Connection between bi-Hamiltonian systems and complete integrability

Question

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 $\left(M,\{ \cdot, \cdot \}_M\right)$ 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).

Answer 1

Your understanding is essentially correct. There are three basic (and closely related) approaches to constructing the integrals of motion required for complete integrability: through separation of variables, through the Lax representation, and through the bi-Hamiltonian representation. The relationship among them is not yet fully understood. See, however, this paper by M. Błaszak, which, in essence, states that any Hamiltonian system that admits separation of variables is (or, rather, can be extended to) bi-Hamiltonian, and this survey paper by G. Falqui and M. Pedroni on separation of variables for bi-Hamiltonian systems. As for the relationship among the Lax representation and bi-Hamiltonian property, see this paper by F. Magri and Y. Kosmann-Schwarzbach and references therein. Now to your questions.

First of all, the bi-Hamiltonian property as you state it, without further restrictions, does not necessarily lead to integrability, and the claim that a bi-Hamiltonian system automatically possesses some integrals of motion does not hold in full generality, as far as I know. I can't think of a specific example right now, but, roughly speaking, if both your Poisson structures are too degenerate (their rank is too low), the recursion can break down and you will not get enough integrals of motion. An example of this for the infinite-dimensional case can be found in the paper Is a bi-Hamiltonian system necessarily integrable? by B.A. Kupershmidt. However, if you put in some additional nondegeneracy assumptions, the answer is yes, and dates back to Magri, Morosi, Gelfand and Dorfman. It is nicely summarized e.g. in Theorem 1.1 of this paper by R.G. Smirnov. The idea behind this is that the integrals of motion are provided by the traces of powers of the ratio of your Poisson structures.

As for the second question, not any Liouville integrable system is bi-Hamiltonian, at least if you impose some fairly reasonable technical assumptions, see the paper Completely integrable bi-Hamiltonian systems by R.L. Fernandes; cf. also the above Smirnov's paper.