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-\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>.
== References ==
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