# 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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