10.1: Example- Equations of Motion in Classical Mechanics
The above standard formulation of the initial-value problem can be used to describe a very large class of time-dependent ODEs found in physics. For example, suppose we have a classical mechanical particle with position \(\vec{r}\), subject to an arbitrary external space-and-time-dependent force \(\vec{f}(\vec{r},t)\) and a friction force \(-\lambda d\vec{r}/dt\) (where \(\lambda\) is a damping coefficient). Newton's second law gives the following equation of motion:
\[m \frac{d^2 \vec{r}}{dt^2} = - \lambda \frac{d\vec{r}}{dt} + \vec{f}(\vec{r}, t).\]
This is a second-order ODE, whereas the standard initial-value problem involves a first-order ODE. However, we can turn it into a first-order ODE with the following trick. Define the velocity vector
\[\vec{v} = \frac{d\vec{r}}{dt},\]
and define the state vector by combining the position and velocity vectors:
\[\vec{y} = \begin{bmatrix}\vec{r} \\ \vec{v}\end{bmatrix}.\]
Then the equation of motion takes the form
\[\frac{d\vec{y}}{dt} = \frac{d}{dt}\begin{bmatrix}\vec{r} \\ \vec{v}\end{bmatrix} = \begin{bmatrix}\vec{v} \\ - (\lambda/m) \vec{v} + \vec{f}(\vec{r}, t)/m\end{bmatrix},\]
which is a first-order ODE, as desired. The quantity on the right-hand side is the derivative function \(\vec{F}(\vec{y},t)\) for the initial-value problem. Its dependence on \(\vec{r}\) and \(\vec{v}\) is simply regarded as a dependence on the upper and lower portions of the state vector \(\vec{y}\). In particular, note that the derivative function does not need to be linear, since \(\vec{f}\) can have any arbitrary nonlinear dependence on \(\vec{r}\), e.g. it could depend on the quantity \(|\vec{r}|\).
The "initial state", \(\vec{y}(t_0)\), requires us to specify both the initial position and velocity of the particle, which is consistent with the fact that the original equation of motion was a second-order equation, requiring two sets of initial values to fully specify a solution. In a similar manner, ODEs of higher order can be converted into first-order form, by defining the higher derivatives as state variables and increasing the size of the state vector.