Let us consider a differential equation
in a Banach space 
, which is invariant under the action T of
a Lie group 
, dim 
, i.e.
for all 
and 
, and so any transformation T(g)
maps any solution to another solution.
The original phase space is foliated by the orbits of , and the well known space reduction method can be used to reduce an
original differential equation admitting symmetry to one
without symmetries (Anosov & Arnold 1985, p.31). The basic
idea is to identify whole orbits of the symmetry group in phase space
with points of the orbit manifold or orbit space. The original vector
field, corresponding to (4), projects onto a vector field in
the orbit space. The differential equation defined by this field has no
symmetry and is called the quotient system.
Figure 2: Decomposition of the movement in space onto movement
along a manifold 
and along the group . Here
,
, and
.
To obtain the quotient system for (4) explicitly, let us
``parameterize'' the orbit space by a manifold 
of codimension k which is, for simplicity, everywhere
transversal to the orbits. Then any point U in the region 
can be uniquely represented in the form
(see Fig. 2)
i.e. (g,V) are coordinates on 
(here we neglect the
possibility of the same group orbit crossing the manifold more than
once; this will be addressed later in Sec. 5).
Differentiating (6) by time and using commutativity
(5), we immediately obtain
The vector field F(V) can also be uniquely decomposed into the two
components, 
which is tangent to
at V, and 
which is tangent to the group
orbit crossing at V,
Substituting this into (7) and equating components along
and along separately, we obtain a differential equation on ,
and another on ,
Note, that Eq. (9) for V does not depend on g, which is a
consequence of commutativity (5). Equation (9) is
the target quotient system, lacking the symmetry of the original system
and determining the motion along the manifold which is separated
from the motion
along the group which can be found afterwards by integrating
(10).
If the transformations T(g) are explicitly defined, the vector fields
and
can be found explicitly; the standard approach is to expand the vectors
tangent to group orbits in the basis of the group representation
generators.
To conclude, we have shown that Eq. (4) can be replaced by (9), which is a generic differential equation, i.e. it is not invariant
under any nontrivial action of , defined on the manifold
; and then the solution of the parent system (4) is
restored through integration of (9), (10) and (6). This
procedure is valid subject to two most important conditions,
is everywhere transversal to the orbits of ,
.
