Constrained systems:  geometric study and symmetry transformations

This Ph. D. dissertation is devoted to the study of some geometric 
aspects of constrained systems and their symmetry transformations.
The main idea in the geometric study is the introduction of what we 
call *linearly constrained systems*;  they are a geometric setting 
for a differential equation where the velocities cannot be isolated
because of a linear operator multiplying them.  We define morphisms 
between these objects, and a stabilization algorithm to solve the 
equation of motion.
The singular lagrangian and the Dirac's hamiltonian formalisms can 
be treated within this framework, and thereby their equivalence can 
be thoroughly studied.  The main tool is an evolution operator K, which 
is a vector field along the Legendre's transformation of a lagrangian;
this operator also relates the constraints of both formalisms.
These results are extended to higher-order singular lagrangians.  A de-
composition of the Legendre-Ostrogradskii's transformation is performed, 
and evolution operators K_r defined.  These operators allow equivalent 
formulations of the dynamics, and also relate the corresponding con-
straints.
Then we study the dynamical symmetry transformations (transformations 
mapping solutions into solutions) in the hamiltonian formalism.  We 
give two characterizations of the generators of such transformations, 
and, as a byproduct, we obtain an algorithm to construct hamiltonian 
gauge generators.  Dirac's conjecture is also discussed.
A similar analysis is performed in the lagrangian formalism, but now 
the point of departure are Noether's transformations.  We see that 
they can be obtained from a kind of "hamiltonian generator" G_H, and 
we give a characterization of it.  Lagrangian gauge generators are 
also considered.
Finally, gauge fixing procedures are studied.  The hamiltonian form-
alism is first considered, and its results are used to perform the 
lagrangian gauge fixing.  This procedure is also extended to higher 
order lagrangians.