5.3 Noether’s theorem

Noether’s theorem states that any continuous transformation that does not change the action is associated with a conserved quantity (we call the quantity a “charge”, \(Q\)). We say that the action is invariant under such a continuous transformation, and we mean that:

\begin{align} S=\int_{t_a}^{t_b}L(q_i,\dot{q}_i,t)dt=\int_{t_a'}^{t_b'}L(q_i',\dot{q}_i',t)dt \tag{5.5}\end{align}

where on the right hand side, since \(t\) is just a dummy integration variable we left it as \(t\) instead of \(t'\) (note however that we did need to change the limits of the integral to be in the transformed time). As a notation, we imply that \(L(q_i,\dot{q}_i,t)\) depends on all the \(q_i\).

Because the continuous transformations are continuous we can consider the case of the infinitesimal version of a given transformation. If the action is invariant under the infinitesimal transformation, it will also be invariant under a series of infinitesimal transformations that lead to a total transformation. Requiring invariance under the infinitesimal transformation:

\begin{align} S&=\int_{t_a}^{t_b}L(q_i,\dot{q}_i,t)dt=\int_{t_a+\delta t_a}^{t_b+\delta t_b} L(q_i+\delta q_i', \dot{q}_i+\delta\dot{q}_i',t)dt\\ \therefore \delta S&=\int_{t_a+\delta t_a}^{t_b+\delta t_b} L(q_i+\delta q_i', \dot{q}_i+\delta\dot{q}_i',t)dt-\int_{t_a}^{t_b}L(q_i,\dot{q}_i,t)dt \tag{5.6}\end{align}

where the \(\delta t_{a(b)}=g(t_a(t_b)) \delta \epsilon\) in the limits of the integral, can in principle depend on \(t\) (through \(g\)) and are thus different at the two end points. If the action is invariant under the transformation, then \(\delta S=0\).

Under the condition that \(\delta S=0\) the following quantity \(Q\) is conserved (does not vary with time):

\begin{align} Q&\equiv\sum_{i=1}^np_i\delta q_i +\left(\sum_{i=1}^n p_i\dot{q}_i-L\right)\delta t\\ &=\sum_{i=1}^n p_if_i +\left(\sum_{i=1}^n p_i\dot{q}_i-L\right)g\end{align}

where we have (re-)introduced \(\delta q_i=f_i\delta\epsilon\) and \(\delta t=g\delta\epsilon\). Note that there are two terms, one is associated with invariance in time, \(g\), and one is associated with invariance in space, \(f\). You may recognize the one related to time as the Jacobi integral.

Note that if a symmetry preserves the Lagrangian, the action will also be preserved. It is often more straightforward to verify if the Lagrangian is invariant under a transformation. It is also more straightforward to only consider variations in the position coordinates.

Consider the variation of the Lagrangian with respect to the coordinate transformation:

\begin{align} q_i'&=q_i+f_i(q_1,\dots ,q_n)\delta\epsilon\\ \delta q_i&=\delta\epsilon f_i(q_1,\dots ,q_n)\\ \delta L&=\sum_{i=1}^n \frac{\partial L}{\partial q_i}\delta q_i+\frac{\partial L}{\partial \dot{q}_i}\delta \dot{q}_i \tag{5.8}\end{align}

Recall the \(L\) must still satisfy Lagrange’s equations, so we have:

\begin{align} p_i&=\frac{\partial L}{\partial \dot{q}_i}\\ \dot{p}_i&=\frac{\partial L}{\partial q_i} \tag{5.9}\end{align}

which we can substitute back into the variation of the Lagrangian:

\begin{align} \delta L&=\sum_{i=1}^n \dot{p}_i\delta q_i+p_i\delta \dot{q}_i\\ &=\frac{d}{dt}\sum_{i=1}^np_i\delta q_i\\ &=\delta\epsilon \frac{d}{dt}\sum_{i=1}^np_if_i(q_1,\dots ,q_n) \tag{5.10}\end{align}

which must be true for any arbitrary \(\epsilon\). Thus, if the variation of the Lagrangian is zero then the quantity:

\begin{align} Q=\sum_{i=1}^np_if_i(q_1,\dots ,q_n) \tag{5.11}\end{align}

is conserved, which is one of the quantities we had before. In principle, the variation of \(L\) can be equal to zero within a total time derivative, since adding a total time derivative to the Lagrangian does not change the equations of motion.

Note that Noether’s theorem only applies to continuous transformations that leave the action or the Lagrangian unchanged. For example, a transformation that mirrors the coordinate system may preserve the Lagrangian, but it is not a continuous transformation, so does not have an associated conserved quantity. There may be other continuous symmetries that are not straightforward coordinate transforms (such as those that we considered in this proof) that may lead to less obvious conserved quantities.