TITLE: What is the role of contact geometry in the hamiltonian mechanics?
QUESTION [20 upvotes]: 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:

In the hamiltonian mechanics, not necessarily only for Hamiltonian of mechanical type, what is the role played by the contact geometry?

Any kind of suggestion is welcome.

REPLY [8 votes]: Form the contact 1-form $\Theta = p \, dq -H \, dt$ on extended phase space $T^* Q \times {\mathbb R}$, the second factor being time and parameterized by $t$, the function  $H = H(q,p,t)$ being the time-dependent energy
or Hamiltonian, and  $p \, dq$ denotes the usual  canonical one-form on $T^* Q$ pulled back to extended phase space by the projection onto the first factor.
(Assume $H \ne 0$ to get $\Theta$  contact.)  Then the Reeb vector field for this 1-form, i.e   the  kernel of $d \Theta$,  is the time-dependent
Hamiltonian vector field, up to scale. 
Arnol'd has a nice discussion of this in his Mathematical Methods in Classical Mechanics.