2
$\begingroup$

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

$\endgroup$
2
$\begingroup$

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.

For reference, the more general result appears as Proposition 5.64 in Olver's Applications of Lie groups to differential equations (Springer, 1993). Olver also requires the PDE to be "totally non-degenerate", a hypothesis whose precise meaning you'll need to look up there. It is satisfied though by those ODEs that can be solved for the highest time derivatives.

$\endgroup$
  • $\begingroup$ Sarlet and Cantrijn also mention "that the meaning of this property as well as the proof of it are much simpler when reinterpreted in the Hamiltonian framework ... it simply follows from the fact that the Poisson bracket of a function with itself is identically zero." $\endgroup$ – Zurab Silagadze Mar 10 '15 at 6:23
  • $\begingroup$ It is rather surprising that this well-known (for professionals) result is not mentioned in standard QFT and Classical Mechanics textbooks. $\endgroup$ – Zurab Silagadze Mar 10 '15 at 6:30
  • 1
    $\begingroup$ I found that Olver's Proposition 5.64 was proved earlier in another form by Khamitova (originally conjectured by Nail Ibragimov): link.springer.com/article/10.1007/BF01018418 (Group structure and the basis of conservation laws). $\endgroup$ – Zurab Silagadze Jun 29 '15 at 5:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.