Let us assume someone is interested in the study of Hamiltonian mechanics.  

What are good examples to illustrate him of the usefulness of contact geometry in this context?


On one hand the Hamiltonian mechanics was time ago expressed in the language of symplectic geometry, but, on the other hand, the contact geometry is often presented like the brother of the symplectic geometry.


My question is:



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> In the hamiltonian mechanics, not necessarily only for Hamiltonian of mechanical type, what is the role played by the contact geometry?
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> 
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Any kind of suggestion is welcome. 



What is the role of contact geometry in the hamiltonian mechanics?
In mechanics you often want to study systems whose Hamiltonian function depends on time (explicitly). For example, you can look at the motion of a charged particle in a *time-dependent* electric field. In such cases you are solving an ODE in the "extended phase space" ($\mathbb{R}^7$ in the above example), and not in $\mathbb{R}^6\simeq T^\vee \mathbb{R}^3$. Also, the translation between Hamiltonian and Lagrangian formulation of mechanics goes via Legendre transform, which fits very nicely in the framework of contact geometry. Contact geometry also enters mechanics through Hamilton-Jacobi theory and the "method of characteristics".


So you can think of contact geometry as the odd-dimensional ("non-stationary") analogue of symplectic geometry. You can go from one to the other by "symplectisation". For example, if you start with a manifold, $M$, then you have a tautological contact structure on $X=\mathbb{P}T^\vee\_M$. The symplectisation of $X$ is $T^\vee \_M-\{ 0 \}$, with the canonical symplectic form. Symplectisation maps contact diffeomorphisms to symplectomorphisms, etc. etc. In the opposite direction, if you are given a symplectic manifold$(M,\omega)$ with $[\omega]=0\in H^2(M,\mathbb{R})$, you can build a line bundle $E\to M$ with a contact structure.


