Xavier Gràcia

Geometric Hamilton-Jacobi Theory

The Hamilton–Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence of a natural symplectic structure on the cotangent bundle. First it is developed for systems described by regular Lagrangians and then extended to systems described by singular Lagrangians with no secondary constraints. We also consider the example of the free relativistic particle, the rigid body and the electron-monopole system.

A generalized geometric framework for constrained systems

A geometric framework for constrained dynamical systems is presented. It allows to describe in a unified way a general type of first order singular differential equations on a manifold; these equations can not be written in normal form since the derivatives appear multiplied by a linear operator, therefore we call them linearly constrained systems. The concepts of constraints and morphisms between linearly constrained systems are defined, and their relationships studied. Finally, a stabilization algorithm is devised and carefully discussed in order to solve the equation of motion. Our formalism includes the presymplectic and the lagrangian formalisms, as well as higher order lagrangians, and we give several applications of it; in particular, a stabilization algorithm for the lagrangian formalism is obtained.

Higher-order Lagrangian systems : geometric structures, dynamics, and constraints

In order to study the connections between Lagrangian and Hamiltonian formalisms constructed from a—perhaps singular—higher‐order Lagrangian, some geometric structures are constructed. Intermediate spaces between those of Lagrangian and Hamiltonian formalisms, partial Ostrogradskiĭ’s transformations and unambiguous evolution operators connecting these spaces are intrinsically defined, and some of their properties studied. Equations of motion, constraints, and arbitrary functions of Lagrangian and Hamiltonian formalisms are thoroughly studied. In particular, all the Lagrangian constraints are obtained from the Hamiltonian ones. Once the gauge transformations are taken into account, the true number of degrees of freedom is obtained, both in the Lagrangian and Hamiltonian formalisms, and also in all the ‘‘intermediate formalisms’’ herein defined.

Some geometric aspects of variational calculus in constrained systems

Abstract We give a geometric description of variational principles in constrained mechanics. For the general case of nonholonomic constraints, a unified variational approach to both vakonomic and nonholonomic frameworks is given, and the corresponding equations of motion are recovered. Special attention is paid to the existence of infinitesimal variations in both cases, and it is proved that these variations coincide when the constraints are integrable. As examples, we give geometric formulations of the equations of motion for the case of optimal control and for vakonomic and nonholonomic mechanics with constraints linear in the velocities.

Gauge generators, Dirac's conjecture, and degrees of freedom for constrained systems

A necessary and sufficient condition for a function to be a generator of dynamical symmetry transformations in hamiltonian formalism is derived and presented in two forms, one of which involves an evolution operator connecting hamiltonian and lagrangian formalisms. As a particular case gauge transformations are studied. A careful distinction between gauge transformations of solutions of equations of motion and gauge transformations of points in phase space allows us to give a definitive clarification of the so-called Dirac conjecture (that is to say: the “ad hoc” addition of all the secondary first class constraints to the hamiltonian). Finally, the gauge fixing procedure is studied in both hamiltonian and lagrangian formalisms, and it is proven, in a nontrivial way, that the true number of degrees of freedom is the same for both formalisms.

GEOMETRIC HAMILTON–JACOBI THEORY FOR NONHOLONOMIC DYNAMICAL SYSTEMS

The geometric formulation of Hamilton–Jacobi theory for systems with nonholonomic constraints is developed, following the ideas of the authors in previous papers. The relation between the solutions of the Hamilton–Jacobi problem with the symplectic structure defined from the Lagrangian function and the constraints is studied. The concept of complete solutions and their relationship with constants of motion, are also studied in detail. Local expressions using quasivelocities are provided. As an example, the nonholonomic free particle is considered.

On an evolution operator connecting Lagrangian and Hamiltonian formalisms

The unambiguous evolution operator K was recently introduced in the theory of constrained systems. By viewing K as a vector field over the Legendre transformation, we give an intrinsic characterization of it through simple and intuitive properties. Some immediate consequences are explored.

Noether’s theorem and gauge transformations: Application to the bosonic string and CP n−12 model

New results on the theory of constrained systems are applied to characterize the generators of Noether’s symmetry transformations. As a byproduct, an algorithm to construct gauge transformations in Hamiltonian formalism is derived. This is illustrated with two relevant examples.

Singular Lagrangians: some geometric structures along the Legendre map

New geometric structures that relate the Lagrangian and Hamiltonian formalisms defined upon a singular Lagrangian are presented. Several vector fields are constructed in velocity space that give new and precise answers to several topics such as the projectability of a vector field to a Hamiltonian vector field, the computation of the kernel of the presymplectic form of a Lagrangian formalism, the construction of the Lagrangian dynamical vector fields and the characterization of dynamical symmetries.

Constrained systems: A unified geometric approach

A general geometric framework is devised in order to contain the presymplectic and Lagrangian formalisms as particular cases. We call these objectsconstrained dynamical systems, since their dynamics usually lead toconstraints. Their most elementary properties are studied, and several related structures, especially morphisms, are defined. In particular, a stabilization algorithm is performed. As a byproduct, the dynamics and constraints of the Lagrangian formalism (with the “second-order condition”) are intrinsically obtained.