Overview
Simple problems in classical mechanics can be represented within a linear algebra framework and computed. The framework, discretizes the time and necessarily involves an approximation to the analytic solution.
Computing Newtons Equations
In this algorithm, time is broken into several segments, where the change in time between each segment is given by {% \Delta t %}. Then, the position and its first and second derivative are arranged in a column vector.
{%
\begin{bmatrix}
\ddot{x}(t+1) \\
\dot{x}(t+1) \\
x(t+1) \\
\end{bmatrix}
%}
Then, we utilize the following approximation
{% y(t_0 + \Delta t) \approx y(t_0) + dy(t_0)/ dt \times \Delta t %}
which can be justified using a
Taylors Series.
Newtons equations can then be modeled through a matrix equation. (see Linear Algebra)
{%
\begin{bmatrix}
\ddot{x}(t+1) \\
\dot{x}(t+1) \\
x(t+1) \\
\end{bmatrix}
=
\begin{bmatrix}
? & ? & ? \\
\Delta t & 1 & 0\\
0 & \Delta t & 1 \\
\end{bmatrix}
\begin{bmatrix}
\ddot{x}(t) \\
\dot{x}(t) \\
x(t) \\
\end{bmatrix}
%}
Here, the top row of the matrix factor represents the coefficients of the position vector constructed above that affects the
acceleration. That is, filling in this row will specify the forces present in the problem, and is problem specific.