Editing Lotka–Volterra equations (section)

===Hamiltonian structure of the system===
Since the quantity <math>V(x,y)</math> is conserved over time, it plays role of a Hamiltonian function of the system.<ref>{{cite journal|last1=Nutku|first1=I. |year=1990|title=Hamiltonian structure of the Lotka-Volterra equations|journal=Physics Letters A|volume=145|issue=1 |pages=27–28|url=https://www.sciencedirect.com/science/article/abs/pii/037596019090270X |doi=10.1016/0375-9601(90)90270-X|bibcode=1990PhLA..145...27N |hdl=11693/26204 |s2cid=121710034 |hdl-access=free}}</ref> To see this we can define [[Poisson bracket]] as follows	
<math> \{f(x,y), g(x,y)\} = -xy \left( \frac{\partial f}{\partial x}\frac{\partial g}{\partial y} - \frac{\partial f}{\partial y} \frac{\partial g}{\partial x} \right) </math>.  Then [[Hamilton's equations]] read
<math display="block">\begin{cases}
\dot{x} =  \{x, V\} = \alpha x - \beta x y, \\ 
\dot{y} =  \{y, V\} = \delta x y - \gamma y.
\end{cases}</math>
The variables <math>x</math> and <math>y</math> are not canonical, since <math> \{x, y\} = -xy \neq 1</math>. However, using transformations<ref>{{cite web |url=http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/Bio-Mathematics-(APPLIED).pdf |title=Lotka-Volterra Dynamics - An introduction |last=Baigent |first=Steve |date=2010-03-02 }}</ref> <math> p = \ln (x)</math> and <math> q = \ln (y)</math> we came up to a canonical form of the [[Hamilton's equations]] featuring the Hamiltonian <math> H(q,p) = V(x(q,p),y(q,p)) = \delta e^p - \gamma p + \beta e^q - \alpha q </math>:
<math display="block">\begin{cases}
\dot{q} =  \frac{\partial H}{\partial p} = \delta e^p - \gamma, \\ 
\dot{p} =  -\frac{\partial H}{\partial q} = \alpha - \beta e^q.
\end{cases}</math>
The [[Poisson bracket]] for the canonical variables <math>(q,p)</math> now takes the standard form <math> \{F(q,p), G(q,p)\} = \left( \frac{\partial F}{\partial q}\frac{\partial G}{\partial p} - \frac{\partial F}{\partial p} \frac{\partial G}{\partial q} \right) </math>.