Contact Dynamics

Reeb Vector Field

There exists a unique vector field $R$ on $M$ such that $\{\begin{array}{l}{ı}_{R}d\alpha =0\\ {ı}_R\alpha =1\end{array}$ The vector field $R$ is called the Reeb vector field determined by $\alpha$. Proof. $\{\begin{array}{l}{ı}_{R}d\alpha =0⟹R\in \ker d\alpha \text{ , which is a line bundle, and}\\ {ı}_R\alpha =1⟹\text{normalizes }R.\end{array}$

Claim

The flow of R preserves the contact form, i.e., if${\rho }_t=\exp tR$ is the isotopy generated by R, then ${\rho }_t^{\ast }\alpha =\alpha ,\forall t\in \mathbb{R}$.

Contactomorphism

A contactomorphism is a diffeomorphism f of a contact manifold $(M,H)$ which preserves the contact structure $(i.e.,f_{\ast }H=H).$

Symplectization

Let $(M,H)$ be a contact manifold with a contact form $\alpha$. Let $\tilde{M}=M\times \mathbb{R}$, and let $\pi :\tilde{M}\to M$, $(p,\tau )\mapsto p$, be the projection. Then $\omega =d(e^{\tau }{\pi }^{\ast }\alpha )$ is a symplectic form on $\tilde{M}$, where $\tau$ is a coordinate on $\mathbb{R}$. And we call $(\tilde{M},\omega )$ the symplectization of $(M,\alpha )$.