Content deleted Content added
m Dating maintenance tags: {{Citation needed}} |
→Example: expressing in matrix form to show it is area preserving |
||
Line 62:
\end{align}</math>
Expanding the term <math>v_{n+1}</math> in the second equation the relations can be expressed in the form
:<math>\begin{bmatrix} x_{n+1} \\v_{n+1}\end{bmatrix} =
\begin{bmatrix}
1-\omega^2 \Delta t^2 & \Delta t \\
-\omega^2 \Delta t & 1
\end{bmatrix} \begin{bmatrix} x_{n} \\ v_{n} \end{bmatrix},</math>
and since the determinant of the matrix is 1 the transformation is area-preserving.
The iteration preserves the modified energy functional <math>E_h(x,v)=\tfrac12\left(v^2+\omega^2\,x^2-\omega^2\Delta t\,vx\right)</math> exactly, leading to stable periodic orbits (for sufficiently small step size) that deviate by <math>O(\Delta t)</math> from the exact orbits. The exact circular frequency <math>\omega</math> increases in the numerical approximation by a factor of <math>1+\tfrac1{24}\omega^2\Delta t^2+O(\Delta t^4)</math>.
| |||