Die Hamiltonschen kanonischen Gleichungen
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\begin{array}{*{20}c}
{\dot qk\left( t \right) = \,\frac{{\partial H}}
{{\partial _{pk} }}\left( {t,q1\left( t \right),...,qN\left( t \right),p1\left( t \right),...pN\left( t \right)} \right)\,\left( {k = 1,...,N} \right),} \\
{\dot pk\left( t \right) = - \frac{{\partial H}}
{{\partial _{qk} }}\left( {t,q1\left( t \right),...,qN\left( t \right),p1\left( t \right),...pN\left( t \right)} \right)\,\left( {k = 1,...,N} \right)} \\
\end{array}
$$
stellen ein gekoppeltes System von gewöhnlichen Differentialgleichungen dar. Durch Zusammenfassung der Orts- und Impulsvariablen zu einem Vektor y(t) = (q(t), p(t)) erhält dieses die Gestalt
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{\mathbf{\dot y}}\left( t \right) = {\mathbf{f}}\left( {t,{\mathbf{y}}\left( t \right)} \right),\,{\text{kurz}}\,{\mathbf{\dot y}} = {\mathbf{f}}\left( {t,{\mathbf{y}}} \right)
$$
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