(a) A mechanical system with $n$ degrees of freedom has the Lagrangian $L(\mathbf{q}, \dot{\mathbf{q}})$, where $\mathbf{q}=\left(q_{1}, \ldots, q_{n}\right)$ are the generalized coordinates and $\dot{\mathbf{q}}=d \mathbf{q} / d t$.

Suppose that $L$ is invariant under the continuous symmetry transformation $\mathbf{q}(t) \mapsto$ $\mathbf{Q}(s, t)$, where $s$ is a real parameter and $\mathbf{Q}(0, t)=\mathbf{q}(t)$. State and prove Noether's theorem for this system.

(b) A particle of mass $m$ moves in a conservative force field with potential energy $V(\mathbf{r})$, where $\mathbf{r}$ is the position vector in three-dimensional space.

Let $(r, \phi, z)$ be cylindrical polar coordinates. $V(\mathbf{r})$ is said to have helical symmetry if it is of the form

$$V(\mathbf{r})=f(r, \phi-k z),$$

for some constant $k$. Show that a particle moving in a potential with helical symmetry has a conserved quantity that is a linear combination of angular and linear momenta.