Symbols and identityΒΆ
Two symbols are used to represent discrete operations:
\[ \begin{align}\begin{aligned}\dif{q}
&
\equiv
q^{n+1}
-
q^{n },\\\ave{q}
&
\equiv
\frac{1}{2}
q^{n+1}
+
\frac{1}{2}
q^{n },\end{aligned}\end{align} \]
which are the central differentiation and the central interpolation at the intermediate time step \(n + \frac{1}{2}\), respectively.
The discrete derivative of the trigonometric functions are
\[ \begin{align}\begin{aligned}\dder{}{t}
\cos \pos
&
=
\frac{
\cos \pos^{n+1}
-
\cos \pos^{n }
}{
\dif{t}
}\\&
=
\frac{
\cos \left( \ave{\pos} + \frac{\dif{\pos}}{2} \right)
-
\cos \left( \ave{\pos} - \frac{\dif{\pos}}{2} \right)
}{
\dif{t}
}\\&
=
-
\frac{2}{\dif{t}}
\sin \frac{\dif{\pos}}{2}
\sin \ave{\pos}\\&
=
-
\frac{2}{\dif{t}}
\frac{\dif{\pos}}{2}
\text{sinc} \frac{\dif{\pos}}{2}
\sin \ave{\pos}\\&
=
-
\ave{\vel}
\text{sinc} \frac{\dif{\pos}}{2}
\sin \ave{\pos},\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned}\dder{}{t}
\sin \pos
&
=
\frac{
\sin \pos^{n+1}
-
\sin \pos^{n }
}{
\dif{t}
}\\&
=
\frac{
\sin \left( \ave{\pos} + \frac{\dif{\pos}}{2} \right)
-
\sin \left( \ave{\pos} - \frac{\dif{\pos}}{2} \right)
}{
\dif{t}
}\\&
=
\frac{2}{\dif{t}}
\sin \frac{\dif{\pos}}{2}
\cos \ave{\pos}\\&
=
\frac{2}{\dif{t}}
\frac{\dif{\pos}}{2}
\text{sinc} \frac{\dif{\pos}}{2}
\cos \ave{\pos}\\&
=
\ave{\vel}
\text{sinc} \frac{\dif{\pos}}{2}
\cos \ave{\pos}.\end{aligned}\end{align} \]