Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 59 / 3 / i

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 $TQ$ asIf $L$ is invariant, ${\xi }_{TQ}L=0$. Along an Euler–Lagrange trajectory, the Noether conserved quantity for a mechanical point symmetry isbecauseThus 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 $Ldt$ 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 ${\xi }_{T^{\ast }Q}$, Cartan's magic formula giveswhere $J_{\xi }=p_i{\xi }^i$. These components define the canonical moment map $J:T^{\ast }Q\to {\mathfrak{g}}^{\ast }$, with $⟨J,\xi ⟩=J_{\xi }$. 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 $T^{\ast }Q$, the cotangent lift does have the canonical components just derived. A Hamiltonian symmetry preserving $H$ gives a conserved generator; the global momentum-map hypotheses must be stated.