I'm currently trying to study contact geometry and stumbled upon the section talking about contact vector fields. Contact vector fields are constructed in a similar manner as the symplectic vector fields so i tried to do the derivation myself, but am stuck at the moment.
We are given the relations
$\mathcal{L}_{X_H}=f_H \eta \quad -H= \eta (X_H) \quad \mathcal{L}_{X_H}\eta=d\eta(X_H, \cdot) + d[\eta(X_H)] \quad dH=d\eta(X_H,\cdot)-\mathcal{L}_{X_H}\eta \quad$ and $ f_H=-\xi(X)$
Where as $\xi$ is the reebs vector field (also called characteristic vector field) and $\eta=dw-p_idx^i$ is the contact form in darboux coordinates.
I wanted to proceed like you do in symplectic geometry, by figuring out the components of the vector field by comparing coefficients.
In the following calculations i will denote the components of the contact hamiltonian vector field with an sub- or superscript H
What I've done so far is
$dH=-dp_i \wedge dx^i(X_H,\cdot)-\xi(H)p_idx^i+\xi(H)dw$
$dH=-p^H_idx^i+x^i_Hdp_i -\xi(H)p_idx^i+\xi(H)dw $
$dH= -(p_i\frac{\partial H}{\partial w}+\frac{\partial H}{\partial q^i})dq^i+\frac{\partial H}{\partial p_i}dp_i + \frac{\partial H}{\partial w}dw $
if $\frac{\partial H}{\partial w}=w_H$ and $\frac{\partial H}{\partial p_i}=x^i_H$, it also holds that:
$\eta(X_H)=dw(X_H)-p_idx^i(X_H)$
$\eta(X_H)=\frac{\partial H}{\partial w}-p_i\frac{\partial H}{\partial p_i}$
$\Rightarrow \frac{\partial H}{\partial w}= p\frac{\partial H}{\partial p} - H$
So if I manage to compare the terms with a version of $dH$ calculated in a different way, so that i can recognize the terms as the wanted $p_i,x^i$ and $w$, i get the wanted contact hamiltonian vector field
$X_H=(p_i\frac{\partial H}{\partial p_i} - H) \frac{\partial}{\partial w} -(p_i\frac{\partial H}{\partial w}+\frac{\partial H}{\partial q^i}) \frac{\partial}{\partial p_i } + \frac{\partial H}{\partial p_i}\frac{\partial}{\partial q^i}$ .
My problem is i have no idea, what the different way of calculating $dH$ is and i also have no idea, why my variables should be conjugate to each other, when I'm considering a system that is not restricted to a Legendre submanifold.
For reference I am reading the script of Alessandro Bravetti "contact geometry and thermodynamics" and "contact hamiltonian mechanics" a paper from Alessandro Bravetti
Edit: After thinking about it a bit. If I calculate $dH$ normally i would get:
$dH=\frac{\partial H}{\partial p_i}dp_i + \frac{\partial H}{\partial x^i}dx^i + \frac{\partial H}{\partial w}dw$.
If i could equate these partial derivatives to be the components of my contact Hamiltonian vector field $X_H$, I would be done (i think). This could be true, if my $p_i$ and $x^i$ are truly conjugate to each other even when not restricted to a Legendre submanifold.
Does anyone have potentially an idea, if my variables are also conjugate to each other, when not restricted to a Legendre submanifold?
Edit 2: Imposing on this question the fact that we are talking about thermodynamic systems, my $x$ and $p$ variables should be indeed conjugate to each other. So the last question would be, why i can equate $\frac{\partial H}{\partial w}$ to be my $w$ component of $X_H$