Apparently it is possible to have chaotic behaviour in a conservative
(hamiltonian) system. A HS conserves phase-space volume, but not
necessarily shape. Think of kneading dough: volume is conserved, but
an initial ball of raisins spreads out.
How to make an example? It needs to be a-symmetric to avoid
integrability. Take a potential like $$U(x,y) = x^2 + y^2 +
\frac{a}{1 + (x-b)^2 + y^2}$$ which looks like a harmonic oscillator
potential well $x^2 + y^2$ at large scale, but has a bump at $(b,0)$.
Can it produce chaotic behaviour?
For simple systems, can the Lyapunov coefficient (LC) be computed
locally? Sure it can, but what matters more is the \emph{average} LC,
no? Need more study here\ldots