Giovanni Rastelli | Università degli Studi di Torino (original) (raw)

Papers by Giovanni Rastelli

Research paper thumbnail of Separation of Variables for Systems of First-Order Partial Differential Equations and the Dirac Equation in Two-Dimensional Manifolds

Symmetries and Overdetermined Systems of Partial Differential Equations, 2008

The problem of solving the Dirac equation on two-dimensional manifolds is approached from the poi... more The problem of solving the Dirac equation on two-dimensional manifolds is approached from the point of separation of variables, with the aim of creating a foundation for analysis in higher dimensions. Beginning from a sound definition of multiplicative separation for ...

Research paper thumbnail of Extended Hamiltonians and shift, ladder functions and operators

Annals of Physics, 2017

In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motio... more In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motion of high degree, or symmetry operators of high order, have been found and studied. Most of these Hamiltonians, in the classical case, can be included in the family of extended Hamiltonians, geometrically characterized by the structure of warped manifold of their configuration manifold. For the extended manifolds, the characteristic constants of motion of high degree are polynomial in the momenta of determined form. We consider here a different form of the constants of motion, based on the factorization procedure developed by S. Kuru, J. Negro and others. We show that an important subclass of the extended Hamiltonians admits factorized constants of motion and we determine their expression. The classical constants may be non-polynomial in the momenta, but the factorization procedure allows, in a type of extended Hamiltonians, their quantization via shift and ladder operators, for systems of any finite dimension.

Research paper thumbnail of On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces

Symmetry, Integrability and Geometry: Methods and Applications

Research paper thumbnail of Extensions of natural Hamiltonians

Given an n-dimensional natural Hamiltonian L on a Riemannian or pseudo-Riemannian manifold, we ca... more Given an n-dimensional natural Hamiltonian L on a Riemannian or pseudo-Riemannian manifold, we call "extension" of L the n+1 dimensional Hamiltonian H = 1 2 p 2 u + α(u)L + β(u) with new canonically conjugated coordinates (u, pu). For suitable L, the functions α and β can be chosen depending on any natural number m such that H admits an extra polynomial first integral in the momenta of degree m, explicitly determined in the form of the mth power of a differential operator applied to a certain function of coordinates and momenta. In particular, if L is maximally superintegrable (MS) then H is MS also. Therefore, the extension procedure allows the creation of new superintegrable systems from old ones. For m=2, the extra first integral generated by the extension procedure determines a second-order symmetry operator of a Laplace-Beltrami quantization of H, modified by taking in account the curvature of the configuration manifold. The extension procedure can be applied to several Hamiltonian systems, including the three-body Calogero and Wolfes systems (without harmonic term), the Tremblay-Turbiner-Winternitz system and n-dimensional anisotropic harmonic oscillators. We propose here a short review of the known results of the theory and some previews of new ones.

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

Symmetry, Integrability and Geometry: Methods and Applications, 2015

The coupling-constant metamorphosis is applied to modified extended Hamiltonians and sufficient c... more The coupling-constant metamorphosis is applied to modified extended Hamiltonians and sufficient conditions are found in order that the transformed high-degree first integral of the transformed Hamiltonian is determined by the same algorithm which computes the corresponding first integral of the original extended Hamiltonian. As examples, we consider the Post-Winternitz system and the 2D caged anisotropic oscillator.

Research paper thumbnail of Block-Separation of Variables: a Form of Partial Separation for Natural Hamiltonians

Research paper thumbnail of Extended Hamiltonians and shift, ladder functions and operators

Annals of Physics, 2017

In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motio... more In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motion of high degree, or symmetry operators of high order, have been found and studied. Most of these Hamiltonians, in the classical case, can be included in the family of extended Hamiltonians, geometrically characterized by the structure of warped manifold of their configuration manifold. For the extended manifolds, the characteristic constants of motion of high degree are polynomial in the momenta of determined form. We consider here a different form of the constants of motion, based on the factorization procedure developed by S. Kuru, J. Negro and others. We show that an important subclass of the extended Hamiltonians admits factorized constants of motion and we determine their expression. The classical constants may be non-polynomial in the momenta, but the factorization procedure allows, in a type of extended Hamiltonians, their quantization via shift and ladder operators, for systems of any finite dimension.

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

Journal of Mathematical Physics , 2016

It is natural to investigate if the quantization of an integrable or superintegrable classical Ha... more It is natural to investigate if the quantization of an integrable or superintegrable classical Hamiltonian systems is still integrable or superintegrable. We study here this problem in the case of natural Hamiltonians with constants of motion quadratic in the momenta. The procedure of quantization here considered, transforms the Hamiltonian into the Laplace-Beltrami operator plus a scalar potential. In order to transform the constants of motion into symmetry operators of the quantum Hamiltonian, additional scalar potentials, known as quantum corrections, must be introduced, depending on the Riemannian structure of the manifold. We give here a complete geometric characterization of the quantum corrections necessary for the case considered. St\"ackel systems are studied in particular details. Examples in conformally and non-conformally flat manifolds are given.

Research paper thumbnail of Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator

Symmetry Integrability and Geometry Methods and Applications, 2016

We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates ... more We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator. Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures. In the examples considered, we have that the Weyl formula always preserves the original superintegrable structure of the system, while the Born-Jordan formula, when producing different operators than the Weyl's one, does not.

Research paper thumbnail of Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures

Journal of Geometric Mechanics, 2015

We give a characterization of linear canonoid transformations on symplectic manifolds and we use ... more We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on so * (3) and so * (4)) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.

Research paper thumbnail of Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures

Journal of Geometric Mechanics, 2015

We give a characterization of linear canonoid transformations on symplectic manifolds and we use ... more We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on so * (3) and so * (4)) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.

Research paper thumbnail of Sistemi di coordinate separabili nel piano euclideo

Research paper thumbnail of EIGENVALUES OF KILLING TENSORS AND ORTHOGONAL SEPARABLE WEBS

Symmetry and Perturbation Theory - Proceedings of the International Conference on SPT 2002, 2002

ABSTRACT

Research paper thumbnail of REMARKS ON CONFORMAL KILLING TENSORS AND SEPARATION OF VARIABLES

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

Research paper thumbnail of Remarks on the connection between the additive separation of the Hamilton–Jacobi equation and the multiplicative separation of the Schrödinger equation. I. The completeness and Robertson conditions

Journal of Mathematical Physics, 2002

The commutation relations of the first-order and second-order operators associated with the first... more The commutation relations of the first-order and second-order operators associated with the first integrals in involution of a Hamiltonian separable system are examined. It is shown that these operators commute if and only if a "pre-Robertson condition" is satisfied. This condition involves the Ricci tensor of the configuration manifold and it is implied by the Robertson condition, which is necessary and sufficient for the separability of the Schrödinger equation.

Research paper thumbnail of Variable separation for natural Hamiltonians with scalar and vector potentials on Riemannian manifolds

Journal of Mathematical Physics, 2001

The additive variable separation in the Hamilton-Jacobi equation is studied for a natural Hamilto... more The additive variable separation in the Hamilton-Jacobi equation is studied for a natural Hamiltonian with scalar and vector potentials on a Riemannian manifold with positive-definite metric. The separation of this Hamiltonian is related to the separation of a suitable geodesic Hamiltonian over an extended Riemannian manifold. Thus the geometrical theory of the geodesic separation is applied and the geometrical characterization of the separation is given in terms of Killing webs, Killing tensors and Killing vectors. The results are applicable to the case of a non-degenerate separation on a manifold with indefinite metric, where no null essential separable coordinates occur.

Research paper thumbnail of Variable-separation theory for the null Hamilton–Jacobi equation

Journal of Mathematical Physics, 2005

The theory of the separation of variables for the null Hamilton-Jacobi equation H = 0 is systemat... more The theory of the separation of variables for the null Hamilton-Jacobi equation H = 0 is systematically revisited and based on Levi-Civita separability conditions with Lagrangian multipliers. The separation of the null equation is shown to be equivalent to the ordinary separation of the image of the original Hamiltonian under a generalized Jacobi-Maupertuis transformation. The general results are applied to the special but fundamental case of the orthogonal separation of a natural Hamiltonian with a fixed value of the energy. The separation is then related to conditions which extend those of Stäckel and Kalnins and Miller ͑for the null geodesic case͒ and it is characterized by the existence of conformal Killing two-tensors of special kind.

Research paper thumbnail of First Integrals of Extended Hamiltonians in n+ 1 Dimensions Generated by Powers of an Operator⋆

Symmetry, Integrability and …, 2011

Abstract. We describe a procedure to construct polynomial in the momenta first integrals of arbit... more Abstract. We describe a procedure to construct polynomial in the momenta first integrals of arbitrarily high degree for natural Hamiltonians H obtained as one-dimensional extensions of natural (geodesic) n-dimensional Hamiltonians L. The Liouville integrability of L implies ...

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 Separation of Variables for Systems of First-Order Partial Differential Equations and the Dirac Equation in Two-Dimensional Manifolds

Symmetries and Overdetermined Systems of Partial Differential Equations, 2008

The problem of solving the Dirac equation on two-dimensional manifolds is approached from the poi... more The problem of solving the Dirac equation on two-dimensional manifolds is approached from the point of separation of variables, with the aim of creating a foundation for analysis in higher dimensions. Beginning from a sound definition of multiplicative separation for ...

Research paper thumbnail of Extended Hamiltonians and shift, ladder functions and operators

Annals of Physics, 2017

In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motio... more In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motion of high degree, or symmetry operators of high order, have been found and studied. Most of these Hamiltonians, in the classical case, can be included in the family of extended Hamiltonians, geometrically characterized by the structure of warped manifold of their configuration manifold. For the extended manifolds, the characteristic constants of motion of high degree are polynomial in the momenta of determined form. We consider here a different form of the constants of motion, based on the factorization procedure developed by S. Kuru, J. Negro and others. We show that an important subclass of the extended Hamiltonians admits factorized constants of motion and we determine their expression. The classical constants may be non-polynomial in the momenta, but the factorization procedure allows, in a type of extended Hamiltonians, their quantization via shift and ladder operators, for systems of any finite dimension.

Research paper thumbnail of On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces

Symmetry, Integrability and Geometry: Methods and Applications

Research paper thumbnail of Extensions of natural Hamiltonians

Given an n-dimensional natural Hamiltonian L on a Riemannian or pseudo-Riemannian manifold, we ca... more Given an n-dimensional natural Hamiltonian L on a Riemannian or pseudo-Riemannian manifold, we call "extension" of L the n+1 dimensional Hamiltonian H = 1 2 p 2 u + α(u)L + β(u) with new canonically conjugated coordinates (u, pu). For suitable L, the functions α and β can be chosen depending on any natural number m such that H admits an extra polynomial first integral in the momenta of degree m, explicitly determined in the form of the mth power of a differential operator applied to a certain function of coordinates and momenta. In particular, if L is maximally superintegrable (MS) then H is MS also. Therefore, the extension procedure allows the creation of new superintegrable systems from old ones. For m=2, the extra first integral generated by the extension procedure determines a second-order symmetry operator of a Laplace-Beltrami quantization of H, modified by taking in account the curvature of the configuration manifold. The extension procedure can be applied to several Hamiltonian systems, including the three-body Calogero and Wolfes systems (without harmonic term), the Tremblay-Turbiner-Winternitz system and n-dimensional anisotropic harmonic oscillators. We propose here a short review of the known results of the theory and some previews of new ones.

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

Symmetry, Integrability and Geometry: Methods and Applications, 2015

The coupling-constant metamorphosis is applied to modified extended Hamiltonians and sufficient c... more The coupling-constant metamorphosis is applied to modified extended Hamiltonians and sufficient conditions are found in order that the transformed high-degree first integral of the transformed Hamiltonian is determined by the same algorithm which computes the corresponding first integral of the original extended Hamiltonian. As examples, we consider the Post-Winternitz system and the 2D caged anisotropic oscillator.

Research paper thumbnail of Block-Separation of Variables: a Form of Partial Separation for Natural Hamiltonians

Research paper thumbnail of Extended Hamiltonians and shift, ladder functions and operators

Annals of Physics, 2017

In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motio... more In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motion of high degree, or symmetry operators of high order, have been found and studied. Most of these Hamiltonians, in the classical case, can be included in the family of extended Hamiltonians, geometrically characterized by the structure of warped manifold of their configuration manifold. For the extended manifolds, the characteristic constants of motion of high degree are polynomial in the momenta of determined form. We consider here a different form of the constants of motion, based on the factorization procedure developed by S. Kuru, J. Negro and others. We show that an important subclass of the extended Hamiltonians admits factorized constants of motion and we determine their expression. The classical constants may be non-polynomial in the momenta, but the factorization procedure allows, in a type of extended Hamiltonians, their quantization via shift and ladder operators, for systems of any finite dimension.

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

Journal of Mathematical Physics , 2016

It is natural to investigate if the quantization of an integrable or superintegrable classical Ha... more It is natural to investigate if the quantization of an integrable or superintegrable classical Hamiltonian systems is still integrable or superintegrable. We study here this problem in the case of natural Hamiltonians with constants of motion quadratic in the momenta. The procedure of quantization here considered, transforms the Hamiltonian into the Laplace-Beltrami operator plus a scalar potential. In order to transform the constants of motion into symmetry operators of the quantum Hamiltonian, additional scalar potentials, known as quantum corrections, must be introduced, depending on the Riemannian structure of the manifold. We give here a complete geometric characterization of the quantum corrections necessary for the case considered. St\"ackel systems are studied in particular details. Examples in conformally and non-conformally flat manifolds are given.

Research paper thumbnail of Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator

Symmetry Integrability and Geometry Methods and Applications, 2016

We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates ... more We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator. Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures. In the examples considered, we have that the Weyl formula always preserves the original superintegrable structure of the system, while the Born-Jordan formula, when producing different operators than the Weyl's one, does not.

Research paper thumbnail of Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures

Journal of Geometric Mechanics, 2015

We give a characterization of linear canonoid transformations on symplectic manifolds and we use ... more We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on so * (3) and so * (4)) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.

Research paper thumbnail of Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures

Journal of Geometric Mechanics, 2015

We give a characterization of linear canonoid transformations on symplectic manifolds and we use ... more We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on so * (3) and so * (4)) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.

Research paper thumbnail of Sistemi di coordinate separabili nel piano euclideo

Research paper thumbnail of EIGENVALUES OF KILLING TENSORS AND ORTHOGONAL SEPARABLE WEBS

Symmetry and Perturbation Theory - Proceedings of the International Conference on SPT 2002, 2002

ABSTRACT

Research paper thumbnail of REMARKS ON CONFORMAL KILLING TENSORS AND SEPARATION OF VARIABLES

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

Research paper thumbnail of Remarks on the connection between the additive separation of the Hamilton–Jacobi equation and the multiplicative separation of the Schrödinger equation. I. The completeness and Robertson conditions

Journal of Mathematical Physics, 2002

The commutation relations of the first-order and second-order operators associated with the first... more The commutation relations of the first-order and second-order operators associated with the first integrals in involution of a Hamiltonian separable system are examined. It is shown that these operators commute if and only if a "pre-Robertson condition" is satisfied. This condition involves the Ricci tensor of the configuration manifold and it is implied by the Robertson condition, which is necessary and sufficient for the separability of the Schrödinger equation.

Research paper thumbnail of Variable separation for natural Hamiltonians with scalar and vector potentials on Riemannian manifolds

Journal of Mathematical Physics, 2001

The additive variable separation in the Hamilton-Jacobi equation is studied for a natural Hamilto... more The additive variable separation in the Hamilton-Jacobi equation is studied for a natural Hamiltonian with scalar and vector potentials on a Riemannian manifold with positive-definite metric. The separation of this Hamiltonian is related to the separation of a suitable geodesic Hamiltonian over an extended Riemannian manifold. Thus the geometrical theory of the geodesic separation is applied and the geometrical characterization of the separation is given in terms of Killing webs, Killing tensors and Killing vectors. The results are applicable to the case of a non-degenerate separation on a manifold with indefinite metric, where no null essential separable coordinates occur.

Research paper thumbnail of Variable-separation theory for the null Hamilton–Jacobi equation

Journal of Mathematical Physics, 2005

The theory of the separation of variables for the null Hamilton-Jacobi equation H = 0 is systemat... more The theory of the separation of variables for the null Hamilton-Jacobi equation H = 0 is systematically revisited and based on Levi-Civita separability conditions with Lagrangian multipliers. The separation of the null equation is shown to be equivalent to the ordinary separation of the image of the original Hamiltonian under a generalized Jacobi-Maupertuis transformation. The general results are applied to the special but fundamental case of the orthogonal separation of a natural Hamiltonian with a fixed value of the energy. The separation is then related to conditions which extend those of Stäckel and Kalnins and Miller ͑for the null geodesic case͒ and it is characterized by the existence of conformal Killing two-tensors of special kind.

Research paper thumbnail of First Integrals of Extended Hamiltonians in n+ 1 Dimensions Generated by Powers of an Operator⋆

Symmetry, Integrability and …, 2011

Abstract. We describe a procedure to construct polynomial in the momenta first integrals of arbit... more Abstract. We describe a procedure to construct polynomial in the momenta first integrals of arbitrarily high degree for natural Hamiltonians H obtained as one-dimensional extensions of natural (geodesic) n-dimensional Hamiltonians L. The Liouville integrability of L implies ...

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 ...