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The two continuous, time-independent symmetries are time translation and a common rotation of both angles. Their Noether charges are respectively the total energy and vertical angular momentum:For the second symmetry, $\theta \mapsto \theta +ϵ$ and $\phi \mapsto \phi +ϵ$, so the Noether theorem gives the sum of their conjugate momenta. Independent rotations are not symmetries of the spring coupling.

There are also discrete symmetries: simultaneous reflection $(\theta ,\phi )\mapsto (-\theta ,-\phi )$, and time reversal in classical mechanics, which reverses both angular velocities. These leave the dynamics invariant but provide no additional infinitesimal Noether charges. Combined with common rotations, the reflections give the full planar orthogonal symmetry of the angular geometry. If $I_1=I_2$, interchange of the two angular coordinates is an additional discrete symmetry of the reduced dynamics.

There is a further useful time-dependent symmetry, up to a total derivative, on a continuous lift of the angles: adding the same constant angular velocity, $(\theta ,\phi )\mapsto (\theta +vt,\phi +vt)$. It changes $L$ byIts conserved quantity is $\boxed{I_1\theta +I_2\phi -tJ}$. It is the integration constant for free common rotation, rather than a new globally single-valued angle charge on the configuration torus. The globally defined continuous charges are $E$ and $J$ above.

The infinitesimal connected internal symmetry of a classical field theory is $\delta {\varphi }_1=ϵ{\varphi }_1$, $\delta {\varphi }_2=-ϵ{\varphi }_2$. For this exact invariance of the Lagrangian density, the Noether current is $J^{\mu }={\sum }_j{\mathcal{L}}_{{\partial }_{\mu }{\varphi }_j}\delta {\varphi }_j/ϵ$. In coordinates $(t,x)$ it isVarying the two fields gives the two wave equations ${\varphi }_{1,tt}-c^2{\varphi }_{1,xx}=0$ and ${\varphi }_{2,tt}-c^2{\varphi }_{2,xx}=0$. Direct differentiation cancels the first-derivative cross terms:The Noether charge $Q=\int J^{0}dx$ is constant when the spatial boundary flux vanishes. The disconnected transformations from part (a) do not have independent infinitesimal generators of this type.

For a complex scalar field with global $U(1)$ invariance and Lagrangian $\mathcal{L}=\frac{1}{2}|{\partial }_t\varphi {|}^2-\frac{1}{2}|\nabla \varphi {|}^2-U(|\varphi |)$, a Q-ball is a localized phase-rotating configuration with conserved Noether charge $Q=\int \mathrm{Im}(\bar{\varphi }{\varphi }_t)$. At fixed charge its real profile can minimizeSuch solutions require a suitable potential and charge range; the complex nature of a field alone does not ensure existence. A common condition is ${\inf }_{f>0}2U(f)/f^2<U^{\prime \prime }(0)$.

A scalar-field shift symmetry acts by $\varphi (x)\mapsto \varphi (x)+c$ for constant $c$. Its Noether charge is the spatial integral of the canonical momentum, up to the normalization of $c$, and interactions invariant under the symmetry contain derivatives of the scalar field.

This nonnegative potential has its only vacuum at $s=0$, since $U(s)=s[(s-1/2{)}^2+3/4]$. For the complex kinetic normalization $\frac{1}{2}|\partial \varphi {|}^2$, the small-amplitude mass squared is $2U^{\prime }(0)=2$, whereas ${\min }_{s>0}2U(s)/s=3/2$. The radial Q-ball equation has a positive localized profile for frequencies $3/2<{\omega }^2<2$, for example in three spatial dimensions. It is a standard non-topological soliton potential: the effective profile potential $U(f^2)-{\omega }^2{f}^2/2$ is negative somewhere, while its quadratic coefficient near zero is positive. The conserved Noether charge, not vacuum winding, supplies the stabilization mechanism.

The odd particle variables form an even antisymmetric tensor through their bilinear. Adding it to orbital angular momentum gives a conserved Lorentz Noether charge. The particle and odd-variable equations make the two time derivatives cancel. After spinning-particle Dirac quantization, the bilinear becomes the spinor Lorentz generator rather than an ordinary commuting classical spin vector.

Translations and Lorentz transformations give Noether charges ${\mathcal{P}}_m=\int P_{m}d\sigma$ and ${\mathcal{J}}_{mn}=\int (X_m{P}_n-X_n{P}_m)d\sigma$. Their time derivatives are boundary fluxes, which vanish for a closed string or a free-ended open string.

For a smooth Skyrme model field $U:{\mathbb{R}}^3\to SU(2)$ with $U(\infty )=1$, compactification gives a map $S^3\to S^3$. With $L_i=U^{†}{\partial }_{i}U$, its degree of a map between oriented manifolds is $B=-(24{\pi }^2{)}^{-1}\int {ϵ}_{ijk}\mathrm{tr}(L_i{L}_j{L}_k)d^{3}x$, with sign chosen so the standard decreasing hedgehog profile has $B=1$. The integrand is the pullback of the normalized volume form on $SU(2)$. Hence it is integer-valued and unchanged by smooth finite-energy homotopies. In the rational map approximation for Skyrmions, a degree-$N$ angular map and profile $f(0)=\pi ,f(\infty )=0$ give $B=-(2N/\pi )\int f^{\prime }{\sin }^{2}fdr=N$. This is a topological conservation law, not the Noether charge of isospin.