and integration by parts give the off-shell identityThus arbitrary local symmetry parameters imply dependencies among the field equations. In gauge theory, these identities accompany constraints rather than independent physical charges for each function. This complements the conserved-current statement of Noether theorem.

For $\delta A_{\mu }={\partial }_{\mu }\alpha$, $\delta \varphi =-ie\alpha \varphi$ and $\delta {\varphi }^{\ast }=ie\alpha {\varphi }^{\ast }$, Noether second theorem givesThis holds without imposing the field equations. On the matter equations it becomes a divergence identity for the electromagnetic equation, expressing compatibility with charge conservation. The Hamiltonian description carries the corresponding Gauss law constraint in gauge theory.

If a time-independent Lagrangian $L(q,\dot{q})$ is invariant under a one-parameter point transformation with infinitesimal generator ${\xi }_i={\partial }_s{Q}_i{|}_{s=0}$, then every Euler-Lagrange trajectory conserves

An infinitesimal transformation that preserves the action up to endpoint terms. For $\delta t=\epsilon \varphi$ and $\delta q_i=\epsilon {\psi }_i$, invariance of $Ldt$ yields the Noether charge ${\sum }_i{p}_i{\psi }_i-H\varphi$, less any boundary-term generator.