Energy conservation¶
To construct a stable scheme, I consider the conservation of energy:
\[E
\equiv
T
+
U
=
const.,\]
where
\[ \begin{align}\begin{aligned}T
&
=
\kene,\\U
&
=
\pene.\end{aligned}\end{align} \]
Kinetic energy part¶
\[ \begin{align}\begin{aligned}\dder{T}{t}
&
=
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\dder{}{t}
\left(
\vel_{\ib}
\vel_{\ic}
\cmat{\ib}{\ic}
\right)\\&
=
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\dder{\vel_{\ib}}{t}
\ave{
\vel_{\ic}
\cmat{\ib}{\ic}
}\\&
+
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\ave{\vel_{\ib}}
\dder{\vel_{\ic}}{t}
\ave{\cmat{\ib}{\ic}}\\&
+
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\ave{\vel_{\ib}}
\,
\ave{\vel_{\ic}}
\dder{}{t}
\cmat{\ib}{\ic}.\end{aligned}\end{align} \]
The first two terms yield
\[m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\dder{\vel_{\ib}}{t}
\left(
\frac{1}{2}
\ave{
\vel_{\ic}
\cmat{\ib}{\ic}
}
+
\frac{1}{2}
\ave{\vel_{\ic}}
\,
\ave{\cmat{\ib}{\ic}}
\right),\]
while the last term leads to
\[ \begin{align}\begin{aligned}&
-
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\ave{\vel_{\ib}}
\,
\ave{\vel_{\ic}}
\left( \ave{\vel_{\ib}} - \ave{\vel_{\ic}} \right)
\text{sinc} \left(
\frac{
\dif{\pos_{\ib}}
}{2}
-
\frac{
\dif{\pos_{\ic}}
}{2}
\right)
\sin \left( \ave{\pos_{\ib}} - \ave{\pos_{\ic}} \right)\\=
&
-
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\ave{\vel_{\ib}}
\,
\ave{\vel_{\ic}}
\,
\ave{\vel_{\ib}}
\text{sinc} \left(
\frac{
\dif{\pos_{\ib}}
}{2}
-
\frac{
\dif{\pos_{\ic}}
}{2}
\right)
\sin \left( \ave{\pos_{\ib}} - \ave{\pos_{\ic}} \right)\\&
+
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\ave{\vel_{\ib}}
\,
\ave{\vel_{\ic}}
\,
\ave{\vel_{\ic}}
\text{sinc} \left(
\frac{
\dif{\pos_{\ib}}
}{2}
-
\frac{
\dif{\pos_{\ic}}
}{2}
\right)
\sin \left( \ave{\pos_{\ib}} - \ave{\pos_{\ic}} \right)\\=
&
-
m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\ave{\vel_{\ib}}
\,
\ave{\vel_{\ic}}
\,
\ave{\vel_{\ib}}
\text{sinc} \left(
\frac{
\dif{\pos_{\ib}}
}{2}
-
\frac{
\dif{\pos_{\ic}}
}{2}
\right)
\sin \left( \ave{\pos_{\ib}} - \ave{\pos_{\ic}} \right).\end{aligned}\end{align} \]
Note that
\[\dder{\pos_{\ia}}{t}
=
\ave{\vel_{\ia}}\]
is assumed.
As a result, I have
\[ \begin{align}\begin{aligned}\dder{T}{t}
&
=
m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\dder{\vel_{\ib}}{t}
\left(
\frac{1}{2}
\ave{
\vel_{\ic}
\cmat{\ib}{\ic}
}
+
\frac{1}{2}
\ave{\vel_{\ic}}
\,
\ave{\cmat{\ib}{\ic}}
\right)\\&
-
m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\ave{\vel_{\ib}}
\,
\ave{\vel_{\ic}}
\,
\ave{\vel_{\ib}}
\text{sinc} \left(
\frac{
\dif{\pos_{\ib}}
-
\dif{\pos_{\ic}}
}{2}
\right)
\sin \left( \ave{\pos_{\ib}} - \ave{\pos_{\ic}} \right)\\&
=
m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\dder{\vel_{\ib}}{t}
\left(
\frac{1}{2}
\ave{
\vel_{\ic}
\cmat{\ic}{\ib}
}
+
\frac{1}{2}
\ave{\vel_{\ic}}
\,
\ave{\cmat{\ic}{\ib}}
\right)\\&
+
m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\ave{\vel_{\ic}}
\,
\ave{\vel_{\ib}}
\,
\ave{\vel_{\ib}}
\text{sinc} \left(
\frac{
\dif{\pos_{\ic}}
}{2}
-
\frac{
\dif{\pos_{\ib}}
}{2}
\right)
\sin \left( \ave{\pos_{\ic}} - \ave{\pos_{\ib}} \right).\end{aligned}\end{align} \]
Potential energy part¶
\[ \begin{align}\begin{aligned}\dder{U}{t}
&
=
-
m g l
\sum_{\ic = 0}^{N - 1}
\left( N - \ic \right)
\dder{}{t}
\sin \pos_{\ic}\\&
=
-
m g l
\sum_{\ic = 0}^{N - 1}
\left( N - \ic \right)
\ave{\vel_{\ic}}
\text{sinc} \frac{\dif{\pos_{\ic}}}{2}
\cos \ave{\pos_{\ic}}.\end{aligned}\end{align} \]
Total energy¶
The change of the total energy results in
\[ \begin{align}\begin{aligned}\dder{E}{t}
&
=
m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\ave{\vel_{\ic}}
\dder{\vel_{\ib}}{t}
\ave{\cmat{\ic}{\ib}}
\left(
\frac{1}{2}
+
\frac{1}{2}
\frac{
\ave{
\vel_{\ic}
\cmat{\ic}{\ib}
}
}{
\ave{\vel_{\ic}}
\,
\ave{\cmat{\ic}{\ib}}
}
\right)\\&
+
m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\ave{\vel_{\ic}}
\,
\ave{\vel_{\ib}}
\,
\ave{\vel_{\ib}}
\text{sinc} \left(
\frac{
\dif{\pos_{\ic}}
}{2}
-
\frac{
\dif{\pos_{\ib}}
}{2}
\right)
\sin \left( \ave{\pos_{\ic}} - \ave{\pos_{\ib}} \right)\\&
-
m g l
\sum_{\ic = 0}^{N - 1}
\left( N - \ic \right)
\ave{\vel_{\ic}}
\text{sinc} \frac{\dif{\pos_{\ic}}}{2}
\cos \ave{\pos_{\ic}}\\&
=
0.\end{aligned}\end{align} \]
Factoring \(\ave{\vel_{\ic}}\) yields
\[\dder{E}{t}
=
\sum_{\ic = 0}^{N - 1}
\ave{\vel_{\ic}}
Q_{\ic}
=
0,\]
where
\[ \begin{align}\begin{aligned}Q_{\ic}
&
\equiv
m l^2
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\dder{\vel_{\ib}}{t}
\ave{\cmat{\ic}{\ib}}
\left(
\frac{1}{2}
+
\frac{1}{2}
\frac{
\ave{
\vel_{\ic}
\cmat{\ic}{\ib}
}
}{
\ave{\vel_{\ic}}
\,
\ave{\cmat{\ic}{\ib}}
}
\right)\\&
+
m l^2
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\ave{\vel_{\ib}}
\,
\ave{\vel_{\ib}}
\text{sinc} \left(
\frac{
\dif{\pos_{\ic}}
}{2}
-
\frac{
\dif{\pos_{\ib}}
}{2}
\right)
\sin \left( \ave{\pos_{\ic}} - \ave{\pos_{\ib}} \right)\\&
-
m g l
\left( N - \ic \right)
\text{sinc} \frac{\dif{\pos_{\ic}}}{2}
\cos \ave{\pos_{\ic}}.\end{aligned}\end{align} \]
Requesting \(Q_{\ic} \equiv 0_{\ic}\) results in the discrete Lagrange’s equation to be handled:
\[\dlag.\]