Symplectic notation is a way to handle the Hamiltonian formalism using matrices. It is often implemented in computerized algorithms, for example for solving for the motion of astrophysical bodies (think of calculating the trajectory of a probe on its way to Mars). Such computerized algorithms are called “symplectic integrators”.
One starts by defining a vector, \(\vec\eta\), of dimension \(2n\) where the first \(n\) coordinates are the generalized coordinates, \(q_i\), and the next \(n\) coordinates are the generalized momenta, \(p_i\).
\begin{align}
\vec\eta\equiv\left(\begin{array}{c}
q_1 \\ \vdots \\ q_n \\ p_1 \\ \vdots \\ p_n \\
\end{array} \right)
\tag{7.28}\end{align}
We define a second vector, \(\die{\vec H}{\eta}\):
\begin{align}
\die{\vec H}{\eta}\equiv\left(\begin{array}{c}
\die{H}{q_1} \\ \vdots \\ \die{H}{q_n} \\ \die{H}{p_1} \\ \vdots \\ \die{H}{p_n} \\
\end{array} \right)
\tag{7.29}\end{align}
and a \(2n\times 2n\) square matrix \(J\):
\begin{align}
J\equiv\left(\begin{array}{cc}
0 &I \\ -I &0 \\
\end{array} \right)
\tag{7.30}\end{align}
where \(I\) is the \(n\times n\) identity matrix. Hamilton’s equations in symplectic notation are thus written as:
\begin{align}
\frac{d\vec\eta}{dt}=J\die{\vec H}{\eta}
\tag{7.31}\end{align}