Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 140 / 4

Let $\pi :T^{\ast }X\to X$ be the cotangent bundle projection and $\lambda$ its canonical one-form on a cotangent bundle, defined by ${\lambda }_{(x,\xi )}(v)=\xi (d\pi (v))$. Use the symplectic form $\omega =-d\lambda$, so in local coordinates $\omega ={\sum }_{j}d{x}_j\wedge d{\xi }_j$.

The graph map $s_{\beta }(x)=(x,{\beta }_x)$ is a smooth embedding, with $s_{\beta }^{\ast }\lambda =\beta$, and henceIts image has dimension $\dim X$, half the dimension of $T^{\ast }X$. By the definition of a Lagrangian submanifold,This is the graph of a closed one-form is Lagrangian criterion. The opposite conventional sign for the symplectic form gives the same criterion.

The cotangent fiber translation $f$ is a diffeomorphism, with inverse $(x,\xi )\mapsto (x,\xi -{\beta }_x)$. Because $\pi \circ f=\pi$, the definition of the canonical one-form on a cotangent bundle givesTaking the exterior derivative yieldsIf $\beta$ is a closed differential form, this is $f^{\ast }\omega =\omega$, proving that the cotangent fiber translation is a symplectomorphism.

Write the exact differential form as $\beta =dh$ and take the cotangent fiber translation family ${\rho }_t(x,\xi )=(x,\xi +td{h}_x)$. Its generating vector field is vertical and in local coordinates isWith $\omega ={\sum }_{j}d{x}_j\wedge d{\xi }_j$ and the Hamiltonian vector field convention from Question 3,Thus ${\rho }_t$ is a Hamiltonian isotopy and $f={\rho }_1$ is a Hamiltonian diffeomorphism. The local flow is explicitly defined for every real $t$, regardless of compactness. With the alternative convention ${\iota }_{X_H}\omega =-dH$, the same family is generated by $H=h\circ \pi$.

The definition in the question does not require compact support for the Hamiltonian function; the displayed Hamiltonian diffeomorphism uses that definition.