@article {2010, title = {On the geometric origin of the bi-Hamiltonian structure of the Calogero-Moser system}, journal = {Int. Math. Res. Not. (2010) 2010:279-296}, number = {arXiv.org;0902.0953v2}, year = {2010}, publisher = {Oxford University Press}, abstract = {We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of gl(n,R). The relation with the Lax formalism is also discussed.}, doi = {10.1093/imrn/rnp130}, url = {http://hdl.handle.net/1963/3800}, author = {Claudio Bartocci and Gregorio Falqui and Igor Mencattini and Giovanni Ortenzi and Marco Pedroni} } @article {2005, title = {Gel\\\'fand-Zakharevich Systems and Algebraic Integrability: the Volterra Lattice Revisited}, number = {SISSA;29/2005/FM}, year = {2005}, abstract = {In this paper we will discuss some features of the bi-Hamiltonian method for solving the Hamilton-Jacobi (H-J) equations by Separation of Variables, and make contact with the theory of Algebraic Complete Integrability and, specifically, with the Veselov-Novikov notion of algebro-geometric (AG) Poisson brackets.}, doi = {10.1070/RD2005v010n04ABEH000322}, url = {http://hdl.handle.net/1963/1689}, author = {Gregorio Falqui and Marco Pedroni} } @article {2004, title = {A geometric approach to the separability of the Neumann-Rosochatius system}, journal = {Differential Geom. Appl. 21 (2004) 349-360}, number = {arXiv.org;nlin/0307021}, year = {2004}, abstract = {We study the separability of the Neumann-Rosochatius system on the n-dimensional sphere using the geometry of bi-Hamiltonian manifolds. Its well-known separation variables are recovered by means of a separability condition relating the Hamiltonian with a suitable (1,1) tensor field on the sphere. This also allows us to iteratively construct the integrals of motion of the system.}, doi = {10.1016/j.difgeo.2004.07.001}, url = {http://hdl.handle.net/1963/2541}, author = {Claudio Bartocci and Gregorio Falqui and Marco Pedroni} } @article {2003, title = {Gaudin models and bending flows: a geometrical point of view}, journal = {J. Phys. A: Math. Gen. 36 (2003) 11655-11676}, number = {SISSA;45/2003/FM}, year = {2003}, publisher = {IOP Publishing}, abstract = {In this paper we discuss the bihamiltonian formulation of the (rational XXX) Gaudin models of spin-spin interaction, generalized to the case of sl(r)-valued spins. In particular, we focus on the homogeneous models. We find a pencil of Poisson brackets that recursively define a complete set of integrals of the motion, alternative to the set of integrals associated with the \\\'standard\\\' Lax representation of the Gaudin model. These integrals, in the case of su(2), coincide wih the Hamiltonians of the \\\'bending flows\\\' in the moduli space of polygons in Euclidean space introduced by Kapovich and Millson. We finally address the problem of separability of these flows and explicitly find separation coordinates and separation relations for the r=2 case.}, doi = {10.1088/0305-4470/36/46/009}, url = {http://hdl.handle.net/1963/2884}, author = {Gregorio Falqui and Fabio Musso} }