[NOTES/ME-06001b]-Using energy conservation to reduce solution to quadratures

sing energy conservation law it is straight forward to solve for the time dependence of position in one dimension. We get the time dependence of position, \(x(t)\), by first solving for velocity \(\dot{x}\), and integrating the resulting equation w.r.t. time. We will write out the steps now. \EqRef{EQ06} gives \begin{eqnarray}\nonumber \dot{x} &=&\sqrt{\frac{2}{m}(E-V(x))} \\ \nonumber \Rightarrow dt &=& \frac{dx}{\sqrt{\frac{2}{m}({E-V(x)})}} \\ \Rightarrow t-t_0 &=& \int_{x_0}^x\frac{dx}{\sqrt{\frac{2}{m}{(E-V(x))}}}. \label{EQ10} \end{eqnarray} This equation gives \(t\) as function of position. Solving this result for \(x(t)\) will give the position as a function of time \(t\). \\ Thus the problem of solving the equation of motion and obtaining the position as function of time reduces to {\tt quadratures}, {\it i.e. } to evaluation of integrals.

• Note that the equation of motion is a second order differential equation in time. On the other hand the energy conservation law leads to a first order differential equation for \(\dot{x(t)}\). So the conservation law leads to a simpler differential equation.