Scope of point symmetries: symmetries belong to the Lagrangian, not to the configuration space, and time translation is the energy case

The symmetry is a property of the Lagrangian. The identity of Infinitesimal point symmetry of a time-dependent Lagrangian and its conserved charge is imposed on the chosen $\mathcal{L}$ through its configuration and velocity gradients ${\mathcal{L}}_q$ and ${\mathcal{L}}_v$; no kinetic/potential decomposition is assumed. A transformation that is a symmetry of the configuration space, of a chart or of a kinematic structure need not leave a chosen $\mathcal{L}$ invariant. The corollary Conserved charges for translation, rotation and boost symmetries of a mechanical Lagrangian exhibits the failure explicitly: for the confining potential $U(q)=\frac{k}{2}\mathrm{\mid }q{\mathrm{\mid }}^2$ the translation and boost identities fail even though the Galilean charts of the framework remain available. A symmetry hypothesis must therefore be verified for the actual model before The charge of an infinitesimal point symmetry is constant along Euler-Lagrange solutions or Noether identity for point transformations with time reparametrisation: the conserved charge $p\cdot \xi -E\tau -B$ can be applied.

Time translation is a separate case. The general theorem includes the case $\xi =0,\tau =1,B=0$, where the charge is the energy $-E$ and the identity holds exactly when ${\mathcal{L}}_t=0$; this agrees with the energy identity of The Hamiltonian on an inverse Legendre branch: differential, equivalence with the Euler-Lagrange equation, and the energy identity in the autonomous case. Time-dependent Lagrangians need not conserve energy, and no conservation is claimed when ${\mathcal{L}}_t\mathrm{\ne }0$.

Conservation is conditional and model-internal. The conserved charges are deductions from the stated symmetry identities along solutions of the Euler–Lagrange equation. Nothing here asserts that an observed conservation law implies a symmetry of a physical model, or that the models possessing these symmetries describe nature; those are separate modelling and empirical questions, and the empirical qualifications of any such use belong to the consuming physical items.