I have recently started studying classical field theory. Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. But I find that often when solving for conserved currents, it is assumed that they are due to invariance of Lagrangian i.e; $\delta L = 0$. Are both the statements same always?
[Physics] Invariance of Action vs. Lagrangian in Noether’s theorem
actionclassical-field-theorylagrangian-formalismnoethers-theoremsymmetry
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For continuous global symmetries, Noether theorem gives you a locally conserved charge density (and an associated current), whose integral over all of space is conserved (i.e. time independent).
For global discrete symmetries, you have to distinguish between the cases where the conserved charge is continuous or discrete. For infinite symmetries like lattice translations the conserved quantity is continuous, albeit a periodic one. So in such case momentum is conserved modulo vectors in the reciprocal lattice. The conservation is local just as in the case of continuous symmetries.
In the case of finite group of symmetries the conserved quantity is itself discrete. You then don't have local conservation laws because the conserved quantity cannot vary continuously in space. Nevertheless, for such symmetries you still have a conserved charge which gives constraints (selection rules) on allowed processes. For example, for parity invariant theories you can give each state of a particle a "parity charge" which is simply a sign, and the total charge has to be conserved for any process, otherwise the amplitude for it is zero.
I) For a mathematical precise treatment of an inverse Noether's Theorem, one should consult e.g. Olver's book (Ref. 1, Thm. 5.58), as user orbifold also writes in his answer (v2). Here we would like give a heuristic and less technical discussion, to convey the heart of the matter, and try to avoid the language of jets and prolongations as much as possible.
In popular terms, we would like to formulate an "inverse Noether machine"
$$ \text{Input: Lagrangian system with known conservation laws} $$ $$ \Downarrow $$ $$ \text{[inverse Noether machine]} $$ $$ \Downarrow $$ $$ \text{Output: (quasi)symmetries of action functional} $$
Since this "machine" is supposed to be a mathematical theorem that should succeed everytime without exceptions (else it is by definition not a theorem!), we might have to narrow down the set/class/category of inputs that we allow into the machine in order not to have halting errors/breakdowns in the machinery.
II) Let us make the following non-necessary restrictions for simplicity:
Let us focus on point mechanics with a local action functional $$ S[q] ~=~ \int\! dt~ L(q(t), \frac{dq(t)}{dt}, \ldots,\frac{d^Nq(t)}{dt^N} ;t), \tag{1} $$ where $N\in\mathbb{N}_0$ is some finite order. Generalization to classical local field theory is straightforward.
Let us restrict to only vertical transformations $\delta q^i$, i.e., any horizontal transformation $\delta t=0$ vanishes. (Olver essentially calls these evolutionary vector fields, and he mentions that it is effectively enough to study those (Ref. 1, Prop. 5.52).)
Let us assume, as Olver also does, that the Lagrangian $L$ and the transformations are real analytic$^{\dagger}$.
The following technical restrictions/extensions are absolutely necessary:
The notion of symmetry $\delta S=0$ should be relaxed to quasisymmetry (QS). By definition a QS of the action $S$ only has to hold modulo boundary terms. (NB: Olver uses a different terminology: He calls a symmetry for a strict symmetry, and a quasisymmetry for a symmetry.)
The notion of QS transformations might only make sense infinitesimally/as a vector field/Lie algebra. There might not exist corresponding finite QS transformations/Lie group. In particular, the QS transformations are allowed to depend on the velocities $\dot{q}$. (Olver refers to this as generalized vector fields (Ref. 1, Def. 5.1).)
III) Noether's Theorem provides a canonical recipe of how to turn a QS of the action $S$ into a conservation law (CL),
$$ \frac{dQ}{dt}~\approx~0,\tag{2}$$
where $Q$ is the full Noether charge. (Here the $\approx$ symbol means equality on-shell, i.e. modulo the equations of motion (eom).)
Remark 1: Apart from time $t$, the QS transformations are only allowed to act on the variables $q^i$ that actively participate in the action principle. If there are passive external parameters, say, coupling constants, etc, the fact that they are constant in the model are just trivial CLs, which should obviously not count as genuine CLs. In particular, $\frac{d1}{dt}=0$ is just a trivial CL.
Remark 2: A CL should by definition hold for all solutions, not just for a particular solution.
Remark 3: A QS of the action $S$ is always implicitly assumed to hold off-shell. (It should be stressed that an on-shell QS of the action
$$ \delta S \approx \text{boundary terms}\tag{3} $$
is a vacuous notion, as the Euler-Lagrange equations remove any bulk term on-shell.)
Remark 4: It should be emphasized that a symmetry of eoms does not always lead to a QS of the Lagrangian, cf. e.g. Ref. 2, Example 1 below, and this Phys.SE post. Hence it is important to trace the off-shell aspects of Noether's Theorem.
Example 1: A symmetry of the eoms is not necessarily a QS of the Lagrangian. Let the Lagrangian be $L=\frac{1}{2}\sum_{i=1}^n \dot{q}^i g_{ij} \dot{q}^j$, where $g_{ij}$ is a constant non-degenerate metric. The eoms $\ddot{q}^i\approx 0$ have a $gl(n,\mathbb{R})$ symmetry $\delta q^{i}=\epsilon^i{}_j~q^{i}$, but only an $o(n,\mathbb{R})$ Lie subalgebra of the $gl(n,\mathbb{R})$ Lie algebra is a QS of the Lagrangian.
IV) Without further assumptions, there is a priori no guarantee that the Noether recipe will turn a QS into a non-trivial CL.
Example 2: Let the Lagrangian $L(q)=0$ be the trivial Lagrangian. The variable $q$ is pure gauge. Then the local gauge symmetry $\delta q(t)=\epsilon(t)$ is a symmetry, although the corresponding CL is trivial.
Example 3: Let the Lagrangian be $L=\frac{1}{2}\sum_{i=1}^3(q^i)^2-q^1q^2q^3$. The eom are $q_1\approx q_2q_3$ and cyclic permutations. It follows that the positions $q^i\in\{ 0,\pm 1\}$ are constant. (Only $1+1+3=5$ out of the $3^3=27$ branches are consistent.) Any function $Q=Q(q)$ is a conserved quantity. The transformation $\delta q^i=\epsilon \dot{q}^i$ is a QS of the action $S$.
If we want to formulate a bijection between QSs and CLs, we must consider equivalence classes of QSs and CLs modulo trivial QSs and CLs, respectively.
- A QS transformation $\delta q^i$ is called trivial if it vanishes on-shell (Ref. 1, p.292).
A CL is called
trivial of first kind if the Noether current $Q$ vanishes on-shell.
trivial of second kind if CL vanishes off-shell.
trivial if it is a linear combination of CLs of first and second kinds (Ref. 1, p.264-265).
V) The most crucial assumption is that the eoms are assumed to be (totally) non-degenerate. Olver writes (Ref. 1, Def. 2.83.): A system of differential equations is called totally non-degenerate if it and all its prolongations are both of maximal rank and locally solvable$^{\ddagger}$.
The non-degeneracy assumption exclude that the action $S$ has a local gauge symmetry. If $N=1$, i.e. $L=L(q,\dot{q},t)$, the non-degeneracy assumption means that the Legendre transformation is regular, so that we may easily construct a corresponding Hamiltonian formulation $H=H(q,p,t)$. The Hamiltonian Lagrangian reads
$$ L_H~=~p_i \dot{q}^i-H.\tag{4} $$
VI) For a Hamiltonian action functional $S_H[p,q] = \int\! dt~ L_H$, there is a canonical way to define an inverse map from a conserved quantity $Q=Q(q,p,t)$ to a transformation of $q^i$ and $p_i$ by using the Noether charge $Q$ as Hamiltonian generator for the transformations, as also explained in e.g. my Phys.SE answer here. Here we briefly recall the proof. The on-shell CL (2) implies
$$ \{Q,H\}+\frac{\partial Q}{\partial t}~=~0\tag{5} $$
off-shell, cf. Remark 2 and this Phys.SE post. The corresponding transformation
$$ \delta q^i~=~ \{q^i,Q\}\epsilon~=~\frac{\partial Q}{\partial p_i}\epsilon\qquad \text{and}\qquad \delta p_i~=~ \{p_i,Q\}\epsilon~=~-\frac{\partial Q}{\partial q^i}\epsilon\tag{6} $$
is a QS of the Hamiltonian Lagrangian
$$\begin{align} \delta L_H ~\stackrel{(4)}{=}~&\dot{q}^i \delta p_i -\dot{p}_i \delta q^i -\delta H+\frac{d}{dt}(p_i \delta q^i)\cr ~\stackrel{(6)+(8)}{=}& -\dot{q}^i\frac{\partial Q}{\partial q^i}\epsilon -\dot{p}_i\frac{\partial Q}{\partial p_i}\epsilon -\{H,Q\}\epsilon + \epsilon \frac{d Q^0}{dt}\cr ~\stackrel{(5)}{=}~& \epsilon \frac{d (Q^0-Q)}{dt}~\stackrel{(9)}{=}~ \epsilon \frac{d f^0}{dt},\end{align}\tag{7} $$
because $\delta L_H$ is a total time derivative. Here $Q^0$ is the bare Noether charge
$$ Q^0~=~ \frac{\partial L_H}{\partial \dot{q}^i} \{q^i,Q\} + \frac{\partial L_H}{\partial \dot{p}_i} \{p_i,Q\} ~=~ p_i \frac{\partial Q}{\partial p_i},\tag{8} $$
and
$$ f^0~=~ Q^0-Q .\tag{9} $$
Hence the corresponding full Noether charge
$$ Q~=~Q^0-f^0\tag{10} $$
is precisely the conserved quantity $Q$ that we began with. Therefore the inverse map works in the Hamiltonian case.
Example 4: The non-relativistic free particle $L_H=p\dot{q}-\frac{p^2}{2m}$ has e.g. the two conserved charges $Q_1=p$ and $Q_2=q-\frac{pt}{m}$.
The inverse Noether Theorem for non-degenerate systems (Ref. 1, Thm. 5.58) can intuitively be understood from the fact, that:
Firstly, there exists an underlying Hamiltonian system $S_H[p,q]$, where the bijective correspondence between QS and CL is evident.
Secondly, by integrating out the momenta $p_i$ we may argue that the same bijective correspondence holds for the original Lagrangian system.
VII) Finally, Ref. 3 lists KdV and sine-Gordon as counterexamples to an inverse Noether Theorem. KdV and sine-Gordon are integrable systems with infinitely many conserved charges $Q_n$, and one can introduce infinitely many corresponding commuting Hamiltonians $\hat{H}_n$ and times $t_n$. According to Olver, KdV and sine-Gordon are not really counterexamples, but just a result of a failure to properly distinguishing between non-trivial and trivial CL. See also Ref. 4.
References:
P.J. Olver, Applications of Lie Groups to Differential Equations, 1993.
V.I. Arnold, Mathematical methods of Classical Mechanics, 2nd eds., 1989, footnote 38 on p. 88.
H. Goldstein, Classical Mechanics; 2nd eds., 1980, p. 594; or 3rd eds., 2001, p. 596.
L.H. Ryder, Quantum Field Theory, 2nd eds., 1996, p. 395.
$^{\dagger}$ Note that if one abandons real analyticity, say for $C^k$ differentiability instead, the analysis may become very technical and cumbersome. Even if one works with the category of smooth $C^\infty$ functions rather than the category of real analytic functions, one could encounter the Lewy phenomenon, where the equations of motion (eom) have no solutions at all! Such situation would render the notion of a conservation law (CL) a bit academic! However, even without solutions, a CL may formally still exists as a formal consequence of eoms. Finally, let us add that if one is only interested in a particular action functional $S$ (as opposed to all action functionals within some class) most often, much less differentiability is usually needed to ensure regularity.
$^{\ddagger}$ Maximal rank is crucial, while locally solvable may not be necessary, cf. previous footnote.
Best Answer
No, they are not the same. To see why, even in classical mechanics, suppose we have symmetry transformation $q \rightarrow q + \epsilon K$ that leaves the Lagrangian invariant. This means that we must have \begin{equation} \lim_{\epsilon \rightarrow 0} \frac{1}{\epsilon}\left(L(q+\epsilon K, \dot{q}+\epsilon\dot{K},t) - L(q,\dot{q},t)\right) = \frac{\partial L}{\partial q}K +\frac{\partial L}{\partial \dot{q}}\dot{K} = 0 \end{equation} $Then$ you use the fact that the equations of motion are satisfied to write $\frac{\partial L}{\partial q}=\frac{d}{dt}\frac{\partial L }{\partial \dot{q}}$ and this implies \begin{equation} \frac{d}{dt}\frac{\partial L}{\partial \dot{q}}K+\frac{\partial L}{\partial \dot{q}}\dot{K}=\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}K\right)=0 \end{equation} i.e. the quantity $\frac{\partial L}{\partial \dot{q}}K$ is conserved.
A symmetry of the action is a transformation that leaves the action invariant whether or not the equations of motion are satisfied. In this case the same procedure yields the condition \begin{equation} \frac{\partial L}{\partial q}K + \frac{\partial L}{\partial \dot{q}}\dot{K} = \frac{dM}{dt} \end{equation} where $M$ is a function of $q, \dot{q}, t$. If such an M exists, we say that the action is invariant under the symmetry transformation.
It's very easy to see that when we impose the equations of motion LHS becomes $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}K\right)$ and we can derive a conserved quantity: \begin{equation} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}K - M\right)=0. \end{equation}
The simplest possible example of a symmetry transformation which is a symmetry of the action but not of the Lagrangian is time translation in systems where the Lagrangian has no explicit time dependence. When we shift the time by an arbitrary small $\epsilon$, the generalized coordinates $q$ change according to $q(t) \rightarrow q(t) + \epsilon \dot{q}(t)$, therefore $K=\dot{q}$. But \begin{equation} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\dot{q}\right)=\frac{\partial L }{\partial q}\dot{q} + \frac{\partial L}{\partial \dot{q}}\ddot{q}=\frac{dL}{dt}\neq 0 \end{equation} In this case $M =L$ and the conserved quantity is \begin{equation} H = \frac{\partial L}{\partial \dot{q}}\dot{q} - L. \end{equation}