= Solution
In <Lagrangian mechanics>, let a one-parameter group on $Q$ have generator $\xi_Q=\xi^i(q)\partial_{q^i}$. Its tangent lift acts on $TQ$ as
$$
\xi_{TQ}=\xi^i\partial_{q^i}+v^j\partial_j\xi^i\partial_{v^i}.
$$
If $L$ is invariant, $\xi_{TQ}L=0$. Along an Euler–Lagrange trajectory, the <Noether conserved quantity for a mechanical point symmetry> is
$$
J^L_\xi=\theta_L(\xi_{TQ})=\frac{\partial L}{\partial v^i}\xi^i,
$$
because
$$
\frac{dJ^L_\xi}{dt}=\dot p_i\xi^i+p_i\partial_j\xi^i v^j=L_{q^i}\xi^i+L_{v^i}\partial_j\xi^iv^j=\xi_{TQ}L=0.
$$
Thus the <conserved quantity> is explicitly derived, not merely attached to the name of a <symmetry>. If the change of $L$ is the total derivative $dB/dt$, then $p_i\xi^i-B$ is conserved. For a general point transformation with time component $\tau$, the conserved expression is $p_i\xi^i-E_L\tau-B$, provided the variation of $L\,dt$ is $dB$. Time-translation invariance of an autonomous Lagrangian therefore yields conservation of energy.
In <Hamiltonian mechanics>, a cotangent-lifted action preserves $\theta$ and hence $\omega$. For its infinitesimal generator $\xi_{T^*Q}$, <Cartan's magic formula> gives
$$
0=\mathcal L_{\xi_{T^*Q}}\theta=\iota_{\xi_{T^*Q}}d\theta+d(\theta(\xi_{T^*Q})),\qquad\iota_{\xi_{T^*Q}}\omega=dJ_\xi,
$$
where $J_\xi=p_i\xi^i$. These components define the canonical <moment map> $J:T^*Q\to\mathfrak g^*$, with $\langle J,\xi\rangle=J_\xi$. If $H$ is invariant, $\{H,J_\xi\}=0$, so $\dot J_\xi=\{J_\xi,H\}=0$. This is the Hamiltonian form of <Noether's theorem>, and the <Legendre transform in mechanics> identifies its charges with the Lagrangian ones.
More generally, a <Hamiltonian group action> is an action whose infinitesimal symplectic generators have globally defined Hamiltonian functions. The existence of those functions is an assumption, not a consequence of every symplectic action. For example, translation on a symplectic two-torus preserves $dq\wedge dp$ but contraction with $\partial_q$ gives $dp$, which is closed and not globally exact. Such a <symmetry> need not have a globally single-valued moment-map component. On $T^*Q$, the cotangent lift does have the canonical components just derived. \b[A Hamiltonian <symmetry> preserving $H$ gives a conserved generator; the global momentum-map hypotheses must be stated.]
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