[NOTES/CM-03002] Symmetries and Conservation Laws

The action principle is an elegant formulation of the laws of motion of a dynamical system. This formalism also provides an important connection between symmetries of Lagrangian and conservation laws. One of the applications of the conservation laws is in integration of equations of motion.

In areas of physics, such as particle physics, where interactions were not known experimentally observed conservation laws and selection rules and corresponding symmetry principles have been guiding principles towards building a theory.

Many times the Lagrangian has a set of continuous geometric transformations which is easy to guess. Such a geometric transformation can be built up as a succession of infinitesimal transformations and the symmetry implies a conservation law.

Symmetry under a continuous transformation

Consider an infinitesimal transformation \begin{equation}\label{EQ01} \delta q_k \Rightarrow q'_k=q_k + \delta q_k \end{equation} where $\delta q_k$ is a specific variation under consideration \begin{equation}\label{EQ02} \delta q_k = \varepsilon \phi_k. \end{equation} where \(\epsilon\) is an infinitesimal quantity and \(\phi_k\) are some functions of generalised coordinates.

We say that the Lagrangian is invariant under transformations \eqref{EQ01}-\eqRef{EQ02} if \begin{equation}\label{EQ03} L(q_k+\varepsilon \phi_k, \dot q_k+\varepsilon \dot\phi_k,t)-L(q_k,\dot q_k,t)=o(\varepsilon^2), \end{equation} {\it i.e.} R.H.S. has {\sf no term proportional} to $\varepsilon$. An infinitesimal transformation type \eqref{EQ01} applied to a path $C$ in configuration space, gives rise to another path $C'$ which is close to $C$. The corresponding variation in action can be computed and is given by

\begin{eqnarray}
\delta\Phi[C]
&=&
\int_{t_1}^{t_2}\sum_k\Big(\frac{\partial{L}}{\partial{q_k}}
\delta q_k(t) + \frac{\partial{L}}{\partial{\dot q_k}}
\delta\dot{q}_k(t)\Big)
\\
&=&\int_{t_1}^{t_2}\sum_k\Big(\frac{\partial{L}}{\partial{q_k}}
\delta q_k(t) + \frac{\partial{L}}{\partial{\dot q_k}}
\dd{t}\big(\delta{q}_k(t)\big)\Big)
\\
&& \qquad \qquad
\HighLight{Integrating the second term by parts}\nonumber
\\
&=&\nonumber
\int_{t_1}^{t_2}\sum_k\Big(\frac{\partial{L}}{\partial{q_k}}
\delta q_k(t)-\Big(\frac
{d} {dt}\frac{\partial{L}}{\partial{\dot
q_k}}\Big) \delta{q}_k(t)\Big)+\sum_k p_k \delta q_k(t)\Big|_{t_1}^{t_2}
\\
&& \qquad\qquad \HighLight{where $p_k=\pp[L]{\dot{q}_k}$}\nonumber
\\
&=&\varepsilon\int_{t_1}^{t_2}\sum_k\Big(\frac{\partial{L}}{\partial{q_k}}-\frac
{d} {dt}\frac{\partial{L}}{\partial{\dot
q_k}}\Big)\phi_k(t)+\varepsilon\sum_kp_k \phi_k(t)\Big|_{t_1}^{t_2}.\label{EQ04}
\end{eqnarray}
\eqref{EQ03} implies $\Delta \Phi[C]=0$.

If $C$ is the classical trajectory on which the EOM are obeyed, the integrand in the first term vanishes and we get \begin{equation} F(t_2)=F(t_1) \label{EQ05} \end{equation} where \begin{equation} F(t)=\sum_k p_k \phi_k(t) =\sum_k\Big(\frac{\partial{L}}{\partial{\dot q_k}}\Big)\phi_k(t) \end{equation} The equation \eqRef{EQ05} shows that $F(t)$ is independent of time, when ever EOM are obeyed, {\it i.e.} $F(t)$ is a constant of motion.

Quasi invariance under a transformation

In the previous section it was assumed that the Lagrangian is invariant under given symmetry transformation. The conservation laws exist in more general case of Lagrangian being invariant up to total time derivative. We shall say that Lagrangian is ``quasi invariant" when, instead of \eqRef{EQ03}, we have \begin{equation} L(q_k+\varepsilon\phi _k,\dot q_k+\varepsilon\dot\phi_k,t)-L(q_k,\dot q_k,t)=\varepsilon\frac{d\Omega}{dt} \end{equation} where $\Omega$ is a function of coordinates. In this case the L.H.S of \eqRef{EQ04} becomes $\sum_k\int_{t_1}^{t_2}\frac{d\Omega}{dt}$ and we get \begin{equation} \varepsilon\Omega(t)\Big |_{t_t}^{t_2}=\varepsilon \int_{t_1}^{t_2}\sum_k\Big(\frac{\partial{L}}{\partial {q_k}}-\frac{d}{dt}\frac{\partial{L}}{\partial{\dot q_k}}\Big)\phi_kdt+\varepsilon\sum p_k \phi_k(t)\Big|_{t_1}^{t_2} \end{equation} Using the EOM the first term inside the integral vansihes. So we get \begin{equation} \Big(\sum_kp_k\phi_k(t)-\Omega(t)\Big)\Big|_{t_1}^{t_2}=0 \end{equation} and the quantity \begin{equation} G(t)=\sum p_k\phi_k-\Omega \end{equation} is a constant of motion.

Noether's Theorem

The result obtained here is summarized as follows: associated with every continuous symmetry transformation of action, there exists a conserved quantity.