A potential $V(r,\phi,z)=f(r,\phi-kz)$ is invariant under the screw transformation $(\phi,z)\mapsto(\phi+ks,z+s)$. The <Noether conserved quantity for a mechanical point symmetry> is
$$
p_z+kp_\phi,
$$
a linear combination of axial <linear momentum> and axial <angular momentum>.

Helical symmetry of a potential

Let
$$
\xi_i(t)=\left.\frac{\partial Q_i(s,t)}{\partial s}\right|_{s=0}
$$
be the infinitesimal generator of the continuous symmetry, and let
$$
p_i=\frac{\partial L}{\partial\dot q_i}
$$
be the <canonical momentum>. <Noether theorem> states that along every trajectory satisfying the <Euler-Lagrange equations>,
$$
\boxed{J=\sum_{i=1}^np_i\xi_i}
$$
is conserved.

Indeed, differentiate the invariance of $L$ with respect to $s$ at zero:
$$
0=\sum_i\left(
\frac{\partial L}{\partial q_i}\xi_i
+\frac{\partial L}{\partial\dot q_i}\dot\xi_i
\right).
$$
On an Euler-Lagrange trajectory, $\partial L/\partial q_i=d p_i/dt$, so
$$
0=\sum_i(\dot p_i\xi_i+p_i\dot\xi_i)
=\frac d{dt}\sum_i p_i\xi_i.
$$
This proves the <Noether conserved quantity for a mechanical point symmetry>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 8E / a / Solution

{c}
{title2=$J=\sum_i p_i\xi_i$}

= Noether's theorem
{c}
{synonym}

If a time-independent <Lagrangian> $L(q,\dot q)$ is invariant under a one-parameter point transformation with infinitesimal generator $\xi_i=\partial_sQ_i|_{s=0}$, then every Euler-Lagrange trajectory conserves
$$
J=\sum_i\frac{\partial L}{\partial\dot q_i}\xi_i.
$$

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