L. Ibort - Academia.edu (original) (raw)

Papers by L. Ibort

Research paper thumbnail of Integrable Systems, Quantum Groups, and Quantum Field Theories

Research paper thumbnail of Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds

Journal of Geometric Mechanics, 2012

It is shown that the geometry of a class of multisymplectic manifolds, that is, smooth manifolds ... more It is shown that the geometry of a class of multisymplectic manifolds, that is, smooth manifolds equipped with a closed nondegenerate form of degree greater than 1, is characterized by their automorphisms. Such a class is distinguished by a local homogeneity property. Thus, locally homogeneous multisymplectic manifolds extend the family of classical geometries possessing a similar property: symplectic, volume and contact. The proof of the first result relies on the characterization of invariant differential forms with respect to the graded Lie algebra of infinitesimal automorphisms and on the study of the local properties of Hamiltonian vector fields on multisymplectic manifolds. In particular it is proved that the group of multisymplectic diffeomorphisms acts transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic diffeomorphisms.

Research paper thumbnail of Locally hamiltonian systems with symmetry and a generalized Noether's theorem

Il Nuovo Cimento B, 1985

Summary An analysis of global aspects of the theory of symmetry groupsG of locally Hamiltonian d... more Summary An analysis of global aspects of the theory of symmetry groupsG of locally Hamiltonian dynamical systems is carried out for particular cases either of the symmetry group, or the differentiable manifoldM supporting the symplectic structure, or the action ofG onM. In every case we obtain a generalization of Noether's theorem. We also look at the classical Noether's theorem for Lagrangian systems from a modern perspective.

Research paper thumbnail of Blow up of mechanical systems with a homogeneous energy

Publicacions Matemàtiques, 1991

Research paper thumbnail of The Feynman problem and the inverse problem for Poisson dynamics

Physics Reports, 1995

We review the Feynman proof of the Lorentz force equations, as well as its generalization to the ... more We review the Feynman proof of the Lorentz force equations, as well as its generalization to the dynamics of particles with internal degrees of freedom. In addition, we discuss the inverse problem for Poisson dynamics and the inverse problem of the calculus of variations. It is proved that the only classical dynamics compatible with localizability and the existence of second order di erential equations on tangent bundles over arbitrary con guration spaces, is necessarily of the Lagrangian type. Furthermore, if the dynamics is independent of the velocity of test particles, it must correspond to that of a particle coupled to an electromagnetic eld and/or a gravitational eld. The same ideas are carried out for particles with internal degrees of freedom. In this case, if we insist on a weak localizability condition and the existence of a second order Hamiltonian di erential equation, then the dynamics results from a singular Lagrangian. (Here we assume in addition that the dynamics satis es a regularity condition.) These results extend those of Feynman and provide the conditions which guarantee the existence of a Lagrangian description. They are applied to systematically discuss Feynman's problem for systems possessing Lie groups as con guration spaces, with internal variables modeled on Lie algebras of groups. Finally, we illustrate what happens when the condition of localizability is dropped. In this regard, we obtain alternative Hamiltonian descriptions of standard dynamical systems. These non{standard solutions are discussed within the framework of Lie{Poisson structures.

Research paper thumbnail of Hyper-K�hler induction and self-duality on Riemann surfaces

Letters in Mathematical Physics, 1992

ABSTRACT

Research paper thumbnail of On the existence of local and global Lagrangians for ordinary differential equations

Journal of Physics A: Mathematical and General, 1990

Necessary and sufficient conditions for the existence of local and global Lagrangians for ordinar... more Necessary and sufficient conditions for the existence of local and global Lagrangians for ordinary differential equations of arbitrary order are described in terms of the geometry of higher-order tangent bundles. The results are applied to the study of gauge invariant differential equations and the second-order differential equation defined by the ( 2 + 1 )-dimensional Yang-Mills Lagrangian with the Chern-Simons term is discussed.

Research paper thumbnail of Non-Noether constants of motion

Journal of Physics A: Mathematical and General, 1983

Mathematical tools of modern differential geometry are used to derive, in an intrinsic formulatio... more Mathematical tools of modern differential geometry are used to derive, in an intrinsic formulation, more general results about non-Noether constants of motion. A relation between two different ways of obtaining such constants is found by making use of ...

Research paper thumbnail of A geometric classification of Lagrangian functions and the reduction of evolution space

Journal of Physics A: Mathematical and General, 1992

A geometric classification of degenerate time-dependent Lagrangian systems is given and the reduc... more A geometric classification of degenerate time-dependent Lagrangian systems is given and the reduction of evolution space is analysed. General properties of semiregular Lagrangians (type II) are discussed and particular attention is paid to the reduction of completely degenerate Lagrangians (type III) which are considered in detail.

Research paper thumbnail of Geometric theory of the equivalence of Lagrangians for constrained systems

Journal of Physics A: Mathematical and General, 1985

... The concept of equivalence we have introduced is a generalisation of the definition given for... more ... The concept of equivalence we have introduced is a generalisation of the definition given for regular systems and is in perfect harmony with the geometric theory developed by Gotay et a1 (1978, 1979) for dealing with pre-symplectic systems. ...

Research paper thumbnail of Canonical transformations theory for presymplectic systems

Journal of Mathematical Physics, 1985

We develop a theory of canonical transformations for presymplectic systems, reducing this concept... more We develop a theory of canonical transformations for presymplectic systems, reducing this concept to that of canonical transformations for regular coisotropic canonical systems. In this way we can also link these with the usual canonical transformations for the symplectic reduced phase space. Furthermore, the concept of a generating function arises in a natural way as well as that of gauge group.

Research paper thumbnail of A note on the existence of graded extensions of Poisson brackets

Journal of Geometry and Physics, 1993

In this note we discuss the existence and projectability of graded extensions of ordinary Poisson... more In this note we discuss the existence and projectability of graded extensions of ordinary Poisson brackets. We will show that there are topological obstructions to both problems. To prove it we use a new algebraic characterization of graded Poisson brackets on graded manifolds based on a characterization of derivations on the exterior algebra of a vector bundle.

Research paper thumbnail of Reduction of degenerate Lagrangian systems

Journal of Geometry and Physics, 1986

The geometrical structure of (finite dimensional) degenerate Lagrangian systems is studied and a ... more The geometrical structure of (finite dimensional) degenerate Lagrangian systems is studied and a reduction scheme, leading to a regular Lagrangian description of these systems on a reduced velocity phase space, is developed. The connection with the canonical approach to the regularization problem of degenerate systems (Dirac 's theory) and the reduction of systems with symmetry (Marsden --Weinstein theory) is investigated. Some examples and applications are discussed.

Research paper thumbnail of On the inverse problem of the calculus of variations for a class of coupled dynamical systems

Inverse Problems, 1991

ABSTRACT The inverse problem of the calculus of variations for a class of coupled dynamical syste... more ABSTRACT The inverse problem of the calculus of variations for a class of coupled dynamical systems-the so-called driven second-order differential equations-are analysed. A family of necessary and sufficient conditions are found that assure the existence of a local Lagrangian function for a given driven second-order differential equation. These conditions are stated geometrically and some of their consequences and examples are discussed.

Research paper thumbnail of On the Inverse Problem of Lagrangian Supermechanics

International Journal of Modern Physics A, 1993

The inverse problem for Lagrangian supermechanics is investigated. A set of necessary and suffici... more The inverse problem for Lagrangian supermechanics is investigated. A set of necessary and sufficient conditions for a system of second order differential equations in superspace to derive from (a possibly non regular) superlagrangian function are given. The harmonic superoscillator is revisited and a family of even and odd alternative superlagrangians are constructed for it. Finally, we comment on the existence of recursion operators.

Research paper thumbnail of Applications of the canonical-transformation theory for presymplectic systems

Il Nuovo Cimento B, 1987

Two applications of the canonical-transformation theory for presymplcctic systems developed in a ... more Two applications of the canonical-transformation theory for presymplcctic systems developed in a previous paper are presented: a new approach to the extended formalism for the time-dependent systems and the relativistic free massive point. For this last system some examples of canonical transformations are constructed explicitly.

Research paper thumbnail of Integrable Systems, Quantum Groups, and Quantum Field Theories

Research paper thumbnail of Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds

Journal of Geometric Mechanics, 2012

It is shown that the geometry of a class of multisymplectic manifolds, that is, smooth manifolds ... more It is shown that the geometry of a class of multisymplectic manifolds, that is, smooth manifolds equipped with a closed nondegenerate form of degree greater than 1, is characterized by their automorphisms. Such a class is distinguished by a local homogeneity property. Thus, locally homogeneous multisymplectic manifolds extend the family of classical geometries possessing a similar property: symplectic, volume and contact. The proof of the first result relies on the characterization of invariant differential forms with respect to the graded Lie algebra of infinitesimal automorphisms and on the study of the local properties of Hamiltonian vector fields on multisymplectic manifolds. In particular it is proved that the group of multisymplectic diffeomorphisms acts transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic diffeomorphisms.

Research paper thumbnail of Locally hamiltonian systems with symmetry and a generalized Noether's theorem

Il Nuovo Cimento B, 1985

Summary An analysis of global aspects of the theory of symmetry groupsG of locally Hamiltonian d... more Summary An analysis of global aspects of the theory of symmetry groupsG of locally Hamiltonian dynamical systems is carried out for particular cases either of the symmetry group, or the differentiable manifoldM supporting the symplectic structure, or the action ofG onM. In every case we obtain a generalization of Noether's theorem. We also look at the classical Noether's theorem for Lagrangian systems from a modern perspective.

Research paper thumbnail of Blow up of mechanical systems with a homogeneous energy

Publicacions Matemàtiques, 1991

Research paper thumbnail of The Feynman problem and the inverse problem for Poisson dynamics

Physics Reports, 1995

We review the Feynman proof of the Lorentz force equations, as well as its generalization to the ... more We review the Feynman proof of the Lorentz force equations, as well as its generalization to the dynamics of particles with internal degrees of freedom. In addition, we discuss the inverse problem for Poisson dynamics and the inverse problem of the calculus of variations. It is proved that the only classical dynamics compatible with localizability and the existence of second order di erential equations on tangent bundles over arbitrary con guration spaces, is necessarily of the Lagrangian type. Furthermore, if the dynamics is independent of the velocity of test particles, it must correspond to that of a particle coupled to an electromagnetic eld and/or a gravitational eld. The same ideas are carried out for particles with internal degrees of freedom. In this case, if we insist on a weak localizability condition and the existence of a second order Hamiltonian di erential equation, then the dynamics results from a singular Lagrangian. (Here we assume in addition that the dynamics satis es a regularity condition.) These results extend those of Feynman and provide the conditions which guarantee the existence of a Lagrangian description. They are applied to systematically discuss Feynman's problem for systems possessing Lie groups as con guration spaces, with internal variables modeled on Lie algebras of groups. Finally, we illustrate what happens when the condition of localizability is dropped. In this regard, we obtain alternative Hamiltonian descriptions of standard dynamical systems. These non{standard solutions are discussed within the framework of Lie{Poisson structures.

Research paper thumbnail of Hyper-K�hler induction and self-duality on Riemann surfaces

Letters in Mathematical Physics, 1992

ABSTRACT

Research paper thumbnail of On the existence of local and global Lagrangians for ordinary differential equations

Journal of Physics A: Mathematical and General, 1990

Necessary and sufficient conditions for the existence of local and global Lagrangians for ordinar... more Necessary and sufficient conditions for the existence of local and global Lagrangians for ordinary differential equations of arbitrary order are described in terms of the geometry of higher-order tangent bundles. The results are applied to the study of gauge invariant differential equations and the second-order differential equation defined by the ( 2 + 1 )-dimensional Yang-Mills Lagrangian with the Chern-Simons term is discussed.

Research paper thumbnail of Non-Noether constants of motion

Journal of Physics A: Mathematical and General, 1983

Mathematical tools of modern differential geometry are used to derive, in an intrinsic formulatio... more Mathematical tools of modern differential geometry are used to derive, in an intrinsic formulation, more general results about non-Noether constants of motion. A relation between two different ways of obtaining such constants is found by making use of ...

Research paper thumbnail of A geometric classification of Lagrangian functions and the reduction of evolution space

Journal of Physics A: Mathematical and General, 1992

A geometric classification of degenerate time-dependent Lagrangian systems is given and the reduc... more A geometric classification of degenerate time-dependent Lagrangian systems is given and the reduction of evolution space is analysed. General properties of semiregular Lagrangians (type II) are discussed and particular attention is paid to the reduction of completely degenerate Lagrangians (type III) which are considered in detail.

Research paper thumbnail of Geometric theory of the equivalence of Lagrangians for constrained systems

Journal of Physics A: Mathematical and General, 1985

... The concept of equivalence we have introduced is a generalisation of the definition given for... more ... The concept of equivalence we have introduced is a generalisation of the definition given for regular systems and is in perfect harmony with the geometric theory developed by Gotay et a1 (1978, 1979) for dealing with pre-symplectic systems. ...

Research paper thumbnail of Canonical transformations theory for presymplectic systems

Journal of Mathematical Physics, 1985

We develop a theory of canonical transformations for presymplectic systems, reducing this concept... more We develop a theory of canonical transformations for presymplectic systems, reducing this concept to that of canonical transformations for regular coisotropic canonical systems. In this way we can also link these with the usual canonical transformations for the symplectic reduced phase space. Furthermore, the concept of a generating function arises in a natural way as well as that of gauge group.

Research paper thumbnail of A note on the existence of graded extensions of Poisson brackets

Journal of Geometry and Physics, 1993

In this note we discuss the existence and projectability of graded extensions of ordinary Poisson... more In this note we discuss the existence and projectability of graded extensions of ordinary Poisson brackets. We will show that there are topological obstructions to both problems. To prove it we use a new algebraic characterization of graded Poisson brackets on graded manifolds based on a characterization of derivations on the exterior algebra of a vector bundle.

Research paper thumbnail of Reduction of degenerate Lagrangian systems

Journal of Geometry and Physics, 1986

The geometrical structure of (finite dimensional) degenerate Lagrangian systems is studied and a ... more The geometrical structure of (finite dimensional) degenerate Lagrangian systems is studied and a reduction scheme, leading to a regular Lagrangian description of these systems on a reduced velocity phase space, is developed. The connection with the canonical approach to the regularization problem of degenerate systems (Dirac 's theory) and the reduction of systems with symmetry (Marsden --Weinstein theory) is investigated. Some examples and applications are discussed.

Research paper thumbnail of On the inverse problem of the calculus of variations for a class of coupled dynamical systems

Inverse Problems, 1991

ABSTRACT The inverse problem of the calculus of variations for a class of coupled dynamical syste... more ABSTRACT The inverse problem of the calculus of variations for a class of coupled dynamical systems-the so-called driven second-order differential equations-are analysed. A family of necessary and sufficient conditions are found that assure the existence of a local Lagrangian function for a given driven second-order differential equation. These conditions are stated geometrically and some of their consequences and examples are discussed.

Research paper thumbnail of On the Inverse Problem of Lagrangian Supermechanics

International Journal of Modern Physics A, 1993

The inverse problem for Lagrangian supermechanics is investigated. A set of necessary and suffici... more The inverse problem for Lagrangian supermechanics is investigated. A set of necessary and sufficient conditions for a system of second order differential equations in superspace to derive from (a possibly non regular) superlagrangian function are given. The harmonic superoscillator is revisited and a family of even and odd alternative superlagrangians are constructed for it. Finally, we comment on the existence of recursion operators.

Research paper thumbnail of Applications of the canonical-transformation theory for presymplectic systems

Il Nuovo Cimento B, 1987

Two applications of the canonical-transformation theory for presymplcctic systems developed in a ... more Two applications of the canonical-transformation theory for presymplcctic systems developed in a previous paper are presented: a new approach to the extended formalism for the time-dependent systems and the relativistic free massive point. For this last system some examples of canonical transformations are constructed explicitly.