  In this paper, following the ideas in Marsden et al.[18], we set up the
regular reduction theory of a regular controlled Lagrangian (RCL) system with
symmetry and momentum map, by using Legendre transformation and Euler-Lagrange
vector field, and this reduction is an extension of symmetric reduction theory
of a regular Lagrangian system under regular controlled Lagrangian equivalence
conditions. Considering the completeness of reduction, in order to describe
uniformly the RCL systems defined on a tangent bundle and on its regular
reduced spaces, we first define a kind of RCL systems on a symplectic fiber
bundle. Then we give a good expression of the dynamical vector field of the RCL
system, such that we can describe the RCL-equivalence for the RCL systems.
Moreover, we introduce regular point and regular orbit reducible RCL systems
with symmetries and momentum maps, by using the reduced Lagrange symplectic
forms and the reduced Euler-Lagrange vector fields, and prove the regular point
and regular orbit reduction theorems for the RCL systems and regular Lagrangian
systems, which explain the relationships between RpCL-equivalence,
RoCL-equivalence for the reducible RCL systems with symmetries and
RCL-equivalence for the associated reduced RCL systems, as well as the
relationship of equivalences of the regular reducible Lagrangian systems,
$R_p$-reduced Lagrangian systems and $R_o$-reduced Lagrangian systems.
