TY - JOUR T1 - On the geometric origin of the bi-Hamiltonian structure of the Calogero-Moser system JF - Int. Math. Res. Not. (2010) 2010:279-296 Y1 - 2010 A1 - Claudio Bartocci A1 - Gregorio Falqui A1 - Igor Mencattini A1 - Giovanni Ortenzi A1 - Marco Pedroni AB - 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. PB - Oxford University Press UR - http://hdl.handle.net/1963/3800 U1 - 8 U2 - LISNU U3 - Interdisciplinary Laboratory for Advanced Studies ER - TY - RPRT T1 - Gel\\\'fand-Zakharevich Systems and Algebraic Integrability: the Volterra Lattice Revisited Y1 - 2005 A1 - Gregorio Falqui A1 - Marco Pedroni AB - 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. JF - Regul. Chaotic Dyn. 10 (2005) 399-412 UR - http://hdl.handle.net/1963/1689 U1 - 2444 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - A geometric approach to the separability of the Neumann-Rosochatius system JF - Differential Geom. Appl. 21 (2004) 349-360 Y1 - 2004 A1 - Claudio Bartocci A1 - Gregorio Falqui A1 - Marco Pedroni AB - 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. UR - http://hdl.handle.net/1963/2541 U1 - 1578 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Separation of variables for Bi-Hamiltonian systems JF - Math. Phys. Anal. Geom. 6 (2003) 139-179 Y1 - 2003 A1 - Gregorio Falqui A1 - Marco Pedroni AB - We address the problem of the separation of variables for the Hamilton-Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special class of bi-Hamiltonian manifolds, called omega-N manifolds, to give intrisic tests of separability (and Staeckel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the omega-N manifold itself. We apply these results to bi-Hamiltonian systems of the Gel\\\'fand-Zakharevich type and we give explicit procedures to find the separated coordinates and the separation relations. PB - SISSA Library UR - http://hdl.handle.net/1963/1598 U1 - 2520 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On a Poisson reduction for Gel\\\'fand-Zakharevich manifolds JF - Rep.Math.Phys.50 (2002), no.3, 395 Y1 - 2002 A1 - Gregorio Falqui A1 - Marco Pedroni PB - SISSA Library UR - http://hdl.handle.net/1963/1602 U1 - 2516 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Bihamiltonian geometry and separation of variables for Toda lattices JF - J. Nonlinear Math. Phys. 8 (2001), suppl., 118-127 Y1 - 2001 A1 - Gregorio Falqui A1 - Franco Magri A1 - Marco Pedroni PB - SISSA Library UR - http://hdl.handle.net/1963/1354 U1 - 3101 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - A bi-Hamiltonian theory for stationary KDV flows and their separability JF - Regul. Chaotic Dyn. 5 (2000), no. 1, 33-52 Y1 - 2000 A1 - Gregorio Falqui A1 - Franco Magri A1 - Marco Pedroni A1 - Jorge P. Zubelli PB - SISSA Library UR - http://hdl.handle.net/1963/1352 U1 - 3103 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - An elementary approach to the polynomial $\\\\tau$-functions of the KP Hierarchy JF - Theor. Math. Phys. 122 (2000) 17-28 Y1 - 2000 A1 - Gregorio Falqui A1 - Franco Magri A1 - Marco Pedroni A1 - Jorge P. Zubelli AB - We give an elementary construction of the solutions of the KP hierarchy associated with polynomial τ-functions starting with a geometric approach to soliton equations based on the concept of a bi-Hamiltonian system. As a consequence, we establish a Wronskian formula for the polynomial τ-functions of the KP hierarchy. This formula, known in the literature, is obtained very directly. PB - Springer UR - http://hdl.handle.net/1963/3223 U1 - 1078 U2 - Mathematics U3 - Mathematical Physics ER - TY - CHAP T1 - A bihamiltonian approach to separation of variables in mechanics T2 - Proceedings of the workshop on nonlinearity, integrability and all that : twenty years after NEEDS \\\'79, Lecce, Italy, July 1 - 10, 1999 / ed. by M. Boiti. - Singapore : World Scientific, 2000. - p. 258-266 Y1 - 1999 A1 - Gregorio Falqui A1 - Franco Magri A1 - Marco Pedroni AB - This paper is a report on a recent approach to the theory of separability of the Hamilton-Jacobi equations from the viewpoint of bihamiltonian geometry. JF - Proceedings of the workshop on nonlinearity, integrability and all that : twenty years after NEEDS \\\'79, Lecce, Italy, July 1 - 10, 1999 / ed. by M. Boiti. - Singapore : World Scientific, 2000. - p. 258-266 PB - World Scientific UR - http://hdl.handle.net/1963/3222 U1 - 1079 U2 - Mathematics U3 - Mathematical Physics ER - TY - CHAP T1 - The method of Poisson pairs in the theory of nonlinear PDEs T2 - Direct and inverse methods in nonlinear evolution equations : Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 5-12, 1999 / Robert Conte ... ; Antonio M. Greco ed. - Berlin : Springer, 2003. - (Lecture Notes in Physics ; 632) Y1 - 1999 A1 - Gregorio Falqui A1 - Franco Magri A1 - Marco Pedroni AB - The aim of these lectures is to show that the methods of classical Hamiltonian mechanics can be profitably used to solve certain classes of nonlinear partial differential equations. The prototype of these equations is the well-known Korteweg-de Vries (KdV) equation.\\nIn these lectures we touch the following subjects:\\ni) the birth and the role of the method of Poisson pairs inside the theory of the KdV equation;\\nii) the theoretical basis of the method of Poisson pairs;\\niii) the Gel\\\'fand-Zakharevich theory of integrable systems on bi-Hamiltonian manifolds;\\niv) the Hamiltonian interpretation of the Sato picture of the KdV flows and of its linearization on an infinite-dimensional Grassmannian manifold.\\nv) the reduction technique(s) and its use to construct classes of solutions;\\nvi) the role of the technique of separation of variables in the study of the reduced systems;\\nvii) some relations intertwining the method of Poisson pairs with the method of Lax pairs. JF - Direct and inverse methods in nonlinear evolution equations : Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 5-12, 1999 / Robert Conte ... ; Antonio M. Greco ed. - Berlin : Springer, 2003. - (Lecture Notes in Physics ; 632) PB - Springer UR - http://hdl.handle.net/1963/1350 U1 - 3105 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - A note on fractional KDV hierarchies. II. The bihamiltonian approach Y1 - 1999 A1 - Paolo Casati A1 - Gregorio Falqui A1 - Marco Pedroni PB - SISSA Library UR - http://hdl.handle.net/1963/1220 U1 - 2723 U2 - Mathematics U3 - Mathematical Physics ER -