symbolic params q_1_1=${q_{1,1}}$ p_1_1=${p_{1,1}}$ q_2_1=${q_{2,1}}$ p_2_1=${p_{2,1}}$ q_3_1=${q_{3,1}}$ p_3_1=${p_{3,1}}$
Hamiltonian
${H} = {{{\frac{1}{2}} {{{{p_{1,1}}}} \cdot {{{p_{1,1}}}}}} + {{\frac{1}{2}} {{{{p_{2,1}}}} \cdot {{{p_{2,1}}}}}} + {{\frac{1}{2}} {{{{p_{3,1}}}} \cdot {{{p_{3,1}}}}}} + {{{0.05}} {{\left({{{\left({{\sqrt{{{\left({{{q_{1,1}}}{-{{q_{2,1}}}}}\right)}} {{\left({{{q_{1,1}}}{-{{q_{2,1}}}}}\right)}}}}{-{1}}}\right)}^{2}} + {{\left({{\sqrt{{{\left({{{q_{1,1}}}{-{{q_{3,1}}}}}\right)}} {{\left({{{q_{1,1}}}{-{{q_{3,1}}}}}\right)}}}}{-{1}}}\right)}^{2}} + {{\left({{\sqrt{{{\left({{{q_{2,1}}}{-{{q_{1,1}}}}}\right)}} {{\left({{{q_{2,1}}}{-{{q_{1,1}}}}}\right)}}}}{-{1}}}\right)}^{2}} + {{\left({{\sqrt{{{\left({{{q_{2,1}}}{-{{q_{3,1}}}}}\right)}} {{\left({{{q_{2,1}}}{-{{q_{3,1}}}}}\right)}}}}{-{1}}}\right)}^{2}} + {{\left({{\sqrt{{{\left({{{q_{3,1}}}{-{{q_{1,1}}}}}\right)}} {{\left({{{q_{3,1}}}{-{{q_{1,1}}}}}\right)}}}}{-{1}}}\right)}^{2}} + {{\left({{\sqrt{{{\left({{{q_{3,1}}}{-{{q_{2,1}}}}}\right)}} {{\left({{{q_{3,1}}}{-{{q_{2,1}}}}}\right)}}}}{-{1}}}\right)}^{2}}}\right)}}}}$
Evolution Equations:
${\frac{\partial {q_{1,1}}}{\partial t}} = {{p_{1,1}}}$
${\frac{\partial {p_{1,1}}}{\partial t}} = {{\frac{1}{4}}{\left({{{{0.8}} {{{q_{2,1}}}}} + {{{0.8}} {{{q_{3,1}}}}}{-{{{1.6}} {{{q_{1,1}}}}}}}\right)}}$
${\frac{\partial {q_{2,1}}}{\partial t}} = {{p_{2,1}}}$
${\frac{\partial {p_{2,1}}}{\partial t}} = {{\frac{1}{4}}{\left({{{{0.8}} {{{q_{1,1}}}}} + {{{0.8}} {{{q_{3,1}}}}}{-{{{1.6}} {{{q_{2,1}}}}}}}\right)}}$
${\frac{\partial {q_{3,1}}}{\partial t}} = {{p_{3,1}}}$
${\frac{\partial {p_{3,1}}}{\partial t}} = {{\frac{1}{4}}{\left({{{{0.8}} {{{q_{1,1}}}}} + {{{0.8}} {{{q_{2,1}}}}}{-{{{1.6}} {{{q_{3,1}}}}}}}\right)}}$
...in a vector:
${\left[\begin{array}{c} \frac{\partial {q_{1,1}}}{\partial t}\\ \frac{\partial {p_{1,1}}}{\partial t}\\ \frac{\partial {q_{2,1}}}{\partial t}\\ \frac{\partial {p_{2,1}}}{\partial t}\\ \frac{\partial {q_{3,1}}}{\partial t}\\ \frac{\partial {p_{3,1}}}{\partial t}\end{array}\right]} = {\left[\begin{array}{c} {p_{1,1}}\\ {\frac{1}{4}}{\left({{{{0.8}} {{{q_{2,1}}}}} + {{{0.8}} {{{q_{3,1}}}}}{-{{{1.6}} {{{q_{1,1}}}}}}}\right)}\\ {p_{2,1}}\\ {\frac{1}{4}}{\left({{{{0.8}} {{{q_{1,1}}}}} + {{{0.8}} {{{q_{3,1}}}}}{-{{{1.6}} {{{q_{2,1}}}}}}}\right)}\\ {p_{3,1}}\\ {\frac{1}{4}}{\left({{{{0.8}} {{{q_{1,1}}}}} + {{{0.8}} {{{q_{2,1}}}}}{-{{{1.6}} {{{q_{3,1}}}}}}}\right)}\end{array}\right]}$
...linearized:
${\left[\begin{array}{c} \frac{\partial {q_{1,1}}}{\partial t}\\ \frac{\partial {p_{1,1}}}{\partial t}\\ \frac{\partial {q_{2,1}}}{\partial t}\\ \frac{\partial {p_{2,1}}}{\partial t}\\ \frac{\partial {q_{3,1}}}{\partial t}\\ \frac{\partial {p_{3,1}}}{\partial t}\end{array}\right]} = {{{\left[\begin{array}{cccccc} 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0\end{array}\right]}} {{\left[\begin{array}{c} {q_{1,1}}\\ {p_{1,1}}\\ {q_{2,1}}\\ {p_{2,1}}\\ {q_{3,1}}\\ {p_{3,1}}\end{array}\right]}}}$
1st order approximate derivative for Forward Euler
${\left[\begin{array}{c} {\frac{1}{\Delta t}}{\left({{-{{q_{1,1}(t)}}} + {{q_{1,1}(t+\Delta t)}}}\right)}\\ {\frac{1}{\Delta t}}{\left({{-{{p_{1,1}(t)}}} + {{p_{1,1}(t+\Delta t)}}}\right)}\\ {\frac{1}{\Delta t}}{\left({{-{{q_{2,1}(t)}}} + {{q_{2,1}(t+\Delta t)}}}\right)}\\ {\frac{1}{\Delta t}}{\left({{-{{p_{2,1}(t)}}} + {{p_{2,1}(t+\Delta t)}}}\right)}\\ {\frac{1}{\Delta t}}{\left({{-{{q_{3,1}(t)}}} + {{q_{3,1}(t+\Delta t)}}}\right)}\\ {\frac{1}{\Delta t}}{\left({{-{{p_{3,1}(t)}}} + {{p_{3,1}(t+\Delta t)}}}\right)}\end{array}\right]} = {{{\left[\begin{array}{cccccc} 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0\end{array}\right]}} {{\left[\begin{array}{c} {q_{1,1}(t)}\\ {p_{1,1}(t)}\\ {q_{2,1}(t)}\\ {p_{2,1}(t)}\\ {q_{3,1}(t)}\\ {p_{3,1}(t)}\end{array}\right]}}}$
${\left[\begin{array}{c} {q_{1,1}(t+\Delta t)}\\ {p_{1,1}(t+\Delta t)}\\ {q_{2,1}(t+\Delta t)}\\ {p_{2,1}(t+\Delta t)}\\ {q_{3,1}(t+\Delta t)}\\ {p_{3,1}(t+\Delta t)}\end{array}\right]} = {{\left[\begin{array}{c} {q_{1,1}(t)}\\ {p_{1,1}(t)}\\ {q_{2,1}(t)}\\ {p_{2,1}(t)}\\ {q_{3,1}(t)}\\ {p_{3,1}(t)}\end{array}\right]} + {{{\Delta t}} \cdot {{\left[\begin{array}{cccccc} 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0\end{array}\right]}} {{\left[\begin{array}{c} {q_{1,1}(t)}\\ {p_{1,1}(t)}\\ {q_{2,1}(t)}\\ {p_{2,1}(t)}\\ {q_{3,1}(t)}\\ {p_{3,1}(t)}\end{array}\right]}}}}$
${\left[\begin{array}{c} {q_{1,1}(t+\Delta t)}\\ {p_{1,1}(t+\Delta t)}\\ {q_{2,1}(t+\Delta t)}\\ {p_{2,1}(t+\Delta t)}\\ {q_{3,1}(t+\Delta t)}\\ {p_{3,1}(t+\Delta t)}\end{array}\right]} = {{{\left({{\left[\begin{array}{cccccc} 1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]} + {{{\Delta t}} \cdot {{\left[\begin{array}{cccccc} 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0\end{array}\right]}}}}\right)}} {{\left[\begin{array}{c} {q_{1,1}(t)}\\ {p_{1,1}(t)}\\ {q_{2,1}(t)}\\ {p_{2,1}(t)}\\ {q_{3,1}(t)}\\ {p_{3,1}(t)}\end{array}\right]}}}$
${\left[\begin{array}{c} {q_{1,1}(t+\Delta t)}\\ {p_{1,1}(t+\Delta t)}\\ {q_{2,1}(t+\Delta t)}\\ {p_{2,1}(t+\Delta t)}\\ {q_{3,1}(t+\Delta t)}\\ {p_{3,1}(t+\Delta t)}\end{array}\right]} = {{{\left[\begin{array}{cccccc} 1& \Delta t& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& \Delta t& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& \Delta t\\ 0& 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{c} {q_{1,1}(t)}\\ {p_{1,1}(t)}\\ {q_{2,1}(t)}\\ {p_{2,1}(t)}\\ {q_{3,1}(t)}\\ {p_{3,1}(t)}\end{array}\right]}}}$
${\left[\begin{array}{c} {q_{1,1}(t+\Delta t)}\\ {p_{1,1}(t+\Delta t)}\\ {q_{2,1}(t+\Delta t)}\\ {p_{2,1}(t+\Delta t)}\\ {q_{3,1}(t+\Delta t)}\\ {p_{3,1}(t+\Delta t)}\end{array}\right]} = {{{\left[\begin{array}{cccccc} 1& 0.1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0.1& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0.1\\ 0& 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{c} {q_{1,1}(t)}\\ {p_{1,1}(t)}\\ {q_{2,1}(t)}\\ {p_{2,1}(t)}\\ {q_{3,1}(t)}\\ {p_{3,1}(t)}\end{array}\right]}}}$
1st order approximate derivative for Backwards Euler
${\left[\begin{array}{c} {\frac{1}{\Delta t}}{\left({{-{{q_{1,1}(t)}}} + {{q_{1,1}(t+\Delta t)}}}\right)}\\ {\frac{1}{\Delta t}}{\left({{-{{p_{1,1}(t)}}} + {{p_{1,1}(t+\Delta t)}}}\right)}\\ {\frac{1}{\Delta t}}{\left({{-{{q_{2,1}(t)}}} + {{q_{2,1}(t+\Delta t)}}}\right)}\\ {\frac{1}{\Delta t}}{\left({{-{{p_{2,1}(t)}}} + {{p_{2,1}(t+\Delta t)}}}\right)}\\ {\frac{1}{\Delta t}}{\left({{-{{q_{3,1}(t)}}} + {{q_{3,1}(t+\Delta t)}}}\right)}\\ {\frac{1}{\Delta t}}{\left({{-{{p_{3,1}(t)}}} + {{p_{3,1}(t+\Delta t)}}}\right)}\end{array}\right]} = {{{\left[\begin{array}{cccccc} 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0\end{array}\right]}} {{\left[\begin{array}{c} {q_{1,1}(t+\Delta t)}\\ {p_{1,1}(t+\Delta t)}\\ {q_{2,1}(t+\Delta t)}\\ {p_{2,1}(t+\Delta t)}\\ {q_{3,1}(t+\Delta t)}\\ {p_{3,1}(t+\Delta t)}\end{array}\right]}}}$
${{\left[\begin{array}{c} {q_{1,1}(t+\Delta t)}\\ {p_{1,1}(t+\Delta t)}\\ {q_{2,1}(t+\Delta t)}\\ {p_{2,1}(t+\Delta t)}\\ {q_{3,1}(t+\Delta t)}\\ {p_{3,1}(t+\Delta t)}\end{array}\right]}{-{{{\Delta t}} \cdot {{\left[\begin{array}{cccccc} 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0\end{array}\right]}} {{\left[\begin{array}{c} {q_{1,1}(t+\Delta t)}\\ {p_{1,1}(t+\Delta t)}\\ {q_{2,1}(t+\Delta t)}\\ {p_{2,1}(t+\Delta t)}\\ {q_{3,1}(t+\Delta t)}\\ {p_{3,1}(t+\Delta t)}\end{array}\right]}}}}} = {\left[\begin{array}{c} {q_{1,1}(t)}\\ {p_{1,1}(t)}\\ {q_{2,1}(t)}\\ {p_{2,1}(t)}\\ {q_{3,1}(t)}\\ {p_{3,1}(t)}\end{array}\right]}$
${{{\left({{\left[\begin{array}{cccccc} 1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]}{-{{{\Delta t}} \cdot {{\left[\begin{array}{cccccc} 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0\end{array}\right]}}}}}\right)}} {{\left[\begin{array}{c} {q_{1,1}(t+\Delta t)}\\ {p_{1,1}(t+\Delta t)}\\ {q_{2,1}(t+\Delta t)}\\ {p_{2,1}(t+\Delta t)}\\ {q_{3,1}(t+\Delta t)}\\ {p_{3,1}(t+\Delta t)}\end{array}\right]}}} = {\left[\begin{array}{c} {q_{1,1}(t)}\\ {p_{1,1}(t)}\\ {q_{2,1}(t)}\\ {p_{2,1}(t)}\\ {q_{3,1}(t)}\\ {p_{3,1}(t)}\end{array}\right]}$
${{{\left[\begin{array}{cccccc} 1& -{\Delta t}& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& -{\Delta t}& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& -{\Delta t}\\ 0& 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{c} {q_{1,1}(t+\Delta t)}\\ {p_{1,1}(t+\Delta t)}\\ {q_{2,1}(t+\Delta t)}\\ {p_{2,1}(t+\Delta t)}\\ {q_{3,1}(t+\Delta t)}\\ {p_{3,1}(t+\Delta t)}\end{array}\right]}}} = {\left[\begin{array}{c} {q_{1,1}(t)}\\ {p_{1,1}(t)}\\ {q_{2,1}(t)}\\ {p_{2,1}(t)}\\ {q_{3,1}(t)}\\ {p_{3,1}(t)}\end{array}\right]}$
${\left[\begin{array}{c} {q_{1,1}(t+\Delta t)}\\ {p_{1,1}(t+\Delta t)}\\ {q_{2,1}(t+\Delta t)}\\ {p_{2,1}(t+\Delta t)}\\ {q_{3,1}(t+\Delta t)}\\ {p_{3,1}(t+\Delta t)}\end{array}\right]} = {{{{\left[\begin{array}{cccccc} 1& -{0.1}& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& -{0.1}& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& -{0.1}\\ 0& 0& 0& 0& 0& 1\end{array}\right]}^{-1}}} {{\left[\begin{array}{c} {q_{1,1}(t)}\\ {p_{1,1}(t)}\\ {q_{2,1}(t)}\\ {p_{2,1}(t)}\\ {q_{3,1}(t)}\\ {p_{3,1}(t)}\end{array}\right]}}}$
Gnuplot Produced by GNUPLOT 5.4 patchlevel 4 -200 -150 -100 -50 0 50 100 150 200 0 100 200 300 400 500 600 700 800 900 1000 1100 q1,1 q1,1 p1,1 p1,1 q2,1 q2,1 p2,1 p2,1 q3,1 q3,1 p3,1 p3,1