What is the motivation of contact Hamiltonian equation I've just checked that this is constructed to mimic the ordinary Hamiltonian equation in symplectic geometry. There are several literatures, and they use
$$
\eta(X_H) = -H\\
\mathrm{d}\eta(X_H,-) = \mathrm{d}H - R(H)\eta
$$ to descirbe the Hamiltonian equation, where $\eta$ is a contact form and $R$ is the associated Reeb vector field.
However, is this formulation approved in the physical region, too? This may be somewhat a little physical question, so maybe here is not the appropriate place to ask.
I'm just wondering whether this is mathematically motivated, or physically motivated. More precisely, is this just really defined to generalize the symplectic Hamiltonian theory to contact geometry? or are there some useful models in physics using this structure? I've also heard that this can be applied to non-conservative system or dissipative system in physics, but I know less about this.
 A: Contact Hamiltonian systems play a role a role in string theory, more specifically in what is known as the AdS-CFT correspondence. See, for example, Transverse Kähler–Ricci Solitons of Five-Dimensional Sasaki–Einstein Spaces.
This is highly specialized stuff, I do not know of any application to mainstream physics.
A: One mathematical motivation is that it shows that the automorphism group of contact manifolds is huge.  To give a bit of an idea of what I mean, compare this to Riemannian metrics.
The isometry group of a given metric is finite dimensional, and in fact, an isometry is uniquely determined by the information on how it acts on the tangent space $T_xM$ of one single point $x$!  Of course, for generic metrics, the isometry group is trivial: an isometry needs to map a point of a certain curvature to a point with the same curvature.  If the metric is a bit random, this condition is very restrictive.
Compare this now to contact topology.  Firstly, we can use the existence of Hamiltonian functions on a chart, to find a local contactomorphism.  This way, we could for example move one point to any other chosen point by patching charts together.  This construction is local from one chart to the next one, but in fact, we can "globalize" it by multiplying the corresponding Hamiltonian functions with suitable bump functions.  This way, we obtain a contactomorphism defined on all of $M$.
Even better:  Choose two sets of $k$ pairwise distinct points $(p_1,\dotsc, p_k)$ and $(q_1,\dotsc,q_k)$ in $M$, then there exists a contactomorphism $\Phi$ with arbitrarily small support around disjoint paths joining $p_j$ to $q_j$ such that $\Phi(p_j) = q_j$.  The contactomorphism group acts $k$-transitively for any finite $k$.
Or, take a smooth family of submanifolds $L_\tau$ that are tangent to the contact structure (for example Legendrians).  Then there exists a family of contactomorphisms $\Phi_\tau$ such that $\Phi_\tau(L_\tau) = L_0$.  In other words all manifolds $L_\tau$ are in a certain sense equivalent.
This way, contact geometry is much closer to topology than to geometry.
