Some topics concerning the theory of singular dynamical systems (original) (raw)

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 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.

Canonical transformations theory for presymplectic systems

Journal of Mathematical Physics, 1985

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.

CANONICALIZATION OF CONSTRAINED HAMILTONIAN EQUATIONS IN A SINGULAR SYSTEM

In this paper, the canonicalization of constrained Hamiltonian system is discussed. Because the constrained Hamiltonian equations are non-canonical, they lead to many limitations in the research. For this purpose, variable transformation is constructed that satisfies the condition of canonical equation, and the new variables can be obtained by a series of derivations. Finally, two examples are given to illustrate the applications of the result.

On the geometry of singular Lagrangians

Journal of Geometry and Physics, 2000

The geometry of Lagrangian systems, whose Legendre map possesses generic singularities, is studied. On its basis the Transition Principle, prescribing the behaviour of phase trajectories on the singular hypersurface, is proposed. As a by-product, the notion of relative Hamiltonian field associated with an arbitrary Lagrangian is introduced.

On singularities of Hamiltonian mappings

Geometry and topology of caustics, 2008

The notion of an implicit Hamiltonian system-an isotropic mapping H : M → (T M,ω) into the tangent bundle endowed with the symplectic structure defined by canonical morphism between tangent and cotangent bundles of M-is studied. The corank one singularities of such systems are classified. Their transversality conditions in the 1-jet space of isotropic mappings are described and the corresponding symplectically invariant algebras of Hamiltonian generating functions are calculated. We have F * ω = 0 andF = π • F,ω = β −1 (dθ).

CONSTRAINT ALGORITHM FOR k-PRESYMPLECTIC HAMILTONIAN SYSTEMS: APPLICATION TO SINGULAR FIELD THEORIES

International Journal of Geometric Methods in Modern Physics, 2009

The k-symplectic formulation of field theories is especially simple, since only tangent and cotangent bundles are needed in its description. Its defining elements show a close relationship with those in the symplectic formulation of mechanics. It will be shown that this relationship also stands in the presymplectic case. In a natural way, one can mimick the presymplectic constraint algorithm to obtain a constraint algorithm that can be applied to k-presymplectic field theory, and more particularly to the Lagrangian and Hamiltonian formulations of field theories defined by a singular Lagrangian, as well as to the unified Lagrangian-Hamiltonian formalism (Skinner-Rusk formalism) for k-presymplectic field theory. Two examples of application of the algorithm are also analyzed.