Non-hamiltonian systems which evolve into hamiltonian by change of coordinates

Question

I am very new to the subject, so please forgive my very naïf question. I learned that there are some non-hamiltonian systems which can become hamiltonian, just by a change of coordinates. I was given SIR (Susceptible - Infected - Removed) model as an example: 
begin{cases} 
frac{dS}{dt} = - alpha SI  
frac{dI}{dt} = alpha SI - beta I  
frac{dR}{dt} = beta I
end{cases}

with $alpha$ and $beta$ being real parameters.

This is clearly non-hamiltonian because it is associated with a vector field which has non zero divergence. However, by choosing $x= log(S)$ and $y= log(I)$, where $S$ and $I$ are the susceptible and infected respectively, the system becomes hamiltonian.

What I found really strange about this result is that I'm used at seeing quantities being preserved by a change of coordinates, while here the property of the system being hamiltonian clearly isn't intrisinc: is there something deeply conceptual that I'm missing about hamiltonian mechanics?

Qmechanic · Accepted Answer

Note that the divergence ${rm div}_{rho}X=rho^{-1}partial_i(rho X^i)$ of a vector field $X=X^ipartial_i$ in general depends on a density $rho$, cf. above comment by user mlk. The possibility of a non-trivial $rho$ makes it more difficult to identify which 1st-order systems are potentially Hamiltonian and which are not.

A 3D phase space can never have a non-degenerate symplectic structure, but if we ignore the last coordinate $R(t)=betaint^t!dt^{prime} I(t^{prime})$, then we have a 2D phase space,
which always has a (local) Hamiltonian formulation, cf. this Phys.SE post.

Concretely, the SIR-model
$$begin{align} dot{S}~=~& - alpha SI~=~{S,H}, cr 
dot{I}~=~& alpha SI - beta I~=~{I,H},cr 
dot{R}~=~&  beta I~=~{R,H},end{align} tag{1}$$
has non-canonical, degenerate Poisson structure
$$begin{align}    {S,I}~=~&SI, cr {I,R}~=~&frac{beta}{alpha}I, cr {S,R}~=~&0,end{align} tag{2} $$
and Hamiltonian
$$ H~=~beta ln S  -alpha (S+I) .tag{3} $$

If we follow OP's suggestion to change coordinates
$$begin{align} s~:=~&ln S, cr
  i~:=~&ln I , cr
 r~:=~&R, end{align}tag{1'}$$
then the fundamental Poisson brackets (2) become constant
$$ begin{align}   {s,i}~=~&1, cr {i,r}~=~&frac{beta}{alpha}, cr {s,r}~=~&0, end{align}tag{2'} $$
and it becomes obvious that the Jacobi identity is satisfied as it should. The Hamiltonian reads
$$ H~=~s -alpha (e^s+e^i) tag{3'} $$
in the new coordinates.

Non-hamiltonian systems which evolve into hamiltonian by change of coordinates

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