The vector case

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 $ q$ represents a configuration, expressed using a coordinate neighborhood on a smooth $ n$-dimensional manifold $ {\cal C}$. Second-order constraints of the form $ g({\ddot q},{\dot q},q) = 0$ or $ g({\ddot q},{\dot q},q,u) = 0$ can be expressed as first-order constraints in a $ 2n$-dimensional state space. Let $ x$ denote the $ 2n$-dimensional phase vector. By extending the method that was applied to the scalar case, $ x$ is defined as $ x = (q,{\dot q})$. For each integer $ i$ such that $ 1 \leq i \leq n$, $ x_i = q_i$. For each $ i$ such that $ n + 1 \leq i \leq 2n$, $ x_i = {\dot q}_{i-n}$. These substitutions can be made directly into an implicit constraint to reduce the order to one.

Suppose that a set of $ n$ differential equations is expressed in parametric form as $ {\ddot q}= h(q,{\dot q},u)$. In the phase space, there are $ 2n$ differential equations. The first $ n$ correspond to the phase space definition $ {\dot x}_i = x_{n+i}$, for each $ i$ such that $ 1 \leq i \leq n$. These hold because $ x_{n+i} = {\dot q}_i$ and $ {\dot x}_i$ is the time derivative of $ {\dot q}_i$ for $ i \leq n$. The remaining $ n$ components of $ {\dot x}=
f(x,u)$ follow directly from $ h$ by substituting the first $ n$ components of $ x$ in the place of $ q$ and the remaining $ n$ in the place of $ {\dot q}$ in the expression $ h(q,{\dot q},u)$. The result can be denoted as $ h(x,u)$ (obtained directly from $ h(q,{\dot q},u)$). This yields the final $ n$ equations as $ {\dot x}_i = h_{i-n}(x,u)$, for each $ i$ such that $ n + 1 \leq i \leq 2n$. These $ 2n$ equations define a phase (or state) transition equation of the form $ {\dot x}=
f(x,u)$. 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 $ X$ is the tangent bundle (defined in (8.9) for $ {\mathbb{R}}^n$ and later in (15.67) for any smooth manifold) of $ {\cal C}$ because $ q$ and $ {\dot q}$ together indicate a tangent space $ T_q({\cal C})$ and a particular tangent vector $ {\dot q}\in T_q({\cal C})$.

Steven M LaValle 2012-04-20