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Papers by Luca Degiovanni

Research paper thumbnail of Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion

Journal of Mathematical Physics, 2017

Research paper thumbnail of Complex variables for separation of Hamilton-Jacobi equation on three-dimensional Minkowski space

Eprint Arxiv Nlin 0612051, Dec 1, 2006

Research paper thumbnail of Extended Hamiltonians, Coupling-Constant Metamorphosis and the Post-Winternitz System

Symmetry, Integrability and Geometry: Methods and Applications, 2015

Research paper thumbnail of A note on the superintegrability of the Toda lattice

The superintegrability of the non-periodic Toda lattice is explained in the framework of systems ... more The superintegrability of the non-periodic Toda lattice is explained in the framework of systems written in action-angles coordinates. Moreover, a simpler form of the first integrals is given.

Research paper thumbnail of Warped product of Hamiltonians and extensions of Hamiltonian systems

Journal of Physics: Conference Series, 2015

The aim of this article is to set the theory of extensions of Hamiltonian systems, developed in a... more The aim of this article is to set the theory of extensions of Hamiltonian systems, developed in a series of previous papers, in the framework of warped products of Hamiltonian systems. Some illustrative examples and plots are given.

Research paper thumbnail of The Tremblay-Turbiner-Winternitz system as extended Hamiltonian

Journal of Mathematical Physics, 2014

Research paper thumbnail of Extensions of Hamiltonian systems dependent on a rational parameter

Journal of Mathematical Physics, 2014

Research paper thumbnail of New Examples of Trihamiltonian Structures Linking Different Lenard Chains

Symmetry and Perturbation Theory - Proceedings of the International Conference on SPT2004, 2005

ABSTRACT The extension of bi-Hamiltonian systems allows to realize a recursion relation between d... more ABSTRACT The extension of bi-Hamiltonian systems allows to realize a recursion relation between different Lenard chains. In particular extensions of either three-particle periodic Toda lattice, or systems on the Euclidean plane separable in parabolic coordinates or in elliptic-hyperbolic ones are presented.

Research paper thumbnail of A Tri-Hamiltonian Route to Spectral Curves

Nonlinear Physics — Theory and Experiment II - Proceedings of the Workshop, 2003

Research paper thumbnail of Superintegrable Extensions of Superintegrable Systems

Symmetry, Integrability and Geometry: Methods and Applications, 2012

Research paper thumbnail of Tri–Hamiltonian Vector Fields, Spectral Curves and Separation Coordinates

Reviews in Mathematical Physics, 2002

We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson... more We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P0,P1,P2), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalise those considered by E. Sklyanin in his algebro-geometric approach, are obtained from the knowledge of: (i) a common Casimir function for the two Poisson pencils (P1-λP0) and (P2-μP0); (ii) a suitable set of vector fields, preserving P0 but transversal to its symplectic leaves. The framework is applied to Lax equations with spectral parameter, for which not only it establishes a theoretical link between the separation techniques of Sklyanin and of Magri, but also provides a more efficient "inverse" procedure to obtain separation variables, not involving the extraction of roots.

Research paper thumbnail of Geometrical classification of Killing tensors on bidimensional flat manifolds

Journal of Mathematical Physics, 2006

Research paper thumbnail of Tri-Hamiltonian extensions of separable systems in the plane

Journal of Mathematical Physics, 2006

Research paper thumbnail of Superintegrable three-body systems on the line

Journal of Mathematical Physics, 2008

Research paper thumbnail of Generalizations of a method for constructing first integrals of a class of natural Hamiltonians and some remarks about quantization

Journal of Physics: Conference Series, 2012

Research paper thumbnail of Complex variables for separation of Hamilton-Jacobi equation on three-dimensional Minkowski space

The real coordinates separating geodesic Hamilton-Jacobi equation on three-dimensional Minkowski ... more The real coordinates separating geodesic Hamilton-Jacobi equation on three-dimensional Minkowski space in several cases cannot be defined in the whole space. We show through an example how to naturally extend them to complex variables defined everywhere (excluding the singular surfaces of each coordinate system only) and still separating the same equation.

Research paper thumbnail of Three and four-body systems in one dimension: Integrability, superintegrability and discrete symmetries

Regular and Chaotic Dynamics, 2011

Abstract Families of three-body Hamiltonian systems in one dimension have been recently proved to... more Abstract Families of three-body Hamiltonian systems in one dimension have been recently proved to be maximally superintegrable by interpreting them as one-body systems in the three-dimensional Euclidean space, examples are the Calogero, Wolfes and Tramblay ...

Research paper thumbnail of Complex variables for separation of the Hamilton-Jacobi equation on real pseudo-Riemannian manifolds

Journal of Mathematical Physics, 2007

In this paper the geometric theory of separation of variables for time-independent Hamilton-Jacob... more In this paper the geometric theory of separation of variables for time-independent Hamilton-Jacobi equation is extended to include the case of complex eigenvalues of a Killing tensor on pseudo-Riemannian manifolds. This task is performed without to complexify the manifold but just considering complex-valued functions on it. The simple formalism introduced allows to extend in a very natural way the classical results on separation of variables (including Levi-Civita criterion and Stäckel-Eisenhart theory) to the complex case. Orthogonal variables only are considered.

Research paper thumbnail of Polynomial constants of motion for Calogero-type systems in three dimensions

Journal of Mathematical Physics, 2011

We give an explicit and concise formula for higher-degree polynomial first integrals of a family ... more We give an explicit and concise formula for higher-degree polynomial first integrals of a family of Calogero-type Hamiltonian systems in dimension three. These first integrals, together with the already known ones, prove the maximal superintegrability of the systems.

Research paper thumbnail of A trihamiltonian extension of the Toda lattice

Journal of Geometry and Physics, 2007

A new Poisson structure on a subspace of the Kupershmidt algebra is defined. This Poisson structu... more A new Poisson structure on a subspace of the Kupershmidt algebra is defined. This Poisson structure, together with other two already known, allows to construct a trihamiltonian recurrence for an extension of the periodic Toda lattice with n particles. Some explicit examples of the construction and of the first integrals found in this way are given.

Research paper thumbnail of Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion

Journal of Mathematical Physics, 2017

Research paper thumbnail of Complex variables for separation of Hamilton-Jacobi equation on three-dimensional Minkowski space

Eprint Arxiv Nlin 0612051, Dec 1, 2006

Research paper thumbnail of Extended Hamiltonians, Coupling-Constant Metamorphosis and the Post-Winternitz System

Symmetry, Integrability and Geometry: Methods and Applications, 2015

Research paper thumbnail of A note on the superintegrability of the Toda lattice

The superintegrability of the non-periodic Toda lattice is explained in the framework of systems ... more The superintegrability of the non-periodic Toda lattice is explained in the framework of systems written in action-angles coordinates. Moreover, a simpler form of the first integrals is given.

Research paper thumbnail of Warped product of Hamiltonians and extensions of Hamiltonian systems

Journal of Physics: Conference Series, 2015

The aim of this article is to set the theory of extensions of Hamiltonian systems, developed in a... more The aim of this article is to set the theory of extensions of Hamiltonian systems, developed in a series of previous papers, in the framework of warped products of Hamiltonian systems. Some illustrative examples and plots are given.

Research paper thumbnail of The Tremblay-Turbiner-Winternitz system as extended Hamiltonian

Journal of Mathematical Physics, 2014

Research paper thumbnail of Extensions of Hamiltonian systems dependent on a rational parameter

Journal of Mathematical Physics, 2014

Research paper thumbnail of New Examples of Trihamiltonian Structures Linking Different Lenard Chains

Symmetry and Perturbation Theory - Proceedings of the International Conference on SPT2004, 2005

ABSTRACT The extension of bi-Hamiltonian systems allows to realize a recursion relation between d... more ABSTRACT The extension of bi-Hamiltonian systems allows to realize a recursion relation between different Lenard chains. In particular extensions of either three-particle periodic Toda lattice, or systems on the Euclidean plane separable in parabolic coordinates or in elliptic-hyperbolic ones are presented.

Research paper thumbnail of A Tri-Hamiltonian Route to Spectral Curves

Nonlinear Physics — Theory and Experiment II - Proceedings of the Workshop, 2003

Research paper thumbnail of Superintegrable Extensions of Superintegrable Systems

Symmetry, Integrability and Geometry: Methods and Applications, 2012

Research paper thumbnail of Tri–Hamiltonian Vector Fields, Spectral Curves and Separation Coordinates

Reviews in Mathematical Physics, 2002

We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson... more We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P0,P1,P2), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalise those considered by E. Sklyanin in his algebro-geometric approach, are obtained from the knowledge of: (i) a common Casimir function for the two Poisson pencils (P1-λP0) and (P2-μP0); (ii) a suitable set of vector fields, preserving P0 but transversal to its symplectic leaves. The framework is applied to Lax equations with spectral parameter, for which not only it establishes a theoretical link between the separation techniques of Sklyanin and of Magri, but also provides a more efficient "inverse" procedure to obtain separation variables, not involving the extraction of roots.

Research paper thumbnail of Geometrical classification of Killing tensors on bidimensional flat manifolds

Journal of Mathematical Physics, 2006

Research paper thumbnail of Tri-Hamiltonian extensions of separable systems in the plane

Journal of Mathematical Physics, 2006

Research paper thumbnail of Superintegrable three-body systems on the line

Journal of Mathematical Physics, 2008

Research paper thumbnail of Generalizations of a method for constructing first integrals of a class of natural Hamiltonians and some remarks about quantization

Journal of Physics: Conference Series, 2012

Research paper thumbnail of Complex variables for separation of Hamilton-Jacobi equation on three-dimensional Minkowski space

The real coordinates separating geodesic Hamilton-Jacobi equation on three-dimensional Minkowski ... more The real coordinates separating geodesic Hamilton-Jacobi equation on three-dimensional Minkowski space in several cases cannot be defined in the whole space. We show through an example how to naturally extend them to complex variables defined everywhere (excluding the singular surfaces of each coordinate system only) and still separating the same equation.

Research paper thumbnail of Three and four-body systems in one dimension: Integrability, superintegrability and discrete symmetries

Regular and Chaotic Dynamics, 2011

Abstract Families of three-body Hamiltonian systems in one dimension have been recently proved to... more Abstract Families of three-body Hamiltonian systems in one dimension have been recently proved to be maximally superintegrable by interpreting them as one-body systems in the three-dimensional Euclidean space, examples are the Calogero, Wolfes and Tramblay ...

Research paper thumbnail of Complex variables for separation of the Hamilton-Jacobi equation on real pseudo-Riemannian manifolds

Journal of Mathematical Physics, 2007

In this paper the geometric theory of separation of variables for time-independent Hamilton-Jacob... more In this paper the geometric theory of separation of variables for time-independent Hamilton-Jacobi equation is extended to include the case of complex eigenvalues of a Killing tensor on pseudo-Riemannian manifolds. This task is performed without to complexify the manifold but just considering complex-valued functions on it. The simple formalism introduced allows to extend in a very natural way the classical results on separation of variables (including Levi-Civita criterion and Stäckel-Eisenhart theory) to the complex case. Orthogonal variables only are considered.

Research paper thumbnail of Polynomial constants of motion for Calogero-type systems in three dimensions

Journal of Mathematical Physics, 2011

We give an explicit and concise formula for higher-degree polynomial first integrals of a family ... more We give an explicit and concise formula for higher-degree polynomial first integrals of a family of Calogero-type Hamiltonian systems in dimension three. These first integrals, together with the already known ones, prove the maximal superintegrability of the systems.

Research paper thumbnail of A trihamiltonian extension of the Toda lattice

Journal of Geometry and Physics, 2007

A new Poisson structure on a subspace of the Kupershmidt algebra is defined. This Poisson structu... more A new Poisson structure on a subspace of the Kupershmidt algebra is defined. This Poisson structure, together with other two already known, allows to construct a trihamiltonian recurrence for an extension of the periodic Toda lattice with n particles. Some explicit examples of the construction and of the first integrals found in this way are given.