Procedure to find a conserved quantity

Question

Given a system such as $$\dot{x}=-kxy$$ $$\dot{y}=kxy-ly$$ with $k,l>0$.How do you find the conserved quantity $E(x,y)$ in general? Secondly , what is the difference between a hamiltonian system and a conserved system?

Answer 1

A conserved quantity $E(x,y)$ requires that $E(x,y)$ is constant on the integral curves. This happens when $E_x \dot x + E_y \dot y =0$. Thus form the differential equation: $$ \frac{\partial E}{\partial x}(-kxy) + \frac{\partial E}{\partial y}( kxy-ly) = 0 $$ You then need to solve for E, either analytically or using other methods..

A Hamiltonian system is simply one which is governed by Hamilton's equations, see here.

We don't say that a system is conserved but rather that some property is conserved for a given system. For example the Hamiltonian function may be conserved.

Answer 2

Since the right-hand sides of both equations have the form $y$ times a function of $x$, take $E = y - f(x)$: $$\frac d {dt} (y - f(x)) = y (k x (f'(x) + 1) - l) = 0, \\ f(x) = \frac {l \ln x} k - x.$$