Differential equations

We now introduce some basic differential equations to model motions. The resulting description is often called a dynamical system. The first step is to describe rigid body velocities in terms of state. Returning to models that involve one or more rigid bodies, the state corresponds to a finite number of parameters. Let

$$x = (x_1,x_2,\ldots,x_n)$$ (8.18)

denote an $n$-dimensional state vector. If each $x_i$ corresponds to a position or orientation parameter for a rigid body, then the state vector puts all bodies in their place. Let

$${\dot x}_i = \frac{dx_i}{dt}$$ (8.19)

represent the time derivative, or velocity, for each parameter.

To obtain the state at any time $t$, the velocities need to be integrated over time. Following (8.4), the integration of each state variable determines the value at time $t$:

$$x_i(t) = x_i(0) + \int_0^t {\dot x}_i(s) ds ,$$ (8.20)

in which $x_i(0)$ is the value of $x_i$ at time $t = 0$.

Two main problems arise with (8.20):

1. The integral almost always must be evaluated numerically.
2. The velocity ${\dot x}_i(t)$ must be specified at each time $t$.