The transformation to the phase space can be extended to differential equations in which there are time derivatives in more than one variable. Suppose that represents a configuration, expressed using a coordinate neighborhood on a smooth -dimensional manifold . Second-order constraints of the form or can be expressed as first-order constraints in a -dimensional state space. Let denote the -dimensional phase vector. By extending the method that was applied to the scalar case, is defined as . For each integer such that , . For each such that , . These substitutions can be made directly into an implicit constraint to reduce the order to one.

Suppose that a set of differential equations is expressed in
parametric form as
. In the phase space, there
are differential equations. The first correspond to the
phase space definition
, for each such that
. These hold because
and
is the time derivative of
for . The remaining
components of
follow directly from by
substituting the first components of in the place of and
the remaining in the place of in the expression
. The result can be denoted as (obtained
directly from
). This yields the final equations as
, for each such that
.
These equations define a *phase (or state) transition
equation* of the form
. Now it is clear that
constraints on acceleration can be manipulated into velocity
constraints on the phase space. This enables the tangent space
concepts from Section 8.3 to express constraints that
involve acceleration. Furthermore, the state space is the
tangent bundle (defined in (8.9) for
and later in (15.67) for any smooth manifold) of
because and together indicate a tangent space
and a particular tangent vector
.

Steven M LaValle 2012-04-20