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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.

The appropriate invariance is invariance of the action element $Ldt$, including the change of the time measure. Under $\delta t=\lambda \varphi$ and $\delta q_i=\lambda {\psi }_i$, differentiation of $q_i^{\prime }(t^{\prime })$ gives $\delta {\dot{q}}_i=\lambda (D_t{\psi }_i-{\dot{q}}_i{D}_t\varphi )$. ThereforeUsing the Euler-Lagrange equations and $D_{t}E=-L_t$, this becomesAn action symmetry therefore yields the Noether chargeFor $\varphi =0$ this reduces to the usual spatial form of Noether's theorem. The same on-shell calculation permits velocity-dependent generators when they define a legitimate variational symmetry. Merely preserving the numerical value of $L$ is insufficient for transformations of time: the Jacobian matrix and determinant $d{t}^{\prime }/dt$ matters. If the action changes by a boundary term $\lambda D_{t}B$, the conserved quantity is $Q-B$.