lotkavolterra2.tex

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\title{Hamiltonian Volterra}

\begin{document}

\maketitle

\begin{eqnarray*}
\dot{x}_j &=& \epsilon_j x_j + \frac{1}{\beta_j} \sum_k a_{jk} x_j x_k\\
\dot{x}_j &=& \epsilon_j x_j + \sum_k a_{jk} x_j x_k\\
a_{jk} &=& - a_{kj}\\
Q_j &\equiv& \int_0^t x_j (\tau ) d\tau\\
\ddot{Q}_j &=& \epsilon_j Q_j + \sum_{k=1}^n a_{jk} \dot{Q}_j \dot{Q}_k\\
H &\equiv& \sum \epsilon_j Q_j - \dot{Q}_j\\
\dot{H} &=& 0\\
P_j &\equiv& \log \dot{Q}_j - \frac{1}{2} \sum a_{jk} Q_k\\
H &=& \sum \epsilon_j Q_j - \sum e^{P_j + \frac{1}{2} \sum a_{jk} Q_k}\\
\dot{P}_j &=& \frac{\partial H}{\partial Q_j}\\
\dot{Q}_j &=& - \frac{\partial H}{\partial P_j}\\
I_j &\equiv& P_j - \frac{1}{2} \sum a_{jk} Q_k - \epsilon_j t\\
\dot{I}_j &=& 0\\
\setof{I_j , I_k } &=& a_{jk}\\
\end{eqnarray*}

\url{http://www.math.illinois.edu/~ruiloja/Meus-papers/HTML/equadiff.pdf}

\end{document}