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Emmy Noether proved the fundamental connection between continuous symmetries of a variational problem and conservation laws. Her 1918 paper also gives identities among the Euler-Lagrange equations for symmetries depending on arbitrary functions, a foundation for understanding gauge symmetry. The distinction between finite-parameter and function-dependent transformations is essential to the two theorems.^[ref]

M. A. Tavel translated Emmy Noether's 1918 paper on invariant variational problems into English; the translation appeared in 1971. It presents the relation between continuous symmetries, conservation laws and identities among the Euler-Lagrange equations. The surviving byline establishes the translator's contribution without requiring an inferred full name or biography.^[ref]

Noether theorem associates conservation laws with continuous action symmetries of an action. The symmetry must preserve the action up to an allowed boundary term; a symmetry of an equation alone is not the full hypothesis. Emmy Noether proved the general relation in her 1918 paper, whose first theorem concerns finite-parameter groups and whose second concerns transformations depending on arbitrary functions.^[ref]

A mechanical version is especially transparent. Let $L(q,\dot{q},t)$ change under an infinitesimal point transformation at fixed time by $\delta L=ϵdB/dt$, and let $\delta q_i=ϵ{\xi }_i$. Along an Euler-Lagrange equation solution, with $p_i=\partial L/\partial {\dot{q}}_i$,Consequently $J={\sum }_i{p}_i{\xi }_i-B$ is constant. A cyclic coordinate gives its conjugate momentum; rotational symmetry gives angular momentum. Allowing the time coordinate to transform gives the energy conservation associated with time-translation symmetry.

The field-theoretic version gives a conserved Noether current on solutions. Noether second theorem instead produces off-shell identities among field equations for arbitrary local gauge parameters. These identities and the associated constraints should not be confused with independent physical conserved charges for every local parameter.