Franco Magri | Università degli Studi di Milano-Bicocca (original) (raw)

Papers by Franco Magri

Research paper thumbnail of Differential equations in the spectral parameter

Communications in Mathematical Physics, 1986

Research paper thumbnail of Recursion Operators in Classical Mechanics

Research paper thumbnail of Poisson-Nijenhuis structures, truncated loop algebras and Sato's KP hierarchy

Research paper thumbnail of The Kowalewski separability conditions

Proceedings of symposia in pure mathematics, 2021

Research paper thumbnail of A geometrical characterization of Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds

Quaderni di Matematica, 2008

Research paper thumbnail of The seeds of Hamiltonian mechanics in the writing of Lagrange

Research paper thumbnail of A geometrical approach to the nonlinear solvable equations

Springer eBooks, Apr 6, 2008

... F. Magri ... This method comes out as a consequence of the change of point of view produced b... more ... F. Magri ... This method comes out as a consequence of the change of point of view produced by the geo-metrical approach, which leads to give preeminence to the Nijenhuis ope-rator A rather than to the integrals which are in involution. ...

Research paper thumbnail of A short introduction to Hamiltonian PDEs. Fifth Workshop on Partial Differential Equations (Rio de Janeiro, 1997)

Research paper thumbnail of The Kowalewski Top Revisited

Cambridge University Press eBooks, Apr 2, 2020

Research paper thumbnail of Haantjes Manifolds

Journal of physics, Mar 5, 2014

ABSTRACT The aim of this paper is to introduce a new category of manifolds, called Haantjes manif... more ABSTRACT The aim of this paper is to introduce a new category of manifolds, called Haantjes manifolds, and to show their interest by a few selected examples.

Research paper thumbnail of An operator approach to Poisson brackets

Annals of Physics, Jul 1, 1976

The aim of this paper is to suggest a general approach to Poisson brackets, based on the study of... more The aim of this paper is to suggest a general approach to Poisson brackets, based on the study of the Lie algebra of potential operators with respect to closed skew-symmetric bilinear forms. This approach allows to extend easily to infinite-dimensional spaces the classical Cartan geometrical approach developed in the phase space. It supplies a simple, unified, and general formalism to deal with such brackets, which contains, as particular cases, the classical and the quantum treatments. The aim of the present paper is to suggest a general theory of Poisson brackets * This work has been sponsored by the Consiglio nazionale delle Ricerche, Gruppo per la Fisica-Matematica.

Research paper thumbnail of A short introduction to Hamiltonian PDEs

Matemática Contemporânea, 1998

Research paper thumbnail of Bihamiltonian Geometry and Separation of Variables for Toda Lattices

Journal of Nonlinear Mathematical Physics, 2001

Research paper thumbnail of A new approach to separation of variables for the Clebsch integrable system. Part I: Reduction to quadratures

arXiv (Cornell University), Feb 5, 2021

The paper describes a new concept of separation of variables with a concrete application to the C... more The paper describes a new concept of separation of variables with a concrete application to the Clebsch integrable case of the Kirchhoff equations. There are two principal novelties: The first is that the separating coordinates are constructed (not guessed) by solving the Kowalewski separability conditions. The second is that the solutions of the equations of motion are written in terms of theta-functions by means of a generalization of the standard Jacobi inversion problem of algebraic geometry. These two novelties are dealt with in two separate parts of the paper. Part I explains the Kowalewski separability conditions and their implementation to the Clebsch case. It is shown that the new separating coordinates lead to quadratures involving Abelian differentials on two different non-hyperelliptic curves (of genus higher than the dimension of the invariant tori). In Part II these quadratures are interpreted as a new generalization of the standard Abel-Jacobi map, and a procedure of its inversion in terms of theta-functions is worked out. The theta-function solution is different from that found long time ago by F. Kötter, since the theta-functions used in this paper have different period matrix.

Research paper thumbnail of A new approach to separation of variables for the Clebsch integrable system. Part II: Inversion of the Abel--Prym map

arXiv (Cornell University), Feb 6, 2021

This is the second part of a paper describing a new concept of separation of variables applied to... more This is the second part of a paper describing a new concept of separation of variables applied to the classical Clebsch integrable case. The quadratures obtained in Part I lead to a new type of the Abel map which contains Abelian integrals on two different algebraic curves. Here we show that this map is from the product of the two curves to the Prym variety of one of them, that it is well defined, although not a bijection. We analyse its properties and formulate a new extention of the Riemann vanishing theorem, which allows to invert the map in terms of theta-functions of higher order. Lastly, we describe how to express the original variables of the Clebsch system in terms of the preimages of the map. This enables one to obtain theta-function solution for the system.

Research paper thumbnail of Haantjes manifolds with symmetry

arXiv (Cornell University), Dec 18, 2017

This paper has two purposes. The first is to introduce the definition of Haantjes manifolds with ... more This paper has two purposes. The first is to introduce the definition of Haantjes manifolds with symmetry. The second is to explain why these manifolds appear in the theory of integrable systems of hydrodynamic type and in topological field theories.

Research paper thumbnail of The Kowalewski's top revisited

arXiv (Cornell University), Sep 6, 2018

The paper is a commentary of one section of the celebrated paper by Sophie Kowalewski on the moti... more The paper is a commentary of one section of the celebrated paper by Sophie Kowalewski on the motion of a rigid body with a fixed point. Its purpose is to show that the results of Kowalewski may be recovered by using the separability conditions obtained by Tullio Levi Civita in 1904.

Research paper thumbnail of Soliton equations, bi-Hamiltonian manifolds and integrability

Research paper thumbnail of Equivalence transformations for nonlinear evolution equations

Journal of Mathematical Physics, Jul 1, 1977

A systematic approach to the study of nonlinear evolution equations based on the theory of the eq... more A systematic approach to the study of nonlinear evolution equations based on the theory of the equivalence transformations is suggested. In this paper it is applied to the Burgers and to the Korteweg–de Vries equations. The main result is that the Hopf–Cole transformation for the Burgers equation and the Miura, Bäcklund, and Hirota transformations for the Korteweg–de Vries equation (together with the linear equations of the inverse scattering theory) are all deduced from a single general equivalence condition.

Research paper thumbnail of Poisson-Nijenhuis Structures and Sato Hierarchy

Reviews in Mathematical Physics, Dec 1, 1991

We show that the direct sum of n copies of a Lie algebra is endowed with a sequence of affine Lie... more We show that the direct sum of n copies of a Lie algebra is endowed with a sequence of affine Lie-Poisson brackets, which are pairwise compatible and define a multi-Hamiltonian structure; to this structure one can associate a recursion operator and a Kac-Moody algebra of Hamiltonian vector fields. If the initial Lie algebra is taken to be an associative algebra of differential operators, a suitable family of Hamiltonian vector fields reproduce either the n-th Gel'fand-Dikii hierarchy (for n finite) or Sato's hierarchy (for n = ∞). Within the same framework, it is also possible to recover a class of integro-differential hierarchies involving a finite number of fields, which generalize the Gel'fand-Dikii equations and are equivalent to Sato's hierarchy.

Research paper thumbnail of Differential equations in the spectral parameter

Communications in Mathematical Physics, 1986

Research paper thumbnail of Recursion Operators in Classical Mechanics

Research paper thumbnail of Poisson-Nijenhuis structures, truncated loop algebras and Sato's KP hierarchy

Research paper thumbnail of The Kowalewski separability conditions

Proceedings of symposia in pure mathematics, 2021

Research paper thumbnail of A geometrical characterization of Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds

Quaderni di Matematica, 2008

Research paper thumbnail of The seeds of Hamiltonian mechanics in the writing of Lagrange

Research paper thumbnail of A geometrical approach to the nonlinear solvable equations

Springer eBooks, Apr 6, 2008

... F. Magri ... This method comes out as a consequence of the change of point of view produced b... more ... F. Magri ... This method comes out as a consequence of the change of point of view produced by the geo-metrical approach, which leads to give preeminence to the Nijenhuis ope-rator A rather than to the integrals which are in involution. ...

Research paper thumbnail of A short introduction to Hamiltonian PDEs. Fifth Workshop on Partial Differential Equations (Rio de Janeiro, 1997)

Research paper thumbnail of The Kowalewski Top Revisited

Cambridge University Press eBooks, Apr 2, 2020

Research paper thumbnail of Haantjes Manifolds

Journal of physics, Mar 5, 2014

ABSTRACT The aim of this paper is to introduce a new category of manifolds, called Haantjes manif... more ABSTRACT The aim of this paper is to introduce a new category of manifolds, called Haantjes manifolds, and to show their interest by a few selected examples.

Research paper thumbnail of An operator approach to Poisson brackets

Annals of Physics, Jul 1, 1976

The aim of this paper is to suggest a general approach to Poisson brackets, based on the study of... more The aim of this paper is to suggest a general approach to Poisson brackets, based on the study of the Lie algebra of potential operators with respect to closed skew-symmetric bilinear forms. This approach allows to extend easily to infinite-dimensional spaces the classical Cartan geometrical approach developed in the phase space. It supplies a simple, unified, and general formalism to deal with such brackets, which contains, as particular cases, the classical and the quantum treatments. The aim of the present paper is to suggest a general theory of Poisson brackets * This work has been sponsored by the Consiglio nazionale delle Ricerche, Gruppo per la Fisica-Matematica.

Research paper thumbnail of A short introduction to Hamiltonian PDEs

Matemática Contemporânea, 1998

Research paper thumbnail of Bihamiltonian Geometry and Separation of Variables for Toda Lattices

Journal of Nonlinear Mathematical Physics, 2001

Research paper thumbnail of A new approach to separation of variables for the Clebsch integrable system. Part I: Reduction to quadratures

arXiv (Cornell University), Feb 5, 2021

The paper describes a new concept of separation of variables with a concrete application to the C... more The paper describes a new concept of separation of variables with a concrete application to the Clebsch integrable case of the Kirchhoff equations. There are two principal novelties: The first is that the separating coordinates are constructed (not guessed) by solving the Kowalewski separability conditions. The second is that the solutions of the equations of motion are written in terms of theta-functions by means of a generalization of the standard Jacobi inversion problem of algebraic geometry. These two novelties are dealt with in two separate parts of the paper. Part I explains the Kowalewski separability conditions and their implementation to the Clebsch case. It is shown that the new separating coordinates lead to quadratures involving Abelian differentials on two different non-hyperelliptic curves (of genus higher than the dimension of the invariant tori). In Part II these quadratures are interpreted as a new generalization of the standard Abel-Jacobi map, and a procedure of its inversion in terms of theta-functions is worked out. The theta-function solution is different from that found long time ago by F. Kötter, since the theta-functions used in this paper have different period matrix.

Research paper thumbnail of A new approach to separation of variables for the Clebsch integrable system. Part II: Inversion of the Abel--Prym map

arXiv (Cornell University), Feb 6, 2021

This is the second part of a paper describing a new concept of separation of variables applied to... more This is the second part of a paper describing a new concept of separation of variables applied to the classical Clebsch integrable case. The quadratures obtained in Part I lead to a new type of the Abel map which contains Abelian integrals on two different algebraic curves. Here we show that this map is from the product of the two curves to the Prym variety of one of them, that it is well defined, although not a bijection. We analyse its properties and formulate a new extention of the Riemann vanishing theorem, which allows to invert the map in terms of theta-functions of higher order. Lastly, we describe how to express the original variables of the Clebsch system in terms of the preimages of the map. This enables one to obtain theta-function solution for the system.

Research paper thumbnail of Haantjes manifolds with symmetry

arXiv (Cornell University), Dec 18, 2017

This paper has two purposes. The first is to introduce the definition of Haantjes manifolds with ... more This paper has two purposes. The first is to introduce the definition of Haantjes manifolds with symmetry. The second is to explain why these manifolds appear in the theory of integrable systems of hydrodynamic type and in topological field theories.

Research paper thumbnail of The Kowalewski's top revisited

arXiv (Cornell University), Sep 6, 2018

The paper is a commentary of one section of the celebrated paper by Sophie Kowalewski on the moti... more The paper is a commentary of one section of the celebrated paper by Sophie Kowalewski on the motion of a rigid body with a fixed point. Its purpose is to show that the results of Kowalewski may be recovered by using the separability conditions obtained by Tullio Levi Civita in 1904.

Research paper thumbnail of Soliton equations, bi-Hamiltonian manifolds and integrability

Research paper thumbnail of Equivalence transformations for nonlinear evolution equations

Journal of Mathematical Physics, Jul 1, 1977

A systematic approach to the study of nonlinear evolution equations based on the theory of the eq... more A systematic approach to the study of nonlinear evolution equations based on the theory of the equivalence transformations is suggested. In this paper it is applied to the Burgers and to the Korteweg–de Vries equations. The main result is that the Hopf–Cole transformation for the Burgers equation and the Miura, Bäcklund, and Hirota transformations for the Korteweg–de Vries equation (together with the linear equations of the inverse scattering theory) are all deduced from a single general equivalence condition.

Research paper thumbnail of Poisson-Nijenhuis Structures and Sato Hierarchy

Reviews in Mathematical Physics, Dec 1, 1991

We show that the direct sum of n copies of a Lie algebra is endowed with a sequence of affine Lie... more We show that the direct sum of n copies of a Lie algebra is endowed with a sequence of affine Lie-Poisson brackets, which are pairwise compatible and define a multi-Hamiltonian structure; to this structure one can associate a recursion operator and a Kac-Moody algebra of Hamiltonian vector fields. If the initial Lie algebra is taken to be an associative algebra of differential operators, a suitable family of Hamiltonian vector fields reproduce either the n-th Gel'fand-Dikii hierarchy (for n finite) or Sato's hierarchy (for n = ∞). Within the same framework, it is also possible to recover a class of integro-differential hierarchies involving a finite number of fields, which generalize the Gel'fand-Dikii equations and are equivalent to Sato's hierarchy.