[NOTES/CM-02004] Integration of EOM by Quadratures

An example
The momentum conjugate to  cyclic coordinates is a constant of motion. This is very helpful in integrating the equations of motion. Consider the example discussed above in previous {\sf Lesson-A03}.  There we had a particle moving in two dimensions in a potential that depends only on the radial distance from the origin. Changing the variables to plane polar coordinates \((r,\theta)\) we  had obtained

\begin{eqnarray}
  \text{K.E.}   &=&  \frac{1}{2}{m\dot{x}^2} + \frac{1}{2}
{m\dot{y}^2}\nonumber \\
                &=& \frac{1}{2} {m\dot r}^2 + \frac{1}{2}m r^2 {\dot \theta}^2\\
   \therefore \qquad         L   &=& \frac{1}{2} {m\dot r}^2 + \frac{1}{2} m r^2
{\dot \theta}^2 -V(r)\label{EQ19}
\end{eqnarray}

Find all cyclic coordinates and conservation laws
 From the expression for Lagrangian in plane polar coordinates it is obvious that \(\theta\) is a cyclic coordinate. Hence the corresponding momentum \(p_\theta\) is conserved. Thus \begin{equation} p_\theta = \pp[L]{\dot{\theta}} = m r^2 {\dot{\theta}} =\text{constant}, \text{say}\,\ell. \label{EQ20} \end{equation} Since the total energy is also  a constant of motion, we have \begin{equation}\label{EQ21} E = \frac{1}{2} m {\dot{r}}^2 + \frac{1}{2} m r^2 {\dot \theta}^2 + V(r) \end{equation}

Solve for generalized velocities
Solving for \(\dot{\theta}\) in terms for \(\ell\) from \EqRef{EQ20} we get \begin{equation}\label{EQ22} \dot{\theta} = \frac{\ell}{mr^2} \end{equation} and substituting in \eqRef{EQ21} we get

\begin{eqnarray}
    E = \frac{1}{2} m {\dot{r}}^2 + \frac{\ell^2}{2m r^2}
                  + V(r)
\end{eqnarray}

This is a relation  between \(r,{\dot r}\) and can be solved for \(\dot{r}\). This gives \begin{equation} \dd[r]{t} = \sqrt{\frac{2}{m}\Big(E -\frac{\ell^2}{2m r^2} - V(r)}\Big) \end{equation}

Integrate the velocities
Integrating we get a relation between \(r\) and \(t\). \begin{equation} t = \int \frac{dr}{\sqrt{\dfrac{2}{m}\Big(E -\dfrac{\ell^2}{2m r^2} - V(r)\Big)}} + \text{const}. \end{equation} This relation can be inverted to give \(r\) as a function of  time, \(r= r(t)\). Substituting in \eqref{EQ22} and integrating  gives \begin{equation} \theta = \int \frac{\ell}{mr^2(t)}\, dt + \text{const}. \end{equation} On integration this equation can be used to get \(\theta\) as a function of \(t\). Thus we see that the conservation laws help in integrating the equations of motion. Sufficiently many conservation laws can reduce the complete solution to quadratures.