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sg.symplectic geometry – Is a homogeneous symplectic isotopy all the time Hamiltonian? retort

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sg.symplectic geometry – Is a homogeneous symplectic isotopy all the time Hamiltonian?

The following statements are written within the Guillermou-Kashiwara-Shapira’s paper “Sheaf quantization of Hamiltonian isotopies and applications to non-displaceability”.

Let $M$ breathe a light-weight manifold and $I$ an launch interval of $mathbb{R}$ containing the inception. assume a homogeneous symplectic isotopy $Phi:dot{T}^*Mtimes Irightarrowdot{T}^*M$ the place $dot{T}^*M=T^*M-0_M$, that’s, for every $tin I$, $phi_t=Phi(cdot,t)$ is a homogeneous symplectomorphism and $phi_0=id_{dot{T}^*M}$. clique $v_Phi=frac{partialPhi}{partial t}:dot{T}^*Mtimes Irightarrow Tdot{T}^*M$, $f=alpha_M(v_Phi):dot{T}^*Mtimes Irightarrowmathbb{R}$ the place $alpha_M$ is the canonical Liouville 1-figure of $dot{T}^*M$ and $f_t=f(cdot,t)$. Then $v_Phi=X_{f_t}$ the place $X_{f_t}$ is a Hamiltonian vector bailiwick in $dot{T}^*M$.

In this assertion, I can’t win why the id $v_Phi=X_{f_t}$ holds. In specific, I do not know the way to make use of the homogenity situation.

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