[NOTES/CM-03006] A short cut to Noether generator and equation for its time variation

A short cut to Noether generator and equation for its time variation

We give a prescription for obtaining conserved quantities associated with a continuous symmetry transformation. In cases where the symmetry is broken, one can also obtain the time variation of the associated generator of symmetry transformation.
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Consider a system with generalized coordinates described by the Lagrangian \(L(q,\dot{q},t)\). Let us consider an infinitesimal transformation \begin{equation} q_k \longrightarrow q_k\Prime = q_k + \epsilon \Delta q_k \end{equation} where \(\epsilon\) is an infinitesimal parameter. Let us take \(\epsilon\) as function of time \(t\), \(\epsilon=\epsilon(t)\) and compute the variation of the Lagrangian. Upto the first order in \(\epsilon\), we compute \begin{equation} \Delta L = L(q\Prime,\dot{q}\Prime,t) - L(q,\dot{q},t) \end{equation} We define \begin{equation} G \stackrel{\text{def}}{\equiv} \pp[\Delta L]{\dot{\epsilon}} \end{equation} which obeys \begin{equation}\label{EQ04} \dd[G]{t} = \pp[(\Delta L)]{\epsilon} \end{equation} If the Lagrangian is invariant under the transformations, the right hand side of \eqref{EQ04} is zero. In such a case we will have the conservation law \begin{equation} \dd[G]{t} = 0. \end{equation} It can be shown that \(G\) is just the conservation quantity given by Noether's theorem.

An Example
We will illustrate the method by means of a simple example of a particle in a potential \(V(x)\). The Lagrangian is \begin{equation} L = \frac{1}{2}m \dot{x}^2 - V(x). \end{equation} Consider translations \(x\to x\Prime = x +\epsilon(t)\). Then

\begin{eqnarray}
\Delta L &=&L(x\Prime, \dot{x}\Prime,t)-L(x,\dot x,t)\\
&=& m \dot{x}\, \Big(\dd[\epsilon]{t}\Big) - V\Prime(x)\, \epsilon(t).
\end{eqnarray}

Then we have \begin{equation} G = \frac{\partial(\Delta L)}{\partial(\partial_t{\epsilon})} = m \dot{x} \end{equation} and \begin{equation} \dd[G]{t} = \pp[(\Delta L)]{\epsilon} = - V\Prime(x). \end{equation} \eqref{EQ04} takes the form \begin{equation} m \ddot{x} =- V\Prime(x). \end{equation} which is just the equation of motion.

In field theory, one takes the parameters dependent on space time and one can then get equation of continuity for broken symmetry cases in a similar fashion. Denoting the Lagrangian density by \(\Lsc\) Thus the current is given by \begin{equation} J^\mu = \pp[\Delta \Lsc]{(\partial_\mu \epsilon)} \end{equation} and its divergence can be computed using \begin{equation} \partial_\mu J^\mu = \pp[\Delta\Lsc]{\epsilon(x,t)}. \end{equation}

The scheme just described, for mechanical systems, is adoption of a method used for fields extensively in mid sixties and was known as GelMann Levy prescription. See, for example,

Reference

See Ch1 Sec 2,"Current Commutators And Divergences In Lagrangian Field Theory" in
Adler S. L. and Dashen R. F. , "Current Algebras and Their Applications" W. A. Benjamin, New York (1968).