Noether identities for variational symmetries

Take a smooth Lagrangian $F(x,u,v)$ and a smooth one-parameter change of the dependent variable that leaves the independent coordinate fixed. Let its infinitesimal generator be $\xi(u)$. Assume the density is invariant to first order for every nearby position and slope, not just on a particular candidate curve. Differentiating invariance gives $F_u\xi+F_v\xi_u v=0$. This is a statement about the Lagrangian before imposing its equations of motion.

Write the momentum as $p=F_v$ evaluated along a twice continuously differentiable curve. Differentiating $p\xi$ and using the invariance identity yields $(p'-F_u)\xi$. The factor in parentheses is the negative of the Euler–Lagrange residual. Therefore the charge is constant on a stationary curve. For vector-valued dependent variables, replace products by the corresponding dot products. Smooth transformations that change the density by a total derivative require an additional boundary-charge correction; strict invariance is the simpler case proved here.

Suppose the Lagrangian has no explicit dependence on the interval coordinate. Define $E=u'F_v-F$. Its total derivative is $u'(p'-F_u)-F_x$, obtained by the product and chain rules. On a stationary curve the first term vanishes, and autonomy makes the second vanish. Thus energy is conserved. For a mechanical Lagrangian equal to kinetic minus potential energy with a standard quadratic kinetic term, this expression is the familiar kinetic plus potential energy. For a general Lagrangian it is a derived conserved expression rather than an automatically positive physical energy.

The independent variable need not be time: in a spatial variational problem the same identity is often called the Beltrami first integral. If explicit coordinate dependence remains, the identity instead gives the rate of change $E'=-F_x$ on solutions. A finite interval with fixed endpoints need not be globally invariant under a time translation as a boundary-value problem; the local conservation identity follows from the density symmetry and equation along the interior curve. Boundary data determine which solution and which value of the conserved constant occur.

A cyclic coordinate is absent from the Lagrangian itself, so its partial derivative is zero and its momentum is constant. In a planar central potential, rotational symmetry similarly gives angular momentum: the generator is the ninety-degree rotation of the position, and its dot product with momentum is the planar cross product. These statements use differentiable continuous symmetries. A discrete symmetry can relate solutions without producing this first-order Noether charge, because it has no infinitesimal generator of the required kind.

Conservation is compatible with minima, maxima, and saddle-type stationary trajectories. The oscillatory action can conserve mechanical energy while failing to be minimal over long time intervals. Dissipative equations introduce another distinction: a damped oscillator's mechanical energy decreases, so the conservative autonomous Lagrangian used here does not describe that damped model unchanged. Specialized variational formulations can handle such systems, but cannot be inferred by silently adding friction to the conservative conservation proof. Always check the actual equation and the exact symmetry being invoked.