Energy conservation¶
The conservation of the total energy states that
\[E
\equiv
T
+
U\]
is constant, where
\[ \begin{align}\begin{aligned}T
&
=
\kene,\\U
&
=
\pene.\end{aligned}\end{align} \]
To derive the energy conservation, I consider to differentiate this equation with respect to time.
Kinetic energy contribution¶
\[ \begin{align}\begin{aligned}\tder{T}{t}
&
=
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\tder{\vel_{\ib}}{t}
\vel_{\ic}
\cos \left( \pos_{\ib} - \pos_{\ic} \right)\\&
+
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib}
\tder{\vel_{\ic}}{t}
\cos \left( \pos_{\ib} - \pos_{\ic} \right)\\&
+
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib}
\vel_{\ic}
\tder{}{t}
\cos \left( \pos_{\ib} - \pos_{\ic} \right),\end{aligned}\end{align} \]
where the first two terms are equal and the sum is
\[m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\tder{\vel_{\ib}}{t}
\vel_{\ic}
\cos \left( \pos_{\ib} - \pos_{\ic} \right),\]
while the last term is
\[ \begin{align}\begin{aligned}&
-
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib}
\vel_{\ic}
\vel_{\ib}
\sin \left( \pos_{\ib} - \pos_{\ic} \right)\\&
+
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib}
\vel_{\ic}
\vel_{\ic}
\sin \left( \pos_{\ib} - \pos_{\ic} \right)\\=
&
-
m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib}
\vel_{\ic}
\vel_{\ib}
\sin \left( \pos_{\ib} - \pos_{\ic} \right).\end{aligned}\end{align} \]
As a result,
\[ \begin{align}\begin{aligned}\tder{T}{t}
&
=
m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\tder{\vel_{\ib}}{t}
\vel_{\ic}
\cos \left( \pos_{\ib} - \pos_{\ic} \right)\\&
-
m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib}
\vel_{\ic}
\vel_{\ib}
\sin \left( \pos_{\ib} - \pos_{\ic} \right)\\&
=
m l^2
\sum_{\ic = 0}^{N - 1}
\vel_{\ic}
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ic, \ib \right) \right\}
\left\{
\tder{\vel_{\ib}}{t}
\cos \left( \pos_{\ic} - \pos_{\ib} \right)
+
\vel_{\ib}
\vel_{\ib}
\sin \left( \pos_{\ic} - \pos_{\ib} \right)
\right\}.\end{aligned}\end{align} \]
Potential energy contribution¶
The potential energy contribution is
\[ \begin{align}\begin{aligned}\tder{U}{t}
&
=
-
m g l
\sum_{\ib = 0}^{N - 1}
\left( N - \ib \right)
\tder{}{t} \left( \sin \pos_{\ib} \right)\\&
=
-
m g l
\sum_{\ic = 0}^{N - 1}
\left( N - \ic \right)
\vel_{\ic}
\cos \pos_{\ic}.\end{aligned}\end{align} \]
Energy conservation¶
The temporal derivative of the total energy results in
\[\tder{E}{t}
=
\sum_{\ia = 0}^{N - 1}
\vel_{\ia}
\left\{
\lag
\right\},\]
which is indeed zero since the relation in the wavy brackets is the left-hand-side of the Lagrange’s equation.