As I understand, the lack of indication on how to obtain first integrals in Arnol'd-Liouville theory is a reason why we are interested in bi-Hamiltonian systems.
Two Poisson brackets
${ cdot,cdot } _{1} , { cdot , cdot } _{2}$ on a manifold $M$ are compatible if their arbitrary linear combination
$lambda { cdot , cdot } _1+mu{cdot,cdot} _2$ is also a Poisson bracket. A bi-Hamiltonian system is one which allows Hamiltonian formulations with respect to two compatible Poisson brackets. It automatically posseses a number of integrals in involution.
The definition of a complete integrability (à la Liouville-Arnol'd) is:
Hamiltonian flows and Poisson maps on a $2n$-dimensional symplectic manifold $left(M,{ cdot, cdot }_Mright)$ with $n$ (smooth real valued) functions $F _1,F _2,dots,F _n$ such that: (i) they are functionally independent (i.e. the gradients $nabla F _k$ are linearly independent everywhere on $M$) and (ii) these functions are in involution (i.e. ${F _k,F _j}=0$) are called completely integrable.
Now, I would like to understand the connections between these two notions, and because I haven't studied the theory, any answer would be helpful. I find reading papers on these subjects too technical at the moment. Specific questions I have in mind are:
Does completely integrable system always allow for a bi-Hamiltonian structure? Is every bi-Hamiltonian system completely integrable? If not, what are examples (or places where to find examples) of systems that posses one property but not the other?
I apologize for any stupid mistakes I might have made above. Feel free to edit (tagging included).
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