# Invariance of the Noether charge

The paper http://epubs.siam.org/doi/abs/10.1137/1023098 (Generalizations of Noether’s Theorem in Classical Mechanics, by Willy Sarlet and Frans Cantrijn) mentions "an interesting property of the Noether-invariant, which is almost never mentioned in the quoted literature, at least not in the context of Lagrangian mechanics". In particular, let $$\hat G=\tau(t,q)\frac{\partial}{\partial t}+\xi^i(t,q)\frac{\partial}{\partial q^i}$$ be the generator of the infinitesimal transformation $$t^\prime=t+\epsilon\tau(t,q),\;\;\;q^{\,\prime i}=q^i+\epsilon\xi^i(t,q), \tag{1}$$ and let $$\hat E=\hat G+\left(\dot{\xi}^i-\dot{q}^i\dot{\tau}\right)\frac{\partial}{\partial \dot{q}^i}.$$ be the generator of the first extended group which gives the change induced in a function of $(t,q^i,\dot{q}^i)$ under the one-parameter group of transformations determined by $\hat G$ in the space $(t,q^i)$. Then, if (1) is a symmetry of the Lagrangian system $L(t,q^i,\dot{q}^i)$, so that $$L\left(t^\prime,q^{\,\prime i},\frac{q^{\,\prime i}}{d t^\prime}\right)\frac{dt^\prime}{dt}\approx L(t,q^i,\dot{q}^i)+\epsilon \frac{df(t,q)}{dt},$$ the conserved Noether charge, corresponding to this symmetry, $$Q=f-L\tau-\left(\xi^i-\dot{q}^i\tau\right)\frac{\partial L}{\partial \dot{q}^i}$$ is left invariant under the action of the first extended group: $$\hat E Q=0. \tag{2}$$

From the point of view of physics, (2) is just what is expected from the symmetry of the system, because (2) indicates that any two solutions of the Euler–Lagrange equations related by the symmetry transformation possess the same value of the associated conserved Noether charge. However, technically (2) does not seem to be obvious or trivial.

Besides Sarlet and Cantrijn's excellent review paper mentioned above, I found the proof and discussion of (2) only in Lutzky's papers http://iopscience.iop.org/0305-4470/11/2/005 (Symmetry groups and conserved quantities for the harmonic oscillator) and http://iopscience.iop.org/0305-4470/12/7/012 (Dynamical symmetries and conserved quantities). Are there any other papers/textbooks where this property of Noether charge is discussed?

P.S. As a result of this discussion, finally I wrote a paper http://arxiv.org/abs/1507.05009

Your question is essentially about a special case of a well-known result. True it is more often stated in the Hamiltonian language, but there is no obstacle to restating it in the Lagrangian language. Instead of an ODE, think more generally of a system of a variational PDE system. Let $\rho_v$ be conservation law corresponding to a symmetry $v$. By this, I mean that $\rho_v$ is an $(n-1)$-form (when we have $n$ independent variables) such that $d\rho_v = 0$ on shell. The conserved charge $Q$ is a special case; since for ODEs $n=1$, it is a $0$-form. If $u$ is another symmetry then the Lie derivative $\mathcal{L}_u \rho_v = \rho_{[u,v]}$ is another conservation law, now corresponding to the symmetry given by the Lie bracket $[u,v]$ of the two symmetries. Naturally, if $u=v$, then $[u,v]=0$ and $\mathcal{L}_u \rho_u = 0$. Your equation $\hat{E}Q = 0$ is precisely of that form. If we are dealing with generalized symmetries, where the components of $u$ depend on higher derivatives of the dynamical fields, it suffices to lift all the calculations to a jet bundle of the appropriate order.