Analytical mechanics

Symmetries and Noether's theorem

Each continuous symmetry of the action yields a conservation law: time-translation invariance is associated with energy, and spatial translation with momentum.

Noether's theorem links two ideas that may seem separate: symmetries of the laws and conserved quantities. If the physical description is unchanged under a continuous transformation, the equations of motion imply a Noether quantity that remains constant.

\delta L=\frac{dF}{dt} \quad\Longrightarrow\quad Q=\sum_i p_i\,\delta q_i-F=\text{constant}

Definition: Continuous symmetry and Noether charge

A continuous transformation $q_i\to q_i+\epsilon\Delta q_i$ is a symmetry of the action if the Lagrangian changes at most by a total derivative $dF/dt$ . Then $Q=\sum_i p_i\Delta q_i-F$ is conserved on solutions, where $p_i=\partial L/\partial\dot q_i$ . The familiar simple form has $F=0$ .

Rotate the coordinate frame around a central potential: the setup is unchanged and angular momentum L is conserved.

Familiar symmetries

Continuous symmetry	Conserved quantity
Time translation	Energy
Spatial translation	Linear momentum
Spatial rotation	Angular momentum

For a particle in a central potential $V(r)$ , rotations about the center leave $L=\tfrac12m\dot{\mathbf r}^{2}-V(r)$ unchanged. Rotational symmetry about the $z$ axis yields conservation of $L_z$ ; in a central potential the full angular-momentum vector is conserved.

Example: From a cyclic angle to angular momentum

A particle of mass $m$ moves in a plane under a potential $V(r)$ . Show that its angular momentum about the perpendicular axis is conserved.

Solution

$L=\tfrac12m(\dot r^2+r^2\dot\phi^2)-V(r)$ has no explicit dependence on $\phi$ . Thus $p_\phi=\partial L/\partial\dot\phi=mr^2\dot\phi=L_z$ is constant by Euler-Lagrange.

Quick check

In general, consider a continuous transformation $q_i\mapsto q_i+\epsilon\Delta q_i$ under which the Lagrangian changes at most by a total derivative, $\epsilon\,dF/dt$ . On solutions of the equations of motion, $Q=\sum_i p_i\Delta q_i-F$ has zero time derivative. If the symmetry leaves time unchanged and $F=0$ , this reduces to $Q=\sum_i p_i\Delta q_i$ . A symmetry may transform time as well as coordinates; then an energy term enters, and the extended form of the theorem is needed.

A conservation law does not require the Lagrangian to be strictly unchanged. If it changes by a total derivative, the action changes only at the boundary; with fixed endpoints the equations of motion are unchanged. This is why Noether’s theorem covers more symmetries than strict invariance alone.

Which conserved quantity is associated with time-translation symmetry?

In a central potential, rotational symmetry about the center conserves what?

References

Cornelius Lanczos (1970). The Variational Principles of Mechanics