Scope of the symmetry-conservation statements: force laws that break kinematic symmetries, residual forces, moving coordinates and the empirical boundary

Symmetries are properties of the model, not of the kinematics. The symmetry criteria of this page are identities imposed on the model's mechanical Lagrangian $L=K-U$ of the declared potential presentation; residual forces enter the charge rates separately (Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions); the affine structure of spacetime, the availability of every inertial frame and the Galilean transformation law do not by themselves make a potential invariant. For the uniform-force potential $U=-{\sum }_i{m}_i\mathbf{g}\cdot x_i$ with $d\equiv 0$, ${\sum }_i{\mathrm{\nabla }}_{x_i}U=-M_{\mathrm{t}\mathrm{o}\mathrm{t}}\mathbf{g}$: the strict translation datum $\tau =0$, ${\xi }_i=a$, $B=0$ satisfies the identity exactly in nonzero dimensionless directions $a\perp \mathbf{g}$, and a direction with $a\cdot \mathbf{g}\mathrm{\ne }0$ gives a momentum and boost charge that fail along motions. The sharpness clauses locate that failure along maximal motions through states where their required directional derivative is nonzero.

A mass-weighted confining potential $U=\frac{k}{2}{\sum }_i{m}_i\mid x_i-x_i^0{\mid }^2$, $k>0$, has ${\sum }_i{\mathrm{\nabla }}_{x_i}U=kM(X-X_0)$, where $X_0=M^{-1}{\sum }_i{m}_i{x}_i^0$. In the global potential model, for any direction with $a\cdot (X-X_0)\mathrm{\ne }0$ the translation condition fails and the momentum charge is not conserved along the maximal motion through that state. If that state's time is nonzero, the boost sharpness clause also applies; at time zero, continuity gives nearby nonzero times with the same nonvanishing directional derivative. Here $k$ has units ${\tau }^{-2}$, so $U$ has energy units and the centre-of-mass equation is $X^{\mathrm{\prime }\mathrm{\prime }}=-k(X-X_0)$.

Rotation is not translation. The rotation condition of Rotation symmetry of the model's Lagrangian and conservation of angular momentum about the origin is compatible with a confining central potential: for $U$ depending only on the labels through rotation-invariant combinations the angular-momentum charge for a dimensionless rotation generator may be conserved in the global potential branch even when the translation charges are not. The conservation statements are separate conditional claims, one per symmetry datum; a model can possess one charge and fail another.

Residual forces destroy charges at a computable rate. With $d\equiv ̸0$ the charge of a strict symmetry is conserved exactly when the residual force pairs to zero with the generator along the motion (The point Noether theorem of the model: conservation of the charge of a strict symmetry, and the residual-force rate clause 2), and for a general datum with time reparametrisation the rate is the residual force paired with $\xi -v\tau$ (The general Noether theorem of the model: point transformations with time reparametrisation, the charge $p\cdot \xi -E\tau -B$, its residual-force rate and its potential gauge clause 3). A friction force along a translation generator, for instance, changes the corresponding momentum charge whenever its total pairing with that generator is nonzero along the motion. The balance theorems of the force page, in particular Mechanical energy balance for a potential presentation: $E^{\mathrm{\prime }}={\mathrm{\partial }}_{t}U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form, remain valid in those situations and carry the residual contribution explicitly; the Noether charges and the balance identities are different deductions, and neither replaces the other.

Moving coordinates and the mechanical energy. In a time-dependent coordinate description the chart energy function of The energy function of a moving chart: exact difference from the inertial mechanical energy and conservation along the chart solutions differs from the inertial mechanical energy by the pairing of the chart's explicit time dependence with the mass-operator velocity, and the chart energy can be conserved while that difference is nonzero; the phrase "the Hamiltonian is conserved" is therefore not by itself a statement about the inertial mechanical energy. For the datum $(\tau ,\xi ,B)=(1,0,0)$, with parameter measured in seconds, the time-translation identity of Time translation and the mechanical energy of the model: conservation in the autonomous global potential branch and the sharp failure for a time-dependent potential holds exactly when ${\mathrm{\partial }}_{t}U\equiv 0$ on $I_0\times Q$. If ${\mathrm{\partial }}_{t}U\equiv ̸0$, this datum fails the identity; in the global potential branch the sharpness clause exhibits a maximal motion through a state where ${\mathrm{\partial }}_{t}U\mathrm{\ne }0$ whose mechanical energy has that nonzero derivative there.

No converse and no empirical claim. The results of this page are conditional deductions inside the force model: symmetry hypotheses on the declared presentation imply conservation of the stated charges along the motions. No converse is asserted — conservation of a quantity along the computed motions does not establish that the model has the corresponding symmetry datum — and no observational content is introduced. If a conservation statement is used in an empirical analysis, the model assumptions, the representative of the potential and any measured inputs enter that analysis as additional premises with their own qualifications; an observation consistent with a conservation law is evidence about the combined model and apparatus, not a proof of either.

Not asserted. The page does not treat discrete symmetries, Noether's second theorem, field-theoretic charges, gauge families whose generators fail to be $C^2$, global existence of the transformation families, or the question whether any physical interaction is invariant under the families considered; those are separate mathematical and empirical developments.