Singular Lagrangian Systems on Jet Bundles

@article{Leon2002SingularLS,
  title={Singular Lagrangian Systems on Jet Bundles},
  author={Manuel de Le'on and J. Mar'in-Solano and Juan Carlos Marrero and Miguel C. Mu{\~n}oz-Lecanda and Narciso Rom'an-Roy},
  journal={Protein Science},
  year={2002},
  volume={50},
  pages={105-169}
}
The jet bundle description of time-dependent mechanics is revisited. The constraint algorithm for singular Lagrangians is discussed and an exhaustive description of the constraint functions is given. By means of auxiliary connections we give a basis of constraint functions in the Lagrangian and Hamiltonian sides. An additional description of constraints is also given considering at the same time compatibility, stability and second order condition problems. Finally, a classification of the… 
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References

SHOWING 1-10 OF 57 REFERENCES
Geometric Theory of Time-Dependent Singular Lagrangians
The geometry of jet bundles is used for obtaining a geometric approach to time-dependent Lagrangian systems, both in the regular and singular case. Generalized symmetries are introduced as being…
NON-STANDARD CONNECTIONS IN CLASSICAL MECHANICS
In the jet-bundle description of first-order classical field theories there are some elements, such as the Lagrangian energy and the construction of the Hamiltonian formalism, which require the prior…
Geometry of Lagrangian First-order Classical Field Theories
We construct a lagrangian geometric formulation for first-order field theories using the canonical structures of first-order jet bundles, which are taken as the phase spaces of the systems in…
Lagrangian theory for presymplectic systems
TLDR
An algorithm is developed which solves simultaneously the problem of the compatibility of the equations of motion, the consistency of their solutions and the SODE problem and gives all the lagrangian constraints, which are completed and improved.
Constraints in Hamiltonian time-dependent mechanics
The key point of the study of constraints in Hamiltonian time-dependent mechanics lies in the fact that a Poisson structure does not provide dynamic equations and a Poisson bracket of constraints…
Generalized Hamiltonian Formalism for Field Theory: Constraint Systems
In the framework of the geometric formulation of field theory, classical fields are represented by sections of fibred manifolds, and their dynamics is phrased in jet manifold terms. The Hamiltonian…
Hamiltonian time-dependent mechanics
The usual formulation of time-dependent mechanics implies a given splitting Y=R×M of an event space Y. This splitting, however, is broken by any time-dependent transformation, including…
A generalized geometric framework for constrained systems
Singular Lagrangian and constrained Hamiltonian systems, generalized canonical formalism
SummaryThe relation between singular Lagrangian systems and the corresponding constrained Hamiltonian systems is studied in detail by making use of the « generalized canonical quantities » (g.c.q.).…
Geometric study of the connection between the Lagrangian and Hamiltonian constraints
...
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