An analogue of the Poisson bracket in contact geometry?

Question

McDuff and Salamon define an analogue of the Poisson bracket in contact geometry on page 135 in the third version of Introduction to Symplectic Topology. The definition is the following.

Let $(M,\alpha)$ be a contact manifold with a globally defined contact one-form $\alpha$. Denote the Reeb vector field by $Y$. For each smooth function $F$ on $M$, there is a unique vector field $X_F$ which satisfies the equations $\alpha(X_F) = F$ and $\iota_{X_F} d \alpha = dF(Y) \alpha - dF $ since $(\xi, d\alpha \big|_\xi)$ is symplectic where $\xi = \ker \alpha$. The authors define a bracket on $C^\infty(M)$ by $$\{F, G \} =\alpha([X_F,X_G]) = X_F \cdot G - dF(Y)G.$$ This gives $C^\infty(M)$ a Lie algebra structure but not a Poisson structure since the term $dF(Y) G$ doesn't give a derivation for a fixed $F$.

My questions is how far does this analogy goes? For example, I can think of a symplectic structure as a special kind of Poisson structure with only one symplectic leaf. Is there any similar concept for this bracket , or what kind of properties does it have, or any further reference?

Answer 1

Cross-posted from mathoverflow.com:

This bracket is sometimes called the Lagrange bracket in the literature (although that terminology is unfortunately not universal, and sometimes refers to something different). It can be characterised as the map $\lbrace\cdot,\cdot\rbrace:C^\infty(M)\times C^\infty(M)\to C^\infty(M)$ that satisfies $[X_F,X_G] = X_{\{F,G\}}$ (where $X_F$ is the vector field on $M$ described in the linked question). The analogy one should consider is that contact form is to Lagrange bracket as symplectic form is to Poisson bracket induced by the symplectic form: the 'characteristic distribution' corresponding to the Lagrange bracket is the distribution spanned by vector fields $X_F$ for all $F\in C^\infty(M)$, and this is just the entire space $M$. So characteristic distributions of Lagrange brackets have one leaf.

The generalisation of the Lagrange bracket, analogous to a general Poisson bracket, is sometimes called the Jacobi bracket. From "On Jacobi Manifolds and Jacobi Bundles" by Marle:

Theorem [Kirillov]. The characteristic distribution of a Jacobi manifold $(M,\Lambda, E)$ is completely integrable in the sense of Stefan and Sussman, thus defines on $M$ a Stefan foliation (i.e., a foliation whose leaves may not be all of the same dimension). Each leaf of this foliation has a unique transitive Jacobi structure such that its canonical injection into $M$ is a Jacobi map.

Remark. The last theorem, together with remark 2.7.3, shows that each leaf of the characteristic foliation of a Jacobi manifold is

• a locally conformally symplectic manifold if its dimension is even,
• a manifold equipped with a contact 1-form if its dimension is odd.