In my differential equations course an example is given from the Lotka-Volterra system of equations:
$$ x'=x-xy$$
$$y'=-\gamma y+xy.\tag{1}$$
This is then transformed by the substitution: $q=\ln x, p=\ln y$.
$$ q'=1-e^p$$
$$p'=-\gamma +e^q.\tag{2}$$
Then without any explanation they say the Hamiltonian is then equal to: $$H(p,q)=\gamma q -e^q+p-e^p\tag{3}$$
How is this Hamiltonian derived?
This is explained in part II of my Phys.SE answer here, which shows that a 2D system always has a Hamiltonian description locally.
It turns out, that before the non-canonical transformation $(x,y) \to (q,p)$, from the first pair of eoms (1) alone, the Hamiltonian and non-canonical Poisson bracket can be derived as $$H~=~\gamma \ln x -x +\ln y -y $$ and $$\{x,y\}_{PB} ~=~ xy,$$ respectively. Next the canonical coordinates $(q,p)$ can be easily determined.
