Kinetic energy¶
To begin with, we consider the velocity of the \(\ia\)-th object on the Cartesian coordinate.
The time derivative of the position vector:
\[\posvec\]
yields
\[-
\vec{e}_x \sum_{\ib = 0}^{\ia} l \vel_{\ib} \sin{ \pos_{\ib} }
+
\vec{e}_y \sum_{\ib = 0}^{\ia} l \vel_{\ib} \cos{ \pos_{\ib} },\]
which is the velocity of the \(\ia\)-th object. The kinetic energy of this object is given by
\[\kinetic_{\ia}
=
\frac{1}{2} m
\left( \sum_{\ib = 0}^{\ia} l \vel_{\ib} \sin{ \pos_{\ib} } \right)^2
+
\frac{1}{2} m
\left( \sum_{\ib = 0}^{\ia} l \vel_{\ib} \cos{ \pos_{\ib} } \right)^2,\]
and thus the total kinetic energy of the system is
\[ \begin{align}\begin{aligned}\kinetic
&
\equiv
\sum_{\ia = 0}^{N - 1} \kinetic_{\ia}\\&
=
\sum_{\ia = 0}^{N - 1}
\frac{1}{2} m
\left( \sum_{\ib = 0}^{\ia} l \vel_{\ib} \sin{ \pos_{\ib} } \right)^2
+
\sum_{\ia = 0}^{N - 1}
\frac{1}{2} m
\left( \sum_{\ib = 0}^{\ia} l \vel_{\ib} \cos{ \pos_{\ib} } \right)^2\\&
=
\sum_{\ia = 0}^{N - 1}
\frac{1}{2} m
\sum_{\ib = 0}^{\ia} l \vel_{\ib} \sin{ \pos_{\ib} }
\sum_{\ic = 0}^{\ia} l \vel_{\ic} \sin{ \pos_{\ic} }
+
\sum_{\ia = 0}^{N - 1}
\frac{1}{2} m
\sum_{\ib = 0}^{\ia} l \vel_{\ib} \cos{ \pos_{\ib} }
\sum_{\ic = 0}^{\ia} l \vel_{\ic} \cos{ \pos_{\ic} }\\&
=
\frac{1}{2} m l^2
\sum_{\ia = 0}^{N - 1}
\sum_{\ib = 0}^{\ia}
\sum_{\ic = 0}^{\ia}
\vel_{\ib} \vel_{\ic}
\cos \left( \pos_{\ib} - \pos_{\ic} \right).\end{aligned}\end{align} \]
To proceed, we use the identity:
\[\sum_{\ia = 0}^{N - 1}
\sum_{\ib = 0}^{\ia}
\sum_{\ic = 0}^{\ia}
Q_{\ib \ic}
=
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{
N - \max \left( \ib, \ic \right)
\right\}
Q_{\ib \ic}.\]
Derivation
First, the base case with \(N = 1\) is true since
\[\sum_{\ia = 0}^{0}
\sum_{\ib = 0}^{\ia}
\sum_{\ic = 0}^{\ia}
Q_{\ib \ic}
=
\sum_{\ib = 0}^{0}
\sum_{\ic = 0}^{0}
\left\{
1 - \max \left( \ib, \ic \right)
\right\}
Q_{\ib \ic}
=
Q_{0 0}.\]
Now we assume that the statement is true for \(N\) and consider with respect to \(N + 1\).
The left-hand-side term of the statement leads to
\[\sum_{\ia = 0}^{N}
\sum_{\ib = 0}^{\ia}
\sum_{\ic = 0}^{\ia}
Q_{\ib \ic}
=
\sum_{\ia = 0}^{N - 1}
\sum_{\ib = 0}^{\ia}
\sum_{\ic = 0}^{\ia}
Q_{\ib \ic}
+
\sum_{\ib = 0}^{N}
\sum_{\ic = 0}^{N}
Q_{\ib \ic},\]
while the right-hand-side term of the statement is
\[ \begin{align}\begin{aligned}\sum_{\ib = 0}^{N}
\sum_{\ic = 0}^{N}
\left\{
N + 1 - \max \left( \ib, \ic \right)
\right\}
Q_{\ib \ic}
&
=
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N}
\left\{
N + 1 - \max \left( \ib, \ic \right)
\right\}
Q_{\ib \ic}
+
\sum_{\ic = 0}^{N}
\left\{
N + 1 - \max \left( N, \ic \right)
\right\}
Q_{N \ic}\\&
=
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{
N + 1 - \max \left( \ib, \ic \right)
\right\}
Q_{\ib \ic}
+
\sum_{\ic = 0}^{N}
\left\{
N + 1 - \max \left( N, \ic \right)
\right\}
Q_{N \ic}
+
\sum_{\ib = 0}^{N - 1}
\left\{
N + 1 - \max \left( \ib, N \right)
\right\}
Q_{\ib N}\\&
=
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{
N - \max \left( \ib, \ic \right)
\right\}
Q_{\ib \ic}
+
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
Q_{\ib \ic}
+
\sum_{\ic = 0}^{N}
Q_{N \ic}
+
\sum_{\ib = 0}^{N - 1}
Q_{\ib N}\\&
=
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{
N - \max \left( \ib, \ic \right)
\right\}
Q_{\ib \ic}
+
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N}
Q_{\ib \ic}
+
\sum_{\ic = 0}^{N}
Q_{N \ic}\\&
=
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{
N - \max \left( \ib, \ic \right)
\right\}
Q_{\ib \ic}
+
\sum_{\ib = 0}^{N}
\sum_{\ic = 0}^{N}
Q_{\ib \ic}.\end{aligned}\end{align} \]
Now the first terms are equal according to the assumption, while the second terms are identical.
As a consequence, we obtain
\[\kinetic
=
\kene.\]
Generalized velocity part¶
First of all, we consider
\[ \begin{align}\begin{aligned}\pder{\kinetic}{\vel_{\ia}}
&
=
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\pder{\vel_{\ib}}{\vel_{\ia}}
\vel_{\ic}
\cos \left( \pos_{\ib} - \pos_{\ic} \right)\\&
+
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib}
\pder{\vel_{\ic}}{\vel_{\ia}}
\cos \left( \pos_{\ib} - \pos_{\ic} \right),\end{aligned}\end{align} \]
where the two terms are equal by interchanging the indices to obtain
\[ \begin{align}\begin{aligned}\pder{\kinetic}{\vel_{\ia}}
&
=
m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib}
\pder{\vel_{\ic}}{\vel_{\ia}}
\cos \left( \pos_{\ib} - \pos_{\ic} \right)\\&
=
m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib}
\delta_{\ic \ia}
\cos \left( \pos_{\ib} - \pos_{\ic} \right)\\&
=
m l^2
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ia \right) \right\}
\vel_{\ib}
\cos \left( \pos_{\ib} - \pos_{\ia} \right)\\&
=
m l^2
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ia, \ib \right) \right\}
\vel_{\ib}
\cos \left( \pos_{\ia} - \pos_{\ib} \right).\end{aligned}\end{align} \]
Differentiating this relation with respect to time leads to
\[ \begin{align}\begin{aligned}\tder{}{t} \pder{\kinetic}{\vel_{\ia}}
&
=
m l^2
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ia, \ib \right) \right\}
\tder{}{t}
\left\{
\vel_{\ib}
\cos \left( \pos_{\ia} - \pos_{\ib} \right)
\right\}\\&
=
m l^2
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ia, \ib \right) \right\}
\tder{\vel_{\ib}}{t}
\cos \left( \pos_{\ia} - \pos_{\ib} \right)\\&
-
m l^2
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ia, \ib \right) \right\}
\vel_{\ib}
\left( \vel_{\ia} - \vel_{\ib} \right)
\sin \left( \pos_{\ia} - \pos_{\ib} \right).\end{aligned}\end{align} \]
Generalized coordinate part¶
\[ \begin{align}\begin{aligned}\pder{\kinetic}{\pos_{\ia}}
&
=
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib} \vel_{\ic}
\pder{}{\pos_{\ia}}
\cos \left( \pos_{\ib} - \pos_{\ic} \right)\\&
=
-
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib} \vel_{\ic}
\left( \pder{\pos_{\ib}}{\pos_{\ia}} - \pder{\pos_{\ic}}{\pos_{\ia}} \right)
\sin \left( \pos_{\ib} - \pos_{\ic} \right)\\&
=
-
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib} \vel_{\ic}
\left( \delta_{\ib \ia} - \delta_{\ic \ia} \right)
\sin \left( \pos_{\ib} - \pos_{\ic} \right)\\&
=
-
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib} \vel_{\ic}
\delta_{\ib \ia}
\sin \left( \pos_{\ib} - \pos_{\ic} \right)
+
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ic \right) \right\}
\vel_{\ib} \vel_{\ic}
\delta_{\ic \ia}
\sin \left( \pos_{\ib} - \pos_{\ic} \right)\\&
=
-
\frac{1}{2} m l^2
\sum_{\ic = 0}^{N - 1}
\left\{ N - \max \left( \ia, \ic \right) \right\}
\vel_{\ia} \vel_{\ic}
\sin \left( \pos_{\ia} - \pos_{\ic} \right)
+
\frac{1}{2} m l^2
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ib, \ia \right) \right\}
\vel_{\ib} \vel_{\ia}
\sin \left( \pos_{\ib} - \pos_{\ia} \right)\\&
=
-
m l^2
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ia, \ib \right) \right\}
\vel_{\ia} \vel_{\ib}
\sin \left( \pos_{\ia} - \pos_{\ib} \right).\end{aligned}\end{align} \]
Total¶
In the Lagrange’s equation, the following terms contribute:
\[ \begin{align}\begin{aligned}\tder{}{t} \pder{\kinetic}{\vel_{\ia}}
-
\pder{\kinetic}{\pos_{\ia}}
&
=
m l^2
\sum_{\ib = 0}^{N - 1}
\left\{
N
-
\max \left( \ia, \ib \right)
\right\}
\tder{\vel_{\ib}}{t}
\cos \left( \pos_{\ia} - \pos_{\ib} \right)\\&
+
m l^2
\sum_{\ib = 0}^{N - 1}
\left\{
N
-
\max \left( \ia, \ib \right)
\right\}
\vel_{\ib}
\vel_{\ib}
\sin \left( \pos_{\ia} - \pos_{\ib} \right).\end{aligned}\end{align} \]