Luis A. Ibort

On the multisymplectic formalism for first order field theories

Abstract The general purpose of this paper is to attempt to clarify the geometrical foundations of first order Lagrangian and Hamiltonian field theories by introducing in a systematic way multisymplectic manifolds, the field theoretical analogues of the symplectic structures used in geometrical mechanics. Much of the confusion surrounding such terms as gauge transformation and symmetry transformation as they are used in the context of Lagrangian theory is thereby eliminated, as we show. We discuss Noethers theorem for general symmetries of Lagrangian and Hamiltonian field theories. The cohomology associated to a group of symmetries of Hamiltonian or Lagrangian field theories is constructed and its relation with the structure of the current algebra is made apparent.

Reduction of degenerate Lagrangian systems

Abstract 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 (Diracs theory) and the reduction of systems with symmetry (Marsden-Weinstein theory) is investigated. Some examples and applications are discussed.

The Feynman problem and the inverse problem for Poisson dynamics

Abstract 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 differential equations on tangent bundles over arbitrary configuration 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 field and/or a gravitational field. 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 differential equation, then the dynamics results from a singular Lagrangian. (Here we assume in addition that the dynamics satisfies 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 Feynmans problem for systems possessing Lie groups as configuration 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.

Non-Noether constants of motion

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 Leverriers method of determining the characteristics polynomial of a matrix in terms of the traces of its increasing powers.

Canonical transformations theory for presymplectic systems

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.

Generalized canonical transformations for time‐dependent systems

We introduce the concept of generalized canonical transformations as symplectomorphisms of the extended phase space. We prove that any such transformation factorizes in a standard canonical transformation times another one that changes only the time variable. The theory of generating functions as well as that of Hamilton–Jacobi is developed. Some further applications are developed.

Geometric theory of the equivalence of Lagrangians for constrained systems

The Lagrangian description of a system is analysed from a geometric viewpoint in order to find a concept for equivalence of singular Lagrangians generalising that of the regular case. Geometric and gauge equivalence of singular Lagrangians are studied and the authors also give some conditions in which second-order differential equations exist satisfying the dynamical equation on the final constraint submanifold.

Noncanonical groups of transformations, anomalies, and cohomology

Two cohomology classes associated to groups of transformations (symplectic or not) of Hamiltonian and Lagrangian systems are studied. A geometrical interpretation of the family of cocycles arising from a class of nonsymplectic actions is given in terms of the Poisson structure of the phase space of the system. These ideas are used to study nongauge (i.e., anomalous) groups of transformations of (locally or globally defined) Lagrangian systems. In particular, well‐known results about the magnetic monopole system are described in this context and some hints relating Yang–Mills anomalies with nonsymplectic groups of transformations are given.

Introduction to Poisson supermanifolds

Abstract A notion of super-Poisson structure in the category of (real) graded manifolds is presented and some of its properties are discussed. Some examples of Poisson supermanifolds are given. The structure of the cotangent supermanifold of a Lie supergroup is described and an extension of the Lie-Poisson reduction theorem for ordinary Lie groups is derived.

Origin and Infinity Manifolds for Mechanical Systems with Homogeneous Potentials

We study manifolds describing the behavior of motions close to the origin and at infinity of configuration space, for mechanical systems with homogeneous potentials. We find an inversion between these behaviors when the sign of the degree of homogeneity is changed. In some cases, the blow up equations can be written in canonical form, by first reducing to a contact structure. A motivation for the use of blow-up techniques is given, and some examples are studied in detail.